DISSERTAÇÃO DE MESTRADO Uma Matemática Voltada para o Professor do Ensino Médio

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  Mestrado Profissional em Matemática em Rede Nacional PROFMAT DISSERTAđấO DE MESTRADO

  Uma Matemática Voltada para o Professor do Ensino Médio Geraldo Ferreira Barbosa Filho

  ❯◆■❱❊❘❙■❉❆❉❊ ❋❊❉❊❘❆▲ ❉❊ ❆▲❆●❖❆❙ ■◆❙❚■❚❯❚❖ ❉❊ ▼❆❚❊▼➪❚■❈❆

  ▼❊❙❚❘❆❉❖ P❘❖❋■❙❙■❖◆❆▲ ❊▼ ▼❆❚❊▼➪❚■❈❆ ❊▼ ❘❊❉❊ ◆❆❈■❖◆❆▲ ●❊❘❆▲❉❖ ❋❊❘❘❊■❘❆ ❇❆❘❇❖❙❆ ❋■▲❍❖

  

❯▼❆ ▼❆❚❊▼➪❚■❈❆ ❱❖▲❚❆❉❆ P❆❘❆ ❖ P❘❖❋❊❙❙❖❘ ❉❖

❊◆❙■◆❖ ▼➱❉■❖

  ▼❆❈❊■Ó ✷✵✶✺

  ❯◆■❱❊❘❙■❉❆❉❊ ❋❊❉❊❘❆▲ ❉❊ ❆▲❆●❖❆❙ ■◆❙❚■❚❯❚❖ ❉❊ ▼❆❚❊▼➪❚■❈❆

  ▼❊❙❚❘❆❉❖ P❘❖❋■❙❙■❖◆❆▲ ❊▼ ▼❆❚❊▼➪❚■❈❆ ❊▼ ❘❊❉❊ ◆❆❈■❖◆❆▲ ●❊❘❆▲❉❖ ❋❊❘❘❊■❘❆ ❇❆❘❇❖❙❆ ❋■▲❍❖

  

❯▼❆ ▼❆❚❊▼➪❚■❈❆ ❱❖▲❚❆❉❆ P❆❘❆ ❖ P❘❖❋❊❙❙❖❘ ❉❖

❊◆❙■◆❖ ▼➱❉■❖

  ❚r❛❜❛❧❤♦ ❛♣r❡s❡♥t❛❞♦ ❛♦ Pr♦❣r❛♠❛ ❞❡ ▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧ ❡♠ ▼❛t❡♠át✐❝❛ ❡♠ ❘❡❞❡ ◆❛❝✐♦♥❛❧ ❞♦ ■♥st✐t✉t♦ ❞❡ ▼❛t❡♠át✐❝❛ ❞❛ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ❆❧❛❣♦❛s ❝♦♠♦ r❡q✉✐s✐t♦ ♣❛r❝✐❛❧ ♣❛r❛ ♦❜t❡♥çã♦ ❞♦ ❣r❛✉ ❞❡ ▼❡str❡ ❡♠ ▼❛t❡♠át✐❝❛✳ ❖r✐❡♥t❛❞♦r✿ Pr♦❢✳ ❉r✳ ❏♦sé ❈❛r❧♦s ❞❡ ▲✐♠❛

  ▼❆❈❊■Ó ✷✵✶✺ Catalogação na fonte Universidade Federal de Alagoas Biblioteca Central Divisão de Tratamento Técnico

  Bibliotecário Responsável: Valter dos Santos Andrade B238m Barbosa Filho, Geraldo Ferreira.

  

Uma matemática voltada para o professor do ensino médio / Geraldo Ferreira

Barbosa Filho. – 2015. 168 f. : il. Orientador: José Carlos de Almeida Lima. Dissertação (Mestrado Profissional em Matemática) – Universidade Federal de Alagoas. Instituto de Matemática. Programa de Pós Graduação de Mestrado Profissional em Matemática em Rede Nacional. Maceió, 2015. Bibliografia: f. 167. Índice: f. 168.

  CDU: 51:377.8

  ✐✐✐ ➚ ♠✐♥❤❛ ❡s♣♦s❛ ❏❛♥❝✐♥❡✐❞❡✱ ❝♦♠♣❛♥❤❡✐r❛ ♣❛r❛ t♦❞♦ ♦ s❡♠♣r❡ ❡ ❧✉③ q✉❡ ❣✉✐❛ ♦ ♠❡✉ ❝❛♠✐♥❤♦✱ ❛♦s ♠❡✉s ✜❧❤♦s ❱✐❝t♦r ❡ ▼❛②❛r❛✱ ✐♥s♣✐✲ r❛çõ❡s ❞❛ ♠✐♥❤❛ ✈✐❞❛✱ à ♠✐♥❤❛ s♦❣r❛ ❚❡r❡③✐♥❤❛✱ ❝❛r✐♥❤♦s❛♠❡♥t❡ ❝❤❛♠❛❞❛ ❞❡ ◆❡♥♦✱ à ♠✐♥❤❛ ❝✉♥❤❛❞❛ ❏❡❛♥❡✱ q✉❡ ❝♦♥s✐❞❡r♦ ❝♦♠♦ ♠✐♥❤❛ ✐r♠ã✱ s❡✉ ❡s♣♦s♦✱ ❊✉❝❧✐❞❡s✱ ❛♦s ❝♦❧❡❣❛s ❞❡ ❝✉rs♦✱ q✉❡ ♥❛s ❤♦r❛s ♠❛✐s ❞✐❢í❝❡✐s ❡stá✈❛♠♦s ❥✉♥t♦s ♣❛r❛ ❛♣♦✐❛r ✉♠ ❛♦ ♦✉tr♦✱ ❛♦s ♣r♦❢❡ss♦r❡s q✉❡ ♥❡st❛ ❥♦r♥❛❞❛ ❝♦♠♣❛rt✐❧❤❛r❛♠ s✉❛s ❡①♣❡r✐ê♥❝✐❛s ❡♠ ♣r♦❧ ❞❛ ♥♦ss❛ ❡✈♦❧✉çã♦ ♥♦ ❝✉rs♦✱ ❡♠ ❡s♣❡❝✐❛❧ ❛♦ ♣r♦❢❡ss♦r ❉r ❏♦sé ❈❛r❧♦s✱ ♠❡✉ ♦r✐❡♥t❛❞♦r ♥♦ tr❛❜❛❧❤♦ ❞❡ ❝♦♥❝❧✉sã♦✳

  

❆❣r❛❞❡❝✐♠❡♥t♦s

  ❆♦ Pr♦❢✳ ❉r✳ ❏♦sé ❈❛r❧♦s ❞❡ ▲✐♠❛✱ ♣❡❧❛ s❡r❡♥✐❞❛❞❡ ❡♠ s✉❛ ✈✐❞❛ ♣r♦✜ss✐♦♥❛❧ ❡ ❝♦✲ ♠✉♥✐❝❛çã♦ ❝♦♠ s❡✉s ❛❧✉♥♦s ❡ s✉❛ ✜❧❤❛ ❉♦✉t♦r❛♥❞❛ ❑❛r❧❛ ❦❛t❡r✐♥❡ ❇❛r❜♦③❛ ❞❡ ▲✐♠❛ ♣❡❧♦ ❣r❛♥❞✐♦s♦ ❛♣♦✐♦ ♥❛ ♦r✐❡♥t❛çã♦ ❞♦s ❝♦♠❛♥❞♦s ♥♦ ▲❛t❡①✳

  ❆♦s ♠❡✉s ❝♦♠♣❛♥❤❡✐r♦s ❞❡ ❝❛♠✐♥❤❛❞❛✱ ❆❧❡①❛♥❞r❡ ❋❛r✐❛✱ ❈❧❛r✐❝② ❆❧✈❡s✱ ❈❧❛✉❞✐♦ ❆❣r❛✱ ❉✐♦❣♦ ▼❡✉r❡r✱ ❉❥❛❧♠❛ ●♦♠❡s✱ ❊❞✈❛♥ ❍♦rá❝✐♦✱ ❊✇❡rt♦♥ ❘♦♦s❡✇❡❧t✱ ❋❛❜rí❝✐❛ ❖♠❡♥❛✱ ❋❡r✲ ♥❛♥❞♦ ❱❛❧ér✐♦✱ ●❡r❛❧❞♦ ❇❛rr♦s✱ ●✐❧♠❛r ❚❡♦❞♦③✐♦✱ ●✐✈❛♥✐❧❞♦ ❙❛♥t♦s✱ ●r❛❝✐♥♦ ❋r❛♥❝✐s❝♦✱ ❏♦sé ❆♣❛r❡❝✐❞♦✱ ❏♦sé ❘❡✐♥❛❧❞♦✱ ▲❡❞✐✈❛❧❞♦ ●♦♠❡s✱ ▼❛r❝✐❡❧ ❞❛ ❙✐❧✈❛✱ ❖r❧❛♥❞♦ ❆♣rí❣✐♦✱ ❘♦s❡ ❙✐❧✈❛✱ ❚❤✐❛❣♦ ▲❡ss❛ ❡ ❱❛♥✐♥❛ ❈❛r✈❛❧❤♦ ♣❡❧❛s ❤♦r❛s ❞❡ ❡st✉❞♦s✱ ♣❡❧❛s ✐♥❢♦r♠❛çõ❡s ❡ ❝♦♠✲ ♣r❡❡♥sã♦✱ ❛ss✐♠ ❝♦♠♦✱ àq✉❡❧❡s q✉❡ ✐♥✐❝✐❛r❛♠ ❛ ❥♦r♥❛❞❛ ❝♦♥♦s❝♦✱ ♠❛s ♣♦r ❛❧❣✉♠ ♠♦t✐✈♦✱ ♥ã♦ ❝♦♥❝❧✉ír❛♠✱ ❊❞❝❛r❧♦s ❞❛ ❙✐❧✈❛✱ ❊♠❛♥♦❡❧ ▼❛r✐❛♥♦✱ ●❡♦✈❛♥ ❱✐❡✐r❛✱ ■s❛❛❝ ●♦♠❡s✱ ❏♦sé ❘♦♠✐❧❞♦✱ ❏✉❝é❧✐♦ ❋❡rr❡✐r❛✱ ▲ú❝✐♦ ❆✉❣✉st♦✱ ◆✐❧③❡r ❏❛❝q✉❡❧✐♥❡ ❡ ❚♦♥② ❋á❜✐♦✳

  ➚ ♠✐♥❤❛ ❢❛♠í❧✐❛✱ ❡♠ ❡s♣❡❝✐❛❧✱ ♠✐♥❤❛ ❡s♣♦s❛✱ ✜❧❤♦s✱ s♦❣r❛✱ ❝✉♥❤❛❞❛ ❡ ❝♦✲❝✉♥❤❛❞♦ ♣❡❧♦s ❞✐❛s✱ ♥♦✐t❡s ❡ ✜♥s ❞❡ s❡♠❛♥❛ ❞❡ ❛✉①í❧✐♦✱ q✉❡ ❢♦✐ ❡ss❡♥❝✐❛❧ ♣❛r❛ ❛ ❝♦♥❝❧✉sã♦ ❞❡ss❡ ❝✉rs♦✳

  ❆♦s ❝♦♠♣❛♥❤❡✐r♦s ❞❡ tr❛❜❛❧❤♦ q✉❡ ♠❡ ❞❡r❛♠ ❛♣♦✐♦ ❝♦♠ ❛ ♣❛rt❡ ♣rát✐❝❛ ❞❡ss❡ tr❛✲ ❜❛❧❤♦✱ ❆❧♠✐r✱ ❇r✉♥✐♦ ❆❧❜✐♥♦✱ ❈❛✐♦ ❈és❛r✱ ▲✉❝✐✈❛❧❞❛ ❇❛r❜♦③❛✱ ❘✐❧❞s♦♥✱ ❙ér❣✐♦ ❈❛❜r❛❧ ❡ ❲✐t❡♠❜❡r❣✱ ❛ss✐♠ ❝♦♠♦ ✉♠ ❛❧✉♥♦ ❡♠ ❡s♣❡❝✐❛❧ ❞❛ ❊s❝♦❧❛ ❊st❛❞✉❛❧ ❉r ❋❡r♥❛♥❞❡s ▲✐♠❛✱ ❆❧✐ss♦♥ ▲♦♣❡s✱ q✉❡ ♣♦ss✉✐ ✉♠ ❢✉t✉r♦ ♣r♦♠✐ss♦r ♥❛ ár❡❛ ❞❛s ❡①❛t❛s✳

  ❆♦s ♠❡✉s ❛❧✉♥♦s ❞❛s ❊s❝♦❧❛s ❊st❛❞✉❛❧ ❉r ❋❡r♥❛♥❞❡s ▲✐♠❛✱ ▼✉♥✐❝✐♣❛❧ ◆✐s❡ ❞❛ ❙✐❧✈❡✐r❛ ❡ ❞♦ ❈♦❧é❣✐♦ ❆♥❝❤✐❡t❛✱ ♣♦r ❝♦♥tr✐❜✉✐r❡♠✱ ❞❡ ❛❧❣✉♠❛ ❢♦r♠❛✱ ❝♦♠ ❡ss❛ ♠✐♥❤❛ ❝♦♥q✉✐st❛✳

  

❘❡s✉♠♦

  ▼✉✐t♦s sã♦ ♦s ❞❡s❛✜♦s q✉❡ ❛ ▼❛t❡♠át✐❝❛ ♣r♦♣õ❡ ❞✉r❛♥t❡ t♦❞❛ ❛ ♥♦ss❛ ✈✐❞❛ ♣r♦✜ss✐♦♥❛❧✱ s❡❥❛ ❝♦♠♦ ❡st✉❞❛♥t❡ ♦✉ ❝♦♠♦ ♣r♦✜ss✐♦♥❛❧ ❞❛ ár❡❛✳ ❆ ❜✉s❝❛ ♣❡❧♦ ❝♦♥❤❡❝✐♠❡♥t♦ é ♥❡❝❡ssár✐❛ ♣❛r❛ ♣♦❞❡r ❞❛r♠♦s ❝♦♥t❛ ❞❛s ♥♦✈✐❞❛❞❡s q✉❡ s✉r❣❡♠ ❛ ❝❛❞❛ ❞✐❛✳ ❙❡♠♣r❡ ♣r❡❝✐s❛♠♦s ❞❡ ♠♦t✐✈❛çã♦ ♣❛r❛ ♣♦❞❡r ✜❝❛r♠♦s ❛t✉❛❧✐③❛❞♦s ❝♦♠ ♦s ❛ss✉♥t♦s ♥❡❝❡ssár✐♦s ♣❛r❛ ♦ ❡♥t❡♥❞✐♠❡♥t♦ ❞❡ss❛s ♥♦✈✐❞❛❞❡s✳ ❆ s❡❣✉✐r✱ ♠♦str❛r❡♠♦s ✉♠ ♠❛t❡r✐❛❧ q✉❡ ♠♦t✐✈❛rá ♦ ♣r♦❢❡ss♦r ❞❡ ♠❛t❡♠át✐❝❛ q✉❡ ❜✉s❝❛ ♦ ❛♣r❡♥❞✐③❛❞♦ ♦ t❡♠♣♦ t♦❞♦✱ r❡s❣❛t❛♥❞♦ ♦ ❡stí♠✉❧♦ q✉❡ ♠✉✐t❛s ✈❡③❡s s❡ ♣❡r❞❡ ♣❡❧♦ ❢❛t♦ ❞♦ ♣r♦✜ss✐♦♥❛❧ ❞❛ ❡❞✉❝❛çã♦ ♥ã♦ ❝♦♥s❡❣✉✐r r❡s♦❧✈❡r ❛❧❣✉♥s ♣r♦❜❧❡♠❛s ♠❛t❡♠át✐❝♦s q✉❡ s✉r❣❡♠ ♥❛ s✉❛ ❝❛♠✐♥❤❛❞❛✳ ❖s ❛ss✉♥t♦s q✉❡ s❡rã♦ ❛❜♦r❞❛❞♦s ❛q✉✐ sã♦ ❞❡ ❣r❛♥❞❡ ✉t✐❧✐❞❛❞❡ ♥❛ ❊❞✉❝❛çã♦ ❇ás✐❝❛ ❡ ♣♦r ❡❧❡s t❡r❡♠ ✉♠❛ ❜❡❧❡③❛ ❡ ❣r❛❝✐♦s✐❞❛❞❡ ♥❛ r❡s♦❧✉çã♦ ❞♦s ♣r♦❜❧❡♠❛s ❞❡✈❡♠ s❡r ❛♣r❡❝✐❛❞♦s ❝♦♠ ♠❛✐♦r ✐♥t❡r❡ss❡✳ ❆ ❧✐♥❣✉❛❣❡♠ ✉t✐❧✐③❛❞❛ ❛q✉✐ é ♠✉✐t♦ ❛❣r❛❞á✈❡❧ ♣❛r❛ ♦ ♣r♦❢❡ss♦r ❞❡ ▼❛t❡♠át✐❝❛✱ ❡st✐♠✉❧❛♥❞♦✲♦ ❛✐♥❞❛ ♠❛✐s ❡♠ q✉❡r❡r ❛❞q✉✐r✐r ♦ ❝♦♥❤❡❝✐♠❡♥t♦✳ ❊st❡ ♠❛t❡r✐❛❧ ❞❡✈❡ s❡r ✉s❛❞♦ ❞❡ ♠❛♥❡✐r❛ ✐♥t❡♥s❛ ❡ ♦st❡♥s✐✈❛ ♣❛r❛ q✉❡ ✈♦❝ê ❝♦♠♦ ♣r♦❢❡ss♦r t❡♥❤❛ t♦t❛❧ ❝♦♥✜❛♥ç❛ ❡ ❞♦♠í♥✐♦ ❞♦ ❝♦♥t❡ú❞♦ ♣❛r❛ ♣r♦♣♦r té❝♥✐❝❛s ❡ ❡str❛té❣✐❛s ♥♦s ♣❧❛♥❡❥❛♠❡♥t♦s ❞❡ s✉❛s ❛✉❧❛s✳

  

❆❜str❛❝t

  ❚❤❡r❡ ❛r❡ ♠❛♥② ❝❤❛❧❧❡♥❣❡s t❤❛t ♠❛t❤❡♠❛t✐❝s ♦✛❡rs t❤r♦✉❣❤♦✉t ♦✉r ✇♦r❦✐♥❣ ❧✐✈❡s✱ ❡✐t❤❡r ❛s ❛ st✉❞❡♥t ♦r ❛s ❛♥ ❛r❡❛ ♦❢ ✇♦r❦✳ ❚❤❡ q✉❡st ❢♦r ❦♥♦✇❧❡❞❣❡ ✐s ♥❡❝❡ss❛r② t♦ ❜❡ ❛❜❧❡ t♦ ❣✐✈❡ ❛❝❝♦✉♥t ♦❢ t❤❡ ♥❡✇ ❞❡✈❡❧♦♣♠❡♥ts t❤❛t ❛r✐s❡ ❡✈❡r② ❞❛②✳ ❆❧✇❛②s ♥❡❡❞ ♦❢ ♠♦t✐✈❛t✐♦♥ ✐♥ ♦r❞❡r t♦ st❛② ✉♣❞❛t❡❞ ✇✐t❤ t❤❡ ♠❛tt❡rs ♥❡❝❡ss❛r② ❢♦r t❤❡ ✉♥❞❡rst❛♥❞✐♥❣ ♦❢ t❤❡s❡ ♥❡✇s✳ ◆❡①t✱ ✇❡ ✇✐❧❧ s❤♦✇ ❛ ♠❛t❡r✐❛❧ t❤❛t ✇✐❧❧ ♠♦t✐✈❛t❡ t❤❡ ♠❛t❤ t❡❛❝❤❡r ✇❤♦ s❡❡❦s ❧❡❛r♥✐♥❣ ❛❧❧ t❤❡ t✐♠❡✱ r❡s❝✉✐♥❣ t❤❡ st✐♠✉❧✉s t❤❛t ♦❢t❡♥ ❣❡ts ❧♦st ❜❡❝❛✉s❡ ♦❢ ♣r♦❢❡ss✐♦♥❛❧ ❡❞✉❝❛t✐♦♥ ❝❛♥ ♥♦t s♦❧✈❡ s♦♠❡ ♠❛t❤❡♠❛t✐❝❛❧ ♣r♦❜❧❡♠s t❤❛t ❛r✐s❡ ✐♥ ②♦✉r ✇❛❧❦✳ ❚❤❡ s✉❜❥❡❝ts t♦ ❜❡ ❞✐s❝✉ss❡❞ ❤❡r❡ ❛r❡ ♦❢ ❣r❡❛t ✉s❡ ✐♥ ❜❛s✐❝ ❡❞✉❝❛t✐♦♥ ❛♥❞ t❤❡② ❤❛✈❡ ❛ ❜❡❛✉t② ❛♥❞ ❣r❛❝❡ ✐♥ s♦❧✈✐♥❣ ♣r♦❜❧❡♠s ♠✉st ❜❡ ❡①❛♠✐♥❡❞ ✇✐t❤ ❣r❡❛t ✐♥t❡r❡st✳ ❚❤❡ ❧❛♥❣✉❛❣❡ ✉s❡❞ ❤❡r❡ ✐s ✈❡r② ♣❧❡❛s❛♥t ❢♦r t❤❡ t❡❛❝❤❡r ♦❢ ▼❛t❤❡♠❛t✐❝s✱ st✐♠✉❧❛t✐♥❣ ✐t ❡✈❡♥ ♠♦r❡ ✐♥ ✇❛♥t✐♥❣ t♦ ❛❝q✉✐r❡ ❦♥♦✇❧❡❞❣❡✳ ❚❤✐s ♠❛t❡r✐❛❧ s❤♦✉❧❞ ❜❡ ✉s❡❞ ✐♥ ❛♥ ✐♥t❡♥s❡ ❛♥❞ ♦✈❡rt ✇❛② ❢♦r ②♦✉ ❛s ❛ t❡❛❝❤❡r ❤❛✈❡ ❢✉❧❧ ❝♦♥✜❞❡♥❝❡ ❛♥❞ ❝♦♥t❡♥ts ♦❢ t❤❡ ❞♦♠❛✐♥ t♦ ♣r♦♣♦s❡ t❡❝❤♥✐q✉❡s ❛♥❞ str❛t❡❣✐❡s ✐♥ ♣❧❛♥♥✐♥❣ t❤❡✐r ❧❡ss♦♥s✳

  

❙✉♠ár✐♦

  ✶ ❚ó♣✐❝♦s ❞❡ ▼❛t❡♠át✐❝❛ ❊❧❡♠❡♥t❛r ✹

  ✶✳✶ Pr✐♥❝í♣✐♦ ❞❛ ■♥❞✉çã♦ ❋✐♥✐t❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺ ✶✳✷ Pr✐♥❝í♣✐♦ ❞❛ ❈❛s❛ ❞♦s P♦♠❜♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸

  ✷ ❆❧❣✉♥s ❆s♣❡❝t♦s ❞❛ ❉✐✈✐s✐❜✐❧✐❞❛❞❡ ♥❛ ➪❧❣❡❜r❛ ❞♦s ◆ú♠❡r♦s ■♥t❡✐r♦s ✶✽ ✷✳✶ ❉✐✈✐s✐❜✐❧✐❞❛❞❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽ ✷✳✷ ❉✐✈✐sã♦ ❊✉❝❧✐❞✐❛♥❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾ ✷✳✸ ▼á①✐♠♦ ❉✐✈✐s♦r ❈♦♠✉♠ ❡ ▼í♥✐♠♦ ▼ú❧t✐♣❧♦ ❈♦♠✉♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵

  ✷✳✸✳✶ ▼á①✐♠♦ ❉✐✈✐s♦r ❈♦♠✉♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶ ✷✳✸✳✷ Pr♦♣r✐❡❞❛❞❡s ❞♦ ♠❞❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶ ✷✳✸✳✸ ▼í♥✐♠♦ ▼ú❧t✐♣❧♦ ❈♦♠✉♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹

  ✷✳✹ ◆ú♠❡r♦s Pr✐♠♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼ ✷✳✺ ❆r✐t♠ét✐❝❛ ❞♦s ❘❡st♦s ❡ ❙✉❛s ❆♣❧✐❝❛çõ❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✵

  ✸ ❆❧❣✉♥s ❚ó♣✐❝♦s ❞❡ ●❡♦♠❡tr✐❛ ❊✉❝❧✐❞✐❛♥❛ ✸✾ ✸✳✶ ❈♦♥❣r✉ê♥❝✐❛ ❞❡ ❚r✐â♥❣✉❧♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✾ ✸✳✷ ❆❧❣✉♠❛s ❆♣❧✐❝❛çõ❡s ❞❡ ❈♦♥❣r✉ê♥❝✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✹ ✸✳✸ ❆ ❉❡s✐❣✉❛❧❞❛❞❡ ❚r✐❛♥❣✉❧❛r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✽ ✸✳✹ ◗✉❛❞r✐❧át❡r♦s ◆♦tá✈❡✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✶

  ✸✳✹✳✶ P❛r❛❧❡❧♦❣r❛♠♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✷ ✸✳✹✳✷ ❚r❛♣é③✐♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✺ ✸✳✹✳✸ ❘❡tâ♥❣✉❧♦✱ ❧♦s❛♥❣♦ ❡ q✉❛❞r❛❞♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✼

  ✸✳✺ ▲✉❣❛r❡s ●❡♦♠étr✐❝♦s ❇ás✐❝♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✵ ✸✳✻ ❚❡♦r❡♠❛ ❞❡ ❚❤❛❧❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✷ ✸✳✼ ❙❡♠❡❧❤❛♥ç❛ ❞❡ ❚r✐â♥❣✉❧♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✻ ✸✳✽ ❆❧❣✉♥s ❚❡♦r❡♠❛s ❞❡ ❙❡♠❡❧❤❛♥ç❛ ❞❡ ❚r✐â♥❣✉❧♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✷ ✸✳✾ ➪r❡❛s ❞❡ ❋✐❣✉r❛s P❧❛♥❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✷

  ❙❯▼➪❘■❖ ✈✐✐✐

  ✸✳✾✳✶ ➪r❡❛ ❞♦ P❛r❛❧❡❧♦❣r❛♠♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✹ ✸✳✾✳✷ ➪r❡❛ ❞♦ ❚r✐â♥❣✉❧♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✹ ✸✳✾✳✸ ➪r❡❛ ❞♦ ❚r❛♣é③✐♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✺ ✸✳✾✳✹ Pr♦♣r✐❡❞❛❞❡s ●❡♦♠étr✐❝❛s ■♠♣♦rt❛♥t❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✻

  ✸✳✶✵ ➪r❡❛ ❞♦ ❈ír❝✉❧♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✾ ✸✳✶✵✳✶ ➪r❡❛s ❞❡ s❡t♦r❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✵

  ✸✳✶✶ Pr♦❜❧❡♠❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✷ ✹ Pr✐♥❝í♣✐♦ ❋✉♥❞❛♠❡♥t❛❧ ❞❡ ❈♦♥t❛❣❡♠

  ✾✾ ✺ ❙♦❧✉çõ❡s ❞♦s Pr♦❜❧❡♠❛s

  ✶✶✵ ✺✳✶ ❚ó♣✐❝♦s ❞❡ ▼❛t❡♠át✐❝❛ ❊❧❡♠❡♥t❛r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶✵

  ✺✳✶✳✶ Pr✐♥❝í♣✐♦ ❞❛ ■♥❞✉çã♦ ❋✐♥✐t❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶✵ ✺✳✶✳✷ Pr✐♥❝í♣✐♦ ❞❛ ❈❛s❛ ❞♦s P♦♠❜♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷✵

  ✺✳✷ ❆❧❣✉♥s ❆s♣❡❝t♦s ❞❛ ❉✐✈✐s✐❜✐❧✐❞❛❞❡ ♥❛ ➪❧❣❡❜r❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷✹ ✺✳✸ ❆❧❣✉♥s ❚ó♣✐❝♦s ❞❡ ●❡♦♠❡tr✐❛ ❊✉❝❧✐❞✐❛♥❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸✽ ✺✳✹ Pr✐♥❝í♣✐♦ ❋✉♥❞❛♠❡♥t❛❧ ❞❡ ❈♦♥t❛❣❡♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺✺

  ✻ ❈♦♥s✐❞❡r❛çõ❡s ✜♥❛✐s ✶✻✹

  ❇✐❜❧✐♦❣r❛✜❛ ✶✻✼

  

■♥tr♦❞✉çã♦

  ❊ss❡ tr❛❜❛❧❤♦ ✈✐s❛ ❝♦♥t❡♠♣❧❛r ♣r♦❢❡ss♦r❡s ❞❡ ▼❛t❡♠át✐❝❛ ❡ ár❡❛s ❛✜♥s✱ ❛tr❛✈és ❞❡ ❝✉rs♦s ❞❡ ❢♦r♠❛çã♦ ♣❛r❛ ♣r♦❢❡ss♦r❡s✱ ♠♦t✐✈❛♥❞♦✲♦s ♣❛r❛ ✉♠ ♠❡❧❤♦r ❞❡s❡♠♣❡♥❤♦ ❡♠ s❛❧❛ ❞❡ ❛✉❧❛✳ ❊st❡ tr❛❜❛❧❤♦ ❢❛③ ♣❛rt❡ ❞❡ ✉♠ s❡♥t✐♠❡♥t♦ ❞❡ ❛♥❣úst✐❛ q✉❡ t✐✈❡ ❤á ❛❧❣✉♥s ❛♥♦s✱ q✉❛♥❞♦ ♣❡r❝❡❜✐ q✉❡ ❡st❛✈❛ ❡♥s✐♥❛♥❞♦ ▼❛t❡♠át✐❝❛ ❞❡ ♠♦❞♦ ❡q✉✐✈♦❝❛❞♦✱ ❝♦♠ r❡s♦❧✉çõ❡s ❞❡ ❡①❡r❝í❝✐♦s✱ s❡♠ ♠✉✐t❛ ♣r❡♦❝✉♣❛çã♦ ❝♦♠ ❞❡✜♥✐çõ❡s✱ ♣r♦♣r✐❡❞❛❞❡s✱ t❡♦r❡♠❛s ❡ ❛♣❧✐❝❛çõ❡s ❞❡✈✐❞❛s ❞❡ss❡s ❝♦♥❤❡❝✐♠❡♥t♦s✳

  ❊ss❡ s❡♥t✐♠❡♥t♦ ♠❡ ❡st✐♠✉❧♦✉ ❛ ❞❡s❡♥✈♦❧✈❡r ❡ss❡ tr❛❜❛❧❤♦✱ ♣♦✐s ♣❡♥s♦ q✉❡ ❡ss❛ ❛♥❣úst✐❛ ♥ã♦ é ♣r✐✈✐❧é❣✐♦ ♠❡✉✳ P♦rt❛♥t♦✱ ♠❡✉s ❝♦❧❡❣❛s t❛♠❜é♠ ❞❡✈❡♠ ♣❛ss❛r ♣♦r ✐ss♦✳ ❊ ❛♥t❡s ❞✐ss♦ s❡ ♣r♦❧♦♥❣❛r ♣❛r❛ ❡❧❡s✱ ❣♦st❛r✐❛ ❞❡ ❝♦♠♣❛rt✐❧❤❛r ❛s ✐❞❡✐❛s ❞❡ss❡ tr❛❜❛❧❤♦✱ q✉❡ ❡s♣❡r♦ s❡r ❞❡ ❣r❛♥❞❡ ✈❛❧✐❛ ♣❛r❛ ♠❡✉s ❝♦♠♣❛♥❤❡✐r♦s✳ ❞❡ ❡♥❝♦r❛❥á✲❧♦ ❛♦s ❞❡s❛✜♦s q✉❡ ❛ ▼❛t❡♠át✐❝❛ ♣r♦♣õ❡ ♣❛r❛ ❛ ❣❡♥t❡✳ P♦r ♠✉✐t❛s ✈❡③❡s ♣❛ss❛♠♦s ♣♦r s✐t✉❛çõ❡s q✉❡ ♥♦s ❞❡✐①❛♠ ❝♦♥str❛♥❣✐❞♦✱ ♣❡❧♦ ❢❛t♦ ❞❡ q✉❡ ♥❛q✉❡❧❡ ♠♦♠❡♥t♦ ❞❡ ❞✐✜❝✉❧❞❛❞❡ ♥ã♦ ❝♦♥s✐❣❛♠♦s ♦❜t❡r ✉♠❛ ✐❞❡✐❛ ♣❛r❛ ❛ r❡s♦❧✉çã♦ ❞❡ ✉♠ ♣r♦❜❧❡♠❛ ♣r♦♣♦st♦✳

  P❛r❛ t♦❞♦ ❝♦♠❡♥tár✐♦ q✉❡ s❡ ♦✉✈❡✱ ❞❡✈❡♠♦s t❡r ❛t❡♥çã♦ ❡♠ ❝❛❞❛ ♣❛❧❛✈r❛ q✉❡ s❡ é ❢❛❧❛❞❛✳ ▼✉✐t❛s ❞❡ss❛s ♣❛❧❛✈r❛s ♣♦❞❡♠ ♥♦s ❧❡✈❛r ❛ ❞✐✈❡rs♦s ❝❛♠✐♥❤♦s✱ ❜❛st❛ ✈♦❝ê ❛ss✐♠✐❧❛r ❛s ✐♥❢♦r♠❛çõ❡s q✉❡ sã♦ út❡✐s ♣❛r❛ ❞❡s❡♥✈♦❧✈❡r ❞❡t❡r♠✐♥❛❞♦s ❛ss✉♥t♦s✳ ❱♦✉ ❝♦♥t❛r ✉♠ ❡①❡♠♣❧♦ q✉❡ ❛❝♦♥t❡❝❡✉ ❝♦♠✐❣♦✳ ❊✉ ❡st❛✈❛ ❛ss✐st✐♥❞♦ ✉♠❛ ❛✉❧❛ ❞❡ ❣❡♦♠❡tr✐❛✱ q✉❛♥❞♦ ♦ ♣r♦❢❡ss♦r ❝♦❧♦❝♦✉ ♥♦ q✉❛❞r♦ ✉♠ tr✐â♥❣✉❧♦ ❝♦♠ ♦s â♥❣✉❧♦s ❞❡ ♠❡❞✐❞❛s x✱ 2x ❡ 3x ♣❛r❛ ❡①♣❧✐❝❛r ❛❧❣✉♠ t❡♦r❡♠❛ ♦✉ ❛❧❣✉♠❛ ♣r♦♣r✐❡❞❛❞❡✱ q✉❛♥❞♦ ✉♠ ❞♦s ♠❡✉s ❝♦❧❡❣❛s ❝❤❛♠♦✉ ❛ ♠✐♥❤❛ ❛t❡♥çã♦ ❛♦ ❞✐③❡r ✲ ❊ss❡ tr✐â♥❣✉❧♦ é r❡tâ♥❣✉❧♦ ✲ ♠❛s ♥ã♦ ❡①♣❧✐❝♦✉ ♦ ♣♦rq✉❡✳ ❋✐q✉❡✐✱ ♣♦r q✉❛s❡ ✷✵ ♠✐♥✉t♦s✱ s❡♠ ❡♥t❡♥❞❡r ❛q✉❡❧❛ ✐♥❢♦r♠❛çã♦ q✉❡ ❡❧❡ ♣❛ss♦✉✱ ❞❡✐①❛♥❞♦✲ ♠❡ ♣❡♥s❛t✐✈♦ ❞❡ ❝♦♠♦ é q✉❡ ❡❧❡ ❡♥①❡r❣❛ tã♦ r❛♣✐❞❛♠❡♥t❡ q✉❡ ❛q✉❡❧❡ tr✐â♥❣✉❧♦ é r❡tâ♥❣✉❧♦✳ ◆❛ ✈❡r❞❛❞❡✱ ♣♦❞❡♠♦s ♣r♦✈❛r ✐ss♦ ❞❡ ❞✐✈❡rs❛s ♠❛♥❡✐r❛s✱ ♠❛s ❛s ❞❡✜♥✐çõ❡s q✉❡ ♥ã♦ sã♦ ❝♦❧♦❝❛❞❛s ❝♦♠ ♠✉✐t❛ ❛✜♥✐❞❛❞❡ ❡♠ ♥♦ss❛ ♠❡♥t❡✱ ♥♦s ❞❡✐①❛♠ ❝❡❣♦ q✉❛♥t♦ ❛ ❡ss❛ ✐♥❢♦r♠❛çã♦✳

  ✆

  ➱ ❢á❝✐❧ ✈❡r q✉❡ s❡♥❞♦ ❛ s♦♠❛ ❞♦s â♥❣✉❧♦s ✐♥t❡r♥♦s ❞❡ ✉♠ tr✐â♥❣✉❧♦ q✉❛❧q✉❡r s❡♠♣r❡ é 180 ✱

  ❙❯▼➪❘■❖ ✷

  ✆ ✆ ✆

  ❡♥tã♦ x 2x 3x ✏ 180 ✱ ♦ q✉❡ ✐♠♣❧✐❝❛ q✉❡ x ✏ 30 ✱ ❡ ❞❡ss❛ ❢♦r♠❛ 3x ✏ 90 ❡ ♣♦rt❛♥t♦✱ ♦ tr✐â♥❣✉❧♦ é r❡tâ♥❣✉❧♦✳ P❡♥s❡✐ ✐ss♦ ❛ ♣r✐♥❝í♣✐♦✱ ♠❛s ❛❝❤♦ q✉❡ ♣❡❧❛ ❛❣✐❧✐❞❛❞❡ q✉❡ ❡❧❡ ❞❡✉ ❛ r❡s♣♦st❛✱ ❛❝r❡❞✐t♦ q✉❡ ♥ã♦ ❢♦✐ ❡ss❛ ❝♦♥t❛ q✉❡ ❡❧❡ ❢❡③ ♥❛ ❤♦r❛✳ ▲♦❣♦ ❡♠ s❡❣✉✐❞❛✱ ❡ss❡ ♠❡s♠♦ ❝♦♠♣❛♥❤❡✐r♦ ❢❡③ ❛ s❡❣✉✐♥t❡ ❛✜r♠❛çã♦✿ ❙❡ ❛ s♦♠❛ ❞❡ ❞♦✐s â♥❣✉❧♦s ✐♥t❡r♥♦s ❞❡ ✉♠ tr✐â♥❣✉❧♦ q✉❛❧q✉❡r ❢♦r ✐❣✉❛❧ ❛♦ t❡r❝❡✐r♦✱ ❡♥tã♦ ♦ tr✐â♥❣✉❧♦ é r❡tâ♥❣✉❧♦✳ ❊ ❡ss❛ ❝♦♥t❛ é ❜❡♠ ♠❛✐s ❢á❝✐❧ ❞❡ ❢❛③❡r ❞♦ q✉❡ ❛ ♣r✐♠❡✐r❛✱ ♣♦✐s x 2x ✏ 3x✱ ♣♦rt❛♥t♦ ♦ tr✐â♥❣✉❧♦ é r❡tâ♥❣✉❧♦✳

  ◆❛ é♣♦❝❛ ❞♦s ●r❡❣♦s✱ ♣♦❞✐❛✲s❡ ❢❛❧❛r ❞♦ ❝á❧❝✉❧♦ ❡ ❞❛ ❣❡♦♠❡tr✐❛ ❝♦♠♦ ♣❛rt❡s ú♥✐❝❛s ❞❡ ✉♠ ❝♦r♣♦ ❞❡ ❝♦♥❤❡❝✐♠❡♥t♦s ❜❡♠ ❞❡❧✐♠✐t❛❞♦ ❡ ♥ã♦ ♠✉✐t♦ ❡①t❡♥s♦✳ ❍♦❥❡ ❡♠ ❞✐❛✱ ♣♦ré♠✱ ❛ q✉❛♥t✐❞❛❞❡ ❞❡ ❝♦♥t❡ú❞♦s ❞❡ ♠❛t❡♠át✐❝❛ q✉❡ s❡ ❝♦♥❤❡❝❡ é ✐♠❡♥s❛ ❡ ❝r❡s❝❡ ❡①♣♦♥❡♥❝✐✲ ❛❧♠❡♥t❡✱ t♦r♥❛♥❞♦✲s❡ ❞✐❢í❝✐❧ ❞❡❝✐❞✐r q✉❛❧ ❞❡✈❡ s❡r ❛ ♠❛t❡♠át✐❝❛ q✉❡ s❡ ❛❝♦♥s❡❧❤❡ ❡♥s✐♥❛r ❡ ❝♦♠♦ ❞❡✈❡ s❡r ❛♣r❡s❡♥t❛❞❛ ♣❛r❛ s✉❛ ♠❡❧❤♦r ❝♦♠♣r❡❡♥sã♦ ❡ s✉❛ ♠❡❧❤♦r ✉t✐❧✐❞❛❞❡ ♣❛r❛ ♦ ❢✉t✉r♦ ❞♦s ❛❧✉♥♦s ❬✾❪✳ ❊✉ ❛❝r❡s❝❡♥t♦✱ ❛✐♥❞❛✱ ❛ ❡ss❛ ❡✈♦❧✉çã♦ q✉❡ ❞✐❛ ❛♣ós ❞✐❛ ❛♣❛r❡✲ ❝❡♠ ♥♦✈❛s t❡❝♥♦❧♦❣✐❛s s❡♥❞♦ q✉❡❜r❛❞❛s ♣r♦♣r✐❡❞❛❞❡s q✉❡ ❛♥t❡s ❡r❛♠ ✐♥q✉❡❜rá✈❡✐s ❡ q✉❡ ♥ós✱ ❞♦❝❡♥t❡s✱ ❛❝❤❛♠♦s q✉❡ s❛❜❡♠♦s ✉♠ ♣♦✉❝♦ ❞❡ ♠❛t❡♠át✐❝❛✱ ♠❛s q✉❡ ♥♦ ❢✉♥❞♦ s♦♠♦s ✐♥s❡❣✉r♦s ❞❛q✉✐❧♦ q✉❡ tr❛♥s♠✐t✐♠♦s ❡♠ s❛❧❛ ❞❡ ❛✉❧❛ ❡ ♣r♦♣♦♠♦s ❝♦♥❤❡❝✐♠❡♥t♦s q✉❡ ♥ã♦ sã♦ ❞❡ ❣r❛♥❞❡ ✈❛❧✐❛ ♣❛r❛ ♦ ❛❧✉♥♦✱ t♦r♥❛♥❞♦✲♦ ✉♠ ❢r✉str❛❞♦ ❝♦♠ ❛ ❞✐s❝✐♣❧✐♥❛✳ ❖ ♣r♦❢❡ss♦r ❲❛❧t❡r ❙♣✐♥❡❧❧✐✱ ❡♠ ❡♥tr❡✈✐st❛ ❛ ❘❡✈✐st❛ ❈á❧❝✉❧♦ ❞❡❝❧❛r❛ q✉❡ s❡♥t❡ ♠❛✐s ❞✐✜❝✉❧❞❛❞❡ ♥♦s t❡♠♣♦s ❞❡ ❤♦❥❡ ❞♦ q✉❡ ❤á ❞♦✐s ❛♥♦s ❡ ❛✐♥❞❛ ♠❛✐s ❞✐✜❝✉❧❞❛❞❡ ❞♦ q✉❡ ❤á q✉❛tr♦ ❛♥♦s✳

  ▲❡♠❜r♦✲♠❡ q✉❡ ❛♦ ❡st✉❞❛r ❣❡♦♠❡tr✐❛ ❛♥❛❧ít✐❝❛ ♥♦ ❡♥s✐♥♦ ♠é❞✐♦✱ ♠❛✐s ❡s♣❡❝✐✜❝❛♠❡♥t❡ ♦ ❡st✉❞♦ ❞❛ ❡q✉❛çã♦ ❞❛ ❝✐r❝✉♥❢❡rê♥❝✐❛✱ ♣❡❞✐ ♣❛r❛ ♦ ♣r♦❢❡ss♦r q✉❡ ❞❡♠♦♥str❛ss❡ ❛ ❡q✉❛çã♦ ❞❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ❡①♣♦st❛ ♣♦r ❡❧❡ ♥♦ q✉❛❞r♦ ♥❛q✉❡❧❛ é♣♦❝❛✳ ❆ r❡s♣♦st❛ q✉❡ ❡✉ ♦❜t✐✈❡ ❞❡ss❡ ♣r♦❢❡ss♦r✱ q✉❡ ❡r❛ ❝♦♥s✐❞❡r❛❞♦ ♦ ♠❡❧❤♦r ❞♦ ❝♦❧é❣✐♦ ❡ ✉♠ ❞♦s ♠❡❧❤♦r❡s ❞♦ ❡st❛❞♦✱ ❢♦✐ q✉❡ ❛ ❞❡♠♦♥str❛çã♦ ❞❡ss❛ ❡q✉❛çã♦ ❞❡♠♦r❛r✐❛ ♥♦ ♠í♥✐♠♦ ❞✉❛s ❛✉❧❛s ❞❡ ✺✵ ♠✐♥✉t♦s ❡ q✉❡ ♥ã♦ ✈❛❧❡r✐❛ ❛ ♣❡♥❛ ❞❡♠♦♥strá✲❧❛✳ ❋✐q✉❡✐ ❝♦♠ ✐ss♦ ❣✉❛r❞❛❞♦ ❡♠ ♠✐♥❤❛ ❝❛❜❡ç❛✱ ❛❝❤❛♥❞♦ q✉❡ ✉♠ ❞✐❛ ♣♦❞❡r✐❛ ❞❡❞✉③í✲❧❛✱ ♣♦✐s ❥á ❤❛✈✐❛ ❢❡✐t♦ ✐ss♦ ❝♦♠ ❛ ❞✐stâ♥❝✐❛ ❡♥tr❡ ❞♦✐s ♣♦♥t♦s✳ ❍♦❥❡✱ ❡✉ ♣❡r❝❡❜♦ q✉❡ ♥ã♦ s❡r✐❛ ♥❡❝❡ssár✐❛ ❞✉❛s ❛✉❧❛s ♣❛r❛ ✉♠❛ ❞❡♠♦♥str❛çã♦ ❞❡ ✉♠❛ ❡q✉❛çã♦ ❞❡ss❛✱ ♣♦✐s ✉s❛ ❛♣❡♥❛s ✉♠ ♣♦✉❝♦ ❞❡ ❣❡♦♠❡tr✐❛ ❡ ♦ ❚❡♦r❡♠❛ ❞❡ P✐tá❣♦r❛s✳ ❖✉ s❡❥❛✱ ✉♠❛ ❝♦✐s❛ s✐♠♣❧❡s ❡ ❛tr❛t✐✈❛ ♣❛r❛ ♠✐♠✳ ❚❡♥❤♦ ❝❡rt❡③❛ ❞❡ q✉❡ s❡ ❡ss❡ ♣r♦❢❡ss♦r ❝♦❧♦❝❛ss❡ ✐ss♦ ❝♦♠♦ ✉♠ ❞❡s❛✜♦ ♣❛r❛ ♠✐♠✱ ❡✉ ♣♦❞❡r✐❛ ♠❡ ❛♣r♦①✐♠❛r ♠❡❧❤♦r ❞❛ ▼❛t❡♠át✐❝❛ ❡ ♠❡ ❛♣❛✐①♦♥❛r ❛✐♥❞❛ ♠❛✐s ♣♦r ❡❧❛✳ P♦ré♠ ❝❛❞❛ ❝♦✐s❛ ❡♠ s❡✉ t❡♠♣♦✱ ❡ ❡✉ ❛❝❤♦ q✉❡ ♦ ♠❡✉ t❡♠♣♦✱ ❛♦s ♣♦✉❝♦s ❡stá ❝❤❡❣❛♥❞♦✳

  ❖ ❡♥s✐♥♦ ❞❡ ▼❛t❡♠át✐❝❛ ❤♦❥❡ ♥♦ ❇r❛s✐❧ ❞✐❢❡r❡ ♣♦✉❝♦ ❞♦ ❡♥s✐♥♦ ♣r❛t✐❝❛❞♦ ❤á ✷✵ ❛♥♦s✳ ❆ ❝❛❞❛ ❛♥♦✱ ❧✐✈r♦s ♥♦✈♦s sã♦ ❡❞✐t❛❞♦s r❡♣❡t✐♥❞♦ q✉❛s❡ s❡♠♣r❡ ♦ ♠❡s♠♦ ❡st✐❧♦ ❡ ♦s ♠❡s♠♦s ❝♦♥t❡ú❞♦s ❞♦s ❛♥t❡r✐♦r❡s✳ ❊①✐st❡♠ ❤♦❥❡ ♥♦ ❇r❛s✐❧ ❜♦♥s ❧✐✈r♦s ❞❡ ▼❛t❡♠át✐❝❛ ❞❡❞✐❝❛❞♦s ❛♦s ❛❧✉♥♦s t❛♥t♦ ❞♦ ❡♥s✐♥♦ ❢✉♥❞❛♠❡♥t❛❧ q✉❛♥t♦ ❞♦ ❡♥s✐♥♦ ♠é❞✐♦✳ ❊♥tr❡t❛♥t♦✱ ♦ q✉❡ ❧❤❡s ❢❛❧t❛ é ✉♠ ✐♥❣r❡❞✐❡♥t❡ q✉❡✱ ♥♦ ♠✉♥❞♦ ❞❡ ❤♦❥❡✱ é ❢✉♥❞❛♠❡♥t❛❧✿ ♦ ❡stí♠✉❧♦ à ❝r✐❛t✐✈✐❞❛❞❡✳

  ❙❯▼➪❘■❖ ✸

  ❊♥t❡♥❞❡♠♦s q✉❡ ♥ã♦ é s✉✜❝✐❡♥t❡ ♣❛r❛ ❛ ❢♦r♠❛çã♦ ❞♦ ❢✉t✉r♦ ❝✐❞❛❞ã♦ ✉♠ ❛♣r❡♥❞✐③❛❞♦ ❜✉r♦❝rát✐❝♦ ❞❛ ▼❛t❡♠át✐❝❛ ❡ ♣❡r❝❡❜❡♠♦s ❛ ✐♠♣♦rtâ♥❝✐❛ ❞❡ ❡st✐♠✉❧❛r ♦s ❛❧✉♥♦s ❞❡s❞❡ ❝❡❞♦ ❛ r❡s♦❧✈❡r ♣r♦❜❧❡♠❛s ♥♦✈♦s ❡ ❞❡s❛✜❛♥t❡s✱ ♣r♦♣✐❝✐❛♥❞♦ ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❛ ✐♠❛❣✐♥❛çã♦ ❡ ❞❛ ❝r✐❛t✐✈✐❞❛❞❡ ❬✶✵❪✳ ❖ ❝♦♥❤❡❝✐♠❡♥t♦ é ❛❞q✉✐r✐❞♦ ❛tr❛✈és ❞❛ ✈♦♥t❛❞❡ ❞❡ ❛♣r❡♥❞❡r ❞♦ ❛❧✉♥♦✳ ◆ã♦ ❡①✐st❡ ✉♠ ♣r♦❢❡ss♦r q✉❡ ❝♦♥s✐❣❛ tr❛♥s♠✐t✐r ❝♦♥❤❡❝✐♠❡♥t♦ ♣❛r❛ ♦ ❛❧✉♥♦ s❡ ❡ss❡ ♥ã♦ ❛❞♠✐t❡ ♦ ❛♣r❡♥❞✐③❛❞♦✳ ❉❡ss❛ ❢♦r♠❛ é ♠✉✐t♦ ❝ô♠♦❞♦ ♣❛r❛ q✉❡♠ ❡st❛r ❛♣r❡♥❞❡♥❞♦✱ ♣♦✐s ♥ã♦ t❡r✐❛ ❡s❢♦rç♦ ♥❡♥❤✉♠ ♣❛r❛ ❛❞q✉✐r✐r ♦ ❝♦♥❤❡❝✐♠❡♥t♦✱ ❡ ❛ss✐♠✱ ♥ã♦ ❞❛r✐❛ ✈❛❧♦r ♣❛r❛ ❛q✉✐❧♦ q✉❡ ❡st❛✈❛ s❡♥❞♦ ❛ss✐♠✐❧❛❞♦✳

  ❆❧é♠ ❞✐ss♦✱ ♣❡♥s♦ q✉❡ ♦ ❛❧✉♥♦ ❞❡✈❡ ❢❛③❡r ♦ s❡✉ ♣❛♣❡❧ ❞❡♥tr♦ ❞❡ s✉❛ ♣r♦✜ssã♦ ❞❡ ❡st✉❞❛♥t❡✳ ❖ q✉❡ q✉❡r♦ ❞✐③❡r é q✉❡ s❡ ♦ ❛❧✉♥♦ ♥ã♦ ❢♦r ❛trás ❞♦ ❝♦♥❤❡❝✐♠❡♥t♦ ♣♦r ✈♦♥t❛❞❡ ♣ró♣r✐❛✱ ♥ã♦ ❡①✐st✐rá ✉♠❛ ♣❡ss♦❛ q✉❡ ❝♦♥s✐❣❛ ❝♦❧♦❝❛r ✐ss♦ ♥❡❧❡✳ ❖ ♣❛♣❡❧ ❞❡ ✉♠ ♣r♦❢❡ss♦r é ❛❥✉❞❛r ♦ ❛❧✉♥♦ ❛ ✈❡r ❛s r❡❧❛çõ❡s ❡♥tr❡ ❛ ▼❛t❡♠át✐❝❛ q✉❡ ❡❧❡ ❡st✉❞❛ ❡ ♦✉tr❛s ❝♦✐s❛s✱ ❛✜r♠❛ ❙♣✐♥❡❧❧✐✳ P♦r ❡①❡♠♣❧♦✱ ♥ú♠❡r♦s ❝♦♠♣❧❡①♦s sã♦ ✉s❛❞♦s ♥❛ ❛♥á❧✐s❡ ❞❡ ❝✐r❝✉✐t♦s ❡❧étr✐❝♦s✱ ❧♦❣❛r✐t♠♦s sã♦ ✉s❛❞♦s ♥❛ ❊s❝❛❧❛ ❘✐❝❤t❡r✱ s❡♥♦ ❡ ❝♦ss❡♥♦ ♥♦s ❝á❧❝✉❧♦s ❞❡ r❛♠♣❛s✱ ❡♥tr❡ ♦✉tr❛s✳ ❖ ❛❧✉♥♦ ♥ã♦ ♣♦❞❡ s❡ ❧✐♠✐t❛r ❛ ✈❡r q✉❡ ✉♠ ❛ss✉♥t♦ ❡♠ ▼❛t❡♠át✐❝❛ só s❡r✈❡ ♣❛r❛ ✉♠ ❞❡t❡r♠✐♥❛❞♦ ❡st✉❞♦✳ ❊ ♦✉tr❛✱ q✉❡♠ ❞❡✈❡ ❡st❛❜❡❧❡❝❡r ❛s r❡❧❛çõ❡s ❡♥tr❡ ❛ ▼❛t❡♠át✐❝❛ ❡ ❛s s✐t✉❛çõ❡s ♥❛s q✉❛✐s ❡❧❛ s❡r✈❡ ♥ã♦ é ♦ ♣r♦❢❡ss♦r✱ ❡ s✐♠ ♦ ❛❧✉♥♦✱ ❛✜r♠❛ ♦ ♣r♦❢❡ss♦r ❲❛❧t❡r ❙♣✐♥❡❧❧✐ ❬✶✶❪✱ ♣r♦❢❡ss♦r ❝♦♠ ♠❛✐s ❞❡ ✹✵ ❛♥♦s ❞❡ ❡①♣❡r✐ê♥❝✐❛ ❡♠ s❛❧❛ ❞❡ ❛✉❧❛✳

  ❖ tr❛❜❛❧❤♦ é ❝♦♠♣♦st♦ ♣♦r q✉❛tr♦ ❝❛♣ít✉❧♦s✱ ♦♥❞❡ sã♦ ❛❜♦r❞❛❞♦s ❛ss✉♥t♦s q✉❡ sã♦ ❞❡ ❡①tr❡♠❛ ✐♠♣♦rtâ♥❝✐❛ ♣❛r❛ ♦ ❡♥s✐♥♦ ♠é❞✐♦✳ ◆♦ ❝❛♣ít✉❧♦ ✶ é ♠♦str❛❞♦ ❞♦✐s ♣r✐♥❝í♣✐♦s ✐♠♣♦rt❛♥tíss✐♠♦s ♥❛ ▼❛t❡♠át✐❝❛✱ t❛♥t♦ ♣❡❧❛ ❢❛❝✐❧✐❞❛❞❡ ❞❡ s❡✉ ❡♥t❡♥❞✐♠❡♥t♦ q✉❛♥t♦ ♣❡❧❛ ❣r❛❝✐♦s✐❞❛❞❡ ♥❛ r❡s♦❧✉çõ❡s ❞♦s ♣r♦❜❧❡♠❛s✳ ❙ã♦ ♦s ♣r✐♥❝í♣✐♦s ❞❛ ■♥❞✉çã♦ ❋✐♥✐t❛ ❡ ❞❛ ❈❛s❛ ❞♦s P♦♠❜♦s✳ ◆♦ ❝❛♣ít✉❧♦ ✷✱ sã♦ ❛❜♦r❞❛❞♦s ❛❧❣✉♥s ❛s♣❡❝t♦s ❞❛ ❞✐✈✐s✐❜✐❧✐❞❛❞❡ ♥❛ á❧❣❡❜r❛ ❞♦s ♥ú♠❡r♦s ✐♥t❡✐r♦s✳ ◆❡ss❡ ❝❛♣ít✉❧♦ s❡rá ♠♦str❛❞♦ ❛ ❞❡✜♥✐çã♦ ❞❡ ❞✐✈✐s✐❜✐❧✐❞❛❞❡✱ ❜❡♠ ❝♦♠♦ ❛❧✲ ❣✉♠❛s ♣r♦♣r✐❡❞❛❞❡s q✉❡ ❡♥✈♦❧✈❡♠ ❛ ❞✐✈✐sã♦ ❞❡ ♥ú♠❡r♦s ✐♥t❡✐r♦s✳ ❙❡rá ❞❛❞♦ ✉♠❛ r❡❧❡✈❛♥t❡ ✐♠♣♦rtâ♥❝✐❛ à ❉✐✈✐sã♦ ❊✉❝❧✐❞✐❛♥❛✱ ❛ss✐♠ ❝♦♠♦✱ ❛♣r❡s❡♥t❛❞♦ ❛s ❞❡✜♥✐çõ❡s ❡ ♣r♦♣r✐❡❞❛❞❡s ❞❡ ▼á①✐♠♦ ❉✐✈✐s♦r ❈♦♠✉♠ ❡ ▼í♥✐♠♦ ▼ú❧t✐♣❧♦ ❈♦♠✉♠ ❡ ♦ ❚❡♦r❡♠❛ ❋✉♥❞❛♠❡♥t❛❧ ❞❛ ❆r✐t♠ét✐❝❛✱ ❡♥✈♦❧✈❡♥❞♦ ❛ ❞❡❝♦♠♣♦s✐çã♦ ❞❡ ✉♠ ♥ú♠❡r♦ ✐♥t❡✐r♦ ❡♠ ♣r♦❞✉t♦s ❞❡ ♥ú♠❡r♦s ♣r✐♠♦s✳ ◆♦ ❝❛♣ít✉❧♦ ✸✱ s❡rá ♠♦str❛❞♦ ❛❧❣✉♥s tó♣✐❝♦s ❞❡ ●❡♦♠❡tr✐❛ ❊✉❝❧✐❞✐❛♥❛✱ ♦♥❞❡ ✈❛❧❡ ❞❡st❛❝❛r ♦s ❝❛s♦s ❞❡ ❝♦♥❣r✉ê♥❝✐❛ ❡ s❡♠❡❧❤❛♥ç❛ ❞❡ tr✐â♥❣✉❧♦s✱ ❛ ❞❡s✐❣✉❛❧❞❛❞❡ tr✐❛♥❣✉❧❛r✱ ❛❧❣✉♥s q✉❛❞r✐❧át❡r♦s ♥♦tá✈❡✐s✱ ♦ ❚❡♦r❡♠❛ ❞❡ ❚❤❛❧❡s ❡ ár❡❛s ❞❡ ❛❧❣✉♠❛s ✜❣✉r❛s ♣❧❛♥❛s✳ ◆♦ ❝❛♣ít✉❧♦ ✹✱ é ❞❛❞♦ ✉♠ ❞❡st❛q✉❡ ❛♦ Pr✐♥❝í♣✐♦ ❋✉♥❞❛♠❡♥t❛❧ ❞❛ ❈♦♥t❛❣❡♠✱ q✉❡ ❝♦♥✈é♠ ❢❛❧❛r é ✉♠ ♣r✐♥❝í♣✐♦ q✉❡ ✉s❛♠♦s ♦ t❡♠♣♦ t♦❞♦✱ ♣♦✐s s❡♠♣r❡ ❡st❛♠♦s t♦♠❛♥❞♦ ❞❡❝✐sõ❡s ❛ ❝❛❞❛ ✐♥st❛♥t❡✱ ❡ ✐ss♦ é ❡ss❡♥❝✐❛❧ ♣❛r❛ ❡ss❡ ♣r✐♥❝í♣✐♦✳ ❋✐♥❛❧✐③❛♥❞♦ ♦ tr❛❜❛❧❤♦✱ s❡rã♦ ❛♣r❡s❡♥t❛❞❛s ❛s s♦❧✉çõ❡s ❞♦s ♣r♦❜❧❡♠❛s ❝♦❧♦❝❛❞♦s ♥♦s q✉❛tr♦ ❝❛♣ít✉❧♦s✳

  ❈❛♣ít✉❧♦

  1

❚ó♣✐❝♦s ❞❡ ▼❛t❡♠át✐❝❛ ❊❧❡♠❡♥t❛r

  ◆❡st❡ ❝❛♣ít✉❧♦✱ ❛ ❢✉♥❞❛♠❡♥t❛çã♦ t❡ór✐❝❛ ❜ás✐❝❛ ♥❡❝❡ssár✐❛ ❛♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❡st❡ tr❛❜❛❧❤♦ s❡rá ❞❡s❡♥✈♦❧✈✐❞❛ ❞❡ ❢♦r♠❛ q✉❡ ♦ ❧❡✐t♦r ♥ã♦ t❡rá ❞✐✜❝✉❧❞❛❞❡ ❡♠ ❝♦♠♣r❡❡♥❞❡r ♦ ♠❡s♠♦✳ ❯♠ ❞♦s ♠❛✐♦r❡s ❞❡s❛✜♦s q✉❡ ♣♦❞❡♠♦s ❡♥❝♦♥tr❛r ♥♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❡ q✉❛❧q✉❡r t❡♠❛ ♠❛t❡♠át✐❝♦ é q✉❛♥❞♦ ♥♦s ❞❡♣❛r❛♠♦s ❝♦♠ ❛ ♥❡❝❡ss✐❞❛❞❡ ❞❡ ❛r❣✉♠❡♥t❛r s♦❜r❡ ❢❛t♦s q✉❡ sã♦ ó❜✈✐♦s✱ ✐st♦ é✱ ✐♠❡❞✐❛t♦s✳

  ◆❡st❡ s❡♥t✐❞♦✱ q✉❛♥❞♦ ♣❡♥s❛♠♦s ❡♠ ❝♦♠♣r❡❡♥❞❡r ✉♠❛ t❡♦r✐❛ ♠❛t❡♠át✐❝❛✱ s❡❥❛ ❞❡ ❝✉♥❤♦ ❡❧❡♠❡♥t❛r ♦✉ ♥ã♦✱ ♣❛ss❛♠♦s ♣❡❧❛ ♥❡❝❡ss✐❞❛❞❡ ❞❡ ❝♦♥s♦❧✐❞❞❛r ♦s ❝♦♥❝❡✐t♦s ❞❡ss❛ t❡♦r✐❛ ❞❡ ❢♦r♠❛ q✉❡ ♣♦ss❛♠♦s ❞❡s❡♥✈♦❧✈❡r ✉♠❛ ❧✐♥❣✉❛❣❡♠ ♠❛t❡♠át✐❝❛ ❝♦♥s✐st❡♥t❡ ❡ q✉❡ s❡❥❛ ❝❛♣❛③ ❞❡ ❞❛r ✉♠ s❡♥t✐❞♦ ❧ó❣✐❝♦ ❛♦s ♥♦ss♦s q✉❡st✐♦♥❛♠❡♥t♦s✳

  ❉❡s❝♦❜r✐ ♥❛ ♣rát✐❝❛ q✉❡ ❛ s♦❧✉çã♦ ❞❡ ✉♠ ♣r♦❜❧❡♠❛ ♦✉ ❛ ❝♦♠♣r❡❡♥sã♦ ❞❡ ✉♠ ❡♥✉♥❝✐❛❞♦ ♣❡r♣❛ss❛✱ ♠✉✐t❛s ✈❡③❡s✱ ♣❡❧♦ ❝♦♥❤❡❝✐♠❡♥t♦ ❤✐stór✐❝♦ ❞❡ ❝♦♠♦ s❡ ❞❡s❡♥✈♦❧✈❡ ❞❡t❡r♠✐♥❛❞♦ ❝♦♥t❡ú❞♦❬✶❪✳

  ❆ s❡❣✉✐r tr❛t❛r❡♠♦s s♦❜r❡ ♦ Pr✐♥❝í♣✐♦ ❞❛ ■♥❞✉çã♦ ❋✐♥✐t❛✱ ❛ss✉♥t♦ q✉❡ ♠❡ ❡♥❝❛♥t♦✉ ♣❡❧❛ ♣r❡❝✐sã♦ ❝♦♠♦ ♠ét♦❞♦ ❡ ♣❡❧❛ s✐♠♣❧✐❝✐❞❛❞❡ ❞❡ s❡✉s ❛r❣✉♠❡♥t♦s✳ ❙❡♠ ❞ú✈✐❞❛ ♥❡♥❤✉♠❛✱ ❡ss❡ ♣r✐♥❝í♣✐♦ ♠❡ ♠♦str♦✉ ♦ q✉❛♥t♦ ❛ ▼❛t❡♠át✐❝❛ ♣♦❞❡ s❡r ❛♣r❡❝✐❛❞❛ ❝♦♠ s❡✉s ❛r❣✉♠❡♥t♦s ❧ó❣✐❝♦s r❡✢❡t✐♥❞♦ ❛ ❝✐ê♥❝✐❛ ♠❛❣♥í✜❝❛ q✉❡ ❡❧❛ é✳

  ❖✉tr♦ ♣r✐♥❝í♣✐♦ q✉❡ ♣r❡♥❞❡✉ ♠✐♥❤❛ ❛t❡♥çã♦ ♣♦r s✉❛ s✐♠♣❧✐❝✐❞❛❞❡ ❧ó❣✐❝❛ é ♦ Pr✐♥❝í♣✐♦ ❞❛ ❈❛s❛ ❞♦s P♦♠❜♦s✳ ❯♠ ♣r✐♥❝í♣✐♦ q✉❡ ❛t❛❝❛ ♣r✐♥❝✐♣❛❧♠❡♥t❡ ♣r♦❜❧❡♠❛s ❞❛ t❡♦r✐❛ ❞♦s ♥ú♠❡r♦s✱ ♠❛s ❝♦♠ ❛♣❧✐❝❛❜✐❧✐❞❛❞❡ ❡♠ ✈ár✐♦s r❛♠♦s ❞❛ ▼❛t❡♠át✐❝❛✳

  ✶✳✶ Pr✐♥❝í♣✐♦ ❞❛ ■♥❞✉çã♦ ❋✐♥✐t❛ ✺

  ✶✳✶ Pr✐♥❝í♣✐♦ ❞❛ ■♥❞✉çã♦ ❋✐♥✐t❛

  ◆♦ ✐♥í❝✐♦ ❞♦ sé❝✉❧♦ ❳❳✱ ♦ ♠❛t❡♠át✐❝♦ ●✐✉s❡♣♣❡ P❡❛♥♦ ✭✶✽✺✽✲✶✾✸✷✮✱ ❝✐t❛❞♦ ♣♦r ❬✽❪✱ ❡st❛❜❡❧❡❝❡✉ ♦s ❛①✐♦♠❛s ♥❡❝❡ssár✐♦s ♣❛r❛ ❞❡s❝r❡✈❡r ❝♦♠ ♣r❡❝✐sã♦ ♦ ❝♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s ♥❛t✉r❛✐s✳ ❖ ú❧t✐♠♦ ❆①✐♦♠❛ ❞❡ P❡❛♥♦ ❞✐③✿

  ❙❡❥❛ ❆ ✉♠ s✉❜❝♦♥❥✉♥t♦ ❞❡ N ✭A ⑨ N✮✳ ❙❡ 1 P A ❡ s❡✱ ❛❧é♠ ❞✐ss♦✱ ❆ ❝♦♥té♠ t♦❞♦s ♦s s✉❝❡ss♦r❡s ❞♦s s❡✉s ❡❧❡♠❡♥t♦s ❡♥tã♦ A ✏ N✳

  ❊ss❡ ❛①✐♦♠❛ é ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ ❛①✐♦♠❛ ❞❡ ✐♥❞✉çã♦ ❡ s❡r✈❡ ♣❛r❛ ❡st❛❜❡❧❡❝❡r ♣r♦✈❛s ❞❡ ❛✜r♠❛çõ❡s ♠❛t❡♠át✐❝❛s ❝♦♠ ♦ r✐❣♦r ♠❛t❡♠át✐❝♦ ♥❡❝❡ssár✐♦✳ ❆❣♦r❛✱ ✐r❡♠♦s ❡♥✉♥❝✐❛r ♦ Pr✐♥❝í♣✐♦ ❞❛ ■♥❞✉çã♦ ❋✐♥✐t❛✳

  ❙❡❥❛ P ♣nq ✉♠❛ s❡♥t❡♥ç❛ ❛❜❡rt❛ s♦❜r❡ N✳ ❙✉♣♦♥❤❛ q✉❡✿ ✭✐✮ ✭❇❛s❡ ❞❛ ■♥❞✉çã♦✮ ❙❡ P ♣n q é ✈❡r❞❛❞❡✐r❛ ❡

  ✭✐✐✮ ♣❛r❛ t♦❞♦ n ➙ n ✱ s❡♠♣r❡ q✉❡ P ♣nq é ✈❡r❞❛❞❡✐r❛✱ s❡❣✉❡ q✉❡ P ♣n 1q é ✈❡r❞❛❞❡✐r❛✳ ❊♥tã♦✱ P ♣nq é ✈❡r❞❛❞❡✐r❛ ♣❛r❛ t♦❞♦ n P N✳ ❊ss❡ ♣r✐♥❝í♣✐♦ ❢♦r♥❡❝❡ ✉♠❛ ❞❛s ♠❛✐s ♣♦❞❡r♦s❛s té❝♥✐❝❛s ❞❡ ❞❡♠♦♥str❛çã♦ ❞❛ ▼❛t❡♠á✲ t✐❝❛✿ ❛ ❞❡♠♦♥str❛çã♦ ♣♦r ✐♥❞✉çã♦✳ ❆ s❡❣✉✐r ✐❧✉str❛r❡♠♦s ✐ss♦ ❝♦♠ ❛❧❣✉♥s ❡①❡♠♣❧♦s✳

  ❊①❡♠♣❧♦ ✶✳✶✳✶✳ ▼♦str❡ q✉❡ ♣❛r❛ t♦❞♦ ✐♥t❡✐r♦s ♣♦s✐t✐✈♦ n t❡♠♦s n ☎ ♣n 1q

  1 2 .

  ☎ ☎ ☎ n ✏

  2 Pr♦✈❛r❡♠♦s ❛ ❛✜r♠❛çã♦ ❛❝✐♠❛ ✉s❛♥❞♦ ♦ Pr✐♥❝í♣✐♦ ❞❛ ■♥❞✉çã♦ ❋✐♥✐t❛✳ ◆♦t❡ q✉❡

  1 ☎ ♣1 1q 1 .

  ✏

  2 P♦rt❛♥t♦✱ P ♣1q é ✈❡r❞❛❞❡✐r❛✳ ❆ss✉♠✐r❡♠♦s✱ ❛❣♦r❛✱ q✉❡ P ♣nq é ✈❡r❞❛❞❡✐r❛ ♣❛r❛ ❛❧❣✉♠ n P N✱ ♦✉ s❡❥❛✱ n

  ☎ ♣n 1q

  1 2 .

  ☎ ☎ ☎ n ✏

  2 ❉❡ ❛❝♦r❞♦ ❝♦♠ ♦ Pr✐♥❝í♣✐♦ ❋✉♥❞❛♠❡♥t❛❧ ❞❛ ■♥❞✉çã♦✱ ❞❡✈❡♠♦s ♠♦str❛r q✉❡ P ♣n 1q

  é ✈❡r❞❛❞❡✐r❛✱ ♦✉ s❡❥❛✱ q✉❡ ♣n 1q ☎ r♣n 1q 1s

  1

  2 .

  ☎ ☎ ☎ n ♣n 1q ✏

  2

  ✶✳✶ Pr✐♥❝í♣✐♦ ❞❛ ■♥❞✉çã♦ ❋✐♥✐t❛ ✻

  ☎ ♣n 1q

  ✏ 3 ☎ ♣3 ☎ n 2

  3 3 ✁ n 3

  3 ☎ n 2 ☎ 3 3 ☎ n ☎ 3 2

  ♣n 1q 3 ♣n 2q 3 ♣n 3q 3 ✁ rn 3 ♣n 1q 3 ♣n 2q 3 s ✏ ♣n 3q 3 ✁ n 3 ✏ n 3

  P❡❧♦ Pr✐♥❝í♣✐♦ ❞❛ ■♥❞✉çã♦ ❋✐♥✐t❛✱ ❞❡✈❡♠♦s ♠♦str❛r q✉❡ P ♣n 1q é ✈❡r❞❛❞❡✐r❛✱ ♦✉ s❡❥❛✱ q✉❡ ♣n 1q 3 ♣n 1 1q 3 ♣n 2 1q 3 ✏ 3t, t P N. ➱ ✈❡r❞❛❞❡✱

  ♣n 1q 3 ♣n 2q 3 ✏ 3k, k P N.

  P♦rt❛♥t♦✱ P ♣1q é ✈❡r❞❛❞❡✐r❛✳ ❆ss✉♠✐r❡♠♦s✱ ❛❣♦r❛✱ q✉❡ P ♣nq é ✈❡r❞❛❞❡✐r❛ ♣❛r❛ ❛❧❣✉♠ n P N✱ ♦✉ s❡❥❛ n 3

  1 3 ♣1 1q 3 ♣1 2q 3 ✏ 36 ✏ 3 ☎ 12.

  ❊①❡♠♣❧♦ ✶✳✶✳✷✳ Pr♦✈❡ q✉❡ n 3 ♣n 1q 3 ♣n 2q 3 é ♠ú❧t✐♣❧♦ ❞❡ 3✳ ▼♦str❛r❡♠♦s✱ ♣♦r ✐♥❞✉çã♦✱ q✉❡ ❛ ❛✜r♠❛çã♦ ❛❝✐♠❛ é ✈❡r❞❛❞❡✐r❛✳ P❡r❝❡❜❛ q✉❡

  2 ✳

  ❖ q✉❡ ❛❝❛❜❛♠♦s ❞❡ ♠♦str❛r é q✉❡ ❞❛❞♦ ✉♠ ♥ú♠❡r♦ ♥❛t✉r❛❧ n✱ ❛ s♦♠❛ ❞♦s n ♣r✐♠❡✐r♦s ♥❛t✉r❛✐s ❝♦♥s❡❝✉t✐✈♦s s❡rá ✐❣✉❛❧ ❛ n

  ❖❜s❡r✈❡ q✉❡ ♣1 2 ☎ ☎ ☎ nq ♣n 1q ✏ n

  2 . P♦rt❛♥t♦ P ♣n 1q é ✈❡r❞❛❞❡✐r❛✳ ❉❡ss❛ ❢♦r♠❛✱ ♣❡❧♦ Pr✐♥❝í♣✐♦ ❞❛ ■♥❞✉çã♦ ❋✐♥✐t❛✱ P ♣nq é ✈❡r❞❛❞❡✐r❛ ♣❛r❛ t♦❞♦ n P N✳

  ✏ ♣n 1q ☎ ♣n 2q

  2 ✡

  2

  ✏ ♣n 1q ☎ ✂ n

  1 ✠

  2

  ✏ ♣n 1q ☎ ✁ n

  2 ♣n 1q

  ☎ ♣n 1q

  3 2 ☎ n 3 2 q.

  ✶✳✶ Pr✐♥❝í♣✐♦ ❞❛ ■♥❞✉çã♦ ❋✐♥✐t❛ ✼

  ▲♦❣♦✱ 3 3 3 2 2 2 3 , n ♣n 1q ♣n 2q ♣n 3q ✏ 3t, t ✏ 3 ☎ n ☎ n 3 P N.

  P♦rt❛♥t♦✱ P ♣n 1q é ✈❡r❞❛❞❡✐r❛✳ ❉❡ss❡ ♠♦❞♦ P ♣nq é ✈❡r❞❛❞❡ ♣❛r❛ t♦❞♦ n P N✱ ♣❡❧♦ 3 3 3 Pr✐♥❝í♣✐♦ ❞❛ ■♥❞✉çã♦ ■♥✜♥✐t❛✳ ❖✉ s❡❥❛✱ n ♣n 1q ♣n 2q é ❞❛ ❢♦r♠❛ 3k, k P N ❡ 3 3 3 ❝♦♥❝❧✉✐✲s❡ q✉❡ n ♣n 1q ♣n 2q é ♠ú❧t✐♣❧♦ ❞❡ 3 ♣❛r❛ t♦❞♦ n P N✳ n : 2

  ❊①❡♠♣❧♦ ✶✳✶✳✸✳ ▼♦str❡ q✉❡✱ ♣❛r❛ t♦❞♦ ♥ú♠❡r♦ ♥❛t✉r❛❧ n✱ M ✏ n ☎ ♣n ✁ 1q ☎ ♣3n 2q é ♠ú❧t✐♣❧♦ ❞❡ ✷✹✳

  ❚❛♠❜é♠ ✉s❛r❡♠♦s ♦ Pr✐♥❝í♣✐♦ ❞❛ ■♥❞✉çã♦ ❋✐♥✐t❛ ♣❛r❛ ♣r♦✈❛r ❡ss❛ ✐❣✉❛❧❞❛❞❡✳ ◆♦t❡ q✉❡ 2 M 1 1 ✏ 1 ☎ ♣1 ✁ 1q ☎ ♣3.1 2q ✏ 0.

  ❉❡ss❛ ❢♦r♠❛✱ M é ♠ú❧t✐♣❧♦ ❞❡ ✷✹ ❡✱ ♣♦rt❛♥t♦✱ P ♣1q é ✈❡r❞❛❞❡✐r❛✳ ❆ss✉♠✐r❡♠♦s ❛❣♦r❛ q✉❡ P ♣nq é ✈❡r❞❛❞❡✐r❛✱ ♦✉ s❡❥❛✱

  M n ✏ 24 ☎ t, t P N. ❉❡ ❛❝♦r❞♦ ❝♦♠ ♦ Pr✐♥❝í♣✐♦ ❋✉♥❞❛♠❡♥t❛❧ ❞❛ ■♥❞✉çã♦✱ ❞❡✈❡♠♦s ♠♦str❛r q✉❡ P ♣n 1q

  é ✈❡r❞❛❞❡✐r❛✱ ♦✉ s❡❥❛✱ M n 1 ✏ 24 ☎ k, k P N.

  ❉❡ ❢❛t♦✱ 2 2 M n 1 n ✁ M ✏ ♣n 1q ☎ r♣n 1q ✁ 1s ☎ r3 ☎ ♣n 1q 2s ✁ n ☎ ♣n ✁ 1q ☎ ♣3n 2q 2 2

  2n ✏ ♣n 1q ☎ rn s ☎ r3n 5s ✁ n ☎ ♣n ✁ 1q ☎ ♣3n 2q 2

  ✏ ♣n 1q ☎ rn ☎ ♣n 2qs ☎ r3n 5s ✁ n ☎ ♣n ✁ 1q ☎ ♣3n 2q ✏ n ☎ ♣n 1q ☎ ♣n 2q ☎ ♣3n 5q ✁ n ☎ ♣n ✁ 1q ☎ ♣n 1q ☎ ♣3n 2q 2 2

  11n 10 n

  2 ✏ n ☎ ♣n 1q ☎ r3n ✁ 3n s ✏ n ☎ ♣n 1q ☎ ♣12n 12q 2 .

  ✏ 12n ☎ ♣n 1q ▲♦❣♦✱ 2 M . n 1 n

  ✁ M ✏ 12n ☎ ♣n 1q ▲❡✈❛♥❞♦✲s❡ ❡♠ ❝♦♥s✐❞❡r❛çã♦ q✉❡ n ❡ n 1 sã♦ ♥ú♠❡r♦s ♥❛t✉r❛✐s ❝♦♥s❡❝✉t✐✈♦s✱ t❡♠♦s q✉❡ ✉♠ ❞❡❧❡s é ♥❡❝❡ss❛r✐❛♠❡♥t❡ ♣❛r✳ ❉❡ss❛ ❢♦r♠❛✱ 2 n

  ✶✳✶ Pr✐♥❝í♣✐♦ ❞❛ ■♥❞✉çã♦ ❋✐♥✐t❛ ✽

  1 ☎ ♣1 1q

  ♣nq ✏ n ☎ ♣n 1q

  ❆❣♦r❛✱ ❛ss✉♠✐r❡♠♦s q✉❡ P ♣nq é ✈❡r❞❛❞❡✐r❛✱ ♦✉ s❡❥❛ P

  F igura ♣bq

  F igura ♣aq

  ♣❛rt❡s✳ ❱❡❥❛ ❛s ✜❣✉r❛s✳ ◆❛ ✜❣✉r❛ ♣aq ❛♣❛r❡❝❡ ♦ ♣❧❛♥♦ ❡♠ ❢♦r♠❛ ❞❡ ♣✐③③❛ s❡♠ ♦ ❝♦rt❡ ❡ ♥❛ ✜❣✉r❛ ♣bq ❛♣❛r❡❝❡ ♦ ♣❧❛♥♦ ❞✐✈✐❞✐❞♦ ❡♠ ❞✉❛s ♣❛rt❡s ❝♦♠ ✉♠ ❝♦rt❡✳

  1 ✏ 2. ▲♦❣♦✱ P ♣1q é ✈❡r❞❛❞❡✐r❛✳ ❱❡❥❛ ♥❛ ♣rát✐❝❛ ❝♦♠♦ ✐ss♦ ❛❝♦♥t❡❝❡✳ ❙✉♣♦♥❤❛ ✉♠ ♣❧❛♥♦ ♥❛ ❢♦r♠❛ ❞❡ ✉♠❛ ♣✐③③❛ ❡ q✉❡r❡♠♦s ❞❛r ✉♠ ❝♦rt❡ ♥❡❧❛✳ ➱ ♦❜✈✐♦ q✉❡ t❡r❡♠♦s ❡ss❡ ♣❧❛♥♦ ✭♣✐③③❛✮ ❞✐✈✐❞✐❞♦ ❡♠ ❞✉❛s

  2

  P ♣1q ✏

  ❉❛í✱ M n 1

  ■r❡♠♦s ♠♦str❛r ♣❡❧♦ Pr✐♥❝í♣✐♦ ❋✉♥❞❛♠❡♥t❛❧ ❞❛ ■♥❞✉çã♦ q✉❡ ❛ ❛✜r♠❛çã♦ ❛❝✐♠❛ é ✈❡r✲ ❞❛❞❡✐r❛✳ ❖❜s❡r✈❡ q✉❡

  1 ✳

  2

  ☎ ♣n 1q

  ❊①❡♠♣❧♦ ✶✳✶✳✹✳ ✭P✐③③❛ ❞❡ ❙t❡✐♥❡r✮ ▼♦str❡ q✉❡ ♦ ♠❛✐♦r ♥ú♠❡r♦ ❞❡ ♣❛rt❡s ♣♦ssí✈❡✐s q✉❡ ♣♦❞❡♠♦s ❞✐✈✐❞✐r ♦ ♣❧❛♥♦ ❝♦♠ n r❡t❛s ❞❡st❡ ♣❧❛♥♦ é ❞❛❞♦ ♣♦r P ♣nq ✏ n

  ❉❡♥♦t❛♥❞♦ ♣♦r k ✏ s t P N✱ t❡♠✲s❡ q✉❡ M n 1 ✏ 24k, k P N. P♦rt❛♥t♦✱ P ♣n 1q é ✈❡r❞❛❞❡✐r❛✳ ❉❡ss❛ ❢♦r♠❛✱ ♣❡❧♦ Pr✐♥❝í♣✐♦ ❞❛ ■♥❞✉çã♦ ❋✐♥✐t❛✱ t❡♠✲s❡ q✉❡ P ♣nq é ✈❡r❞❛❞❡✐r❛ ♣❛r❛ t♦❞♦ n✳

  M n 1 ✏ 24s M n ✏ 24s 24t ✏ 24 ☎ ♣s tq.

  ✁ M n ✏ 24s, s P N. ❉♦♥❞❡✱ s❡❣✉❡ q✉❡

  1,

  ✶✳✶ Pr✐♥❝í♣✐♦ ❞❛ ■♥❞✉çã♦ ❋✐♥✐t❛ ✾

  ❡ ✐r❡♠♦s ♠♦str❛r✱ ♣❡❧♦ Pr✐♥❝í♣✐♦ ❞❛ ■♥❞✉çã♦ ❋✐♥✐t❛✱ q✉❡ P ♣n 1q é ✈❡r❞❛❞❡✐r❛✱ ♦✉ s❡❥❛✱ ♣n 1q ☎ ♣n 2q

  P 1.

  ♣n 1q ✏

  2 n ☎ ♣n 1q

  1 ❚r❛ç❛♥❞♦✲s❡ n r❡t❛s✱ ♦❜t❡♠♦s✱ ♣♦r ❤✐♣ót❡s❡✱ ♣❛rt❡s✳ ❖ q✉❡ s❡ ❞❡✈❡

  2 ❡♥tr❛r ❡♠ ❞✐s❝✉sã♦ é q✉❛❧ s❡r✐❛ ❛ ❢♦r♠❛ ❞❡ ❝♦rt❛r ♦ ♣❧❛♥♦ ❝♦♠ n 1 r❡t❛s ♣❛r❛ ♦❜t❡r♠♦s ♦ ♠❛✐♦r ♥ú♠❡r♦ ❞❡ ♣❛rt❡s ♣♦ssí✈❡✐s✳

  ■r❡♠♦s ♦❜t❡r ♦ ♠❛✐♦r ♥ú♠❡r♦ ❞❡ ♣❛rt❡s s❡ ❛ r❡t❛ ❞❡ ♦r❞❡♠ n 1 ❝♦rt❛r ❝❛❞❛ ✉♠❛ ❞❛s ♦✉tr❛s r❡t❛s ❡♠ ♣♦♥t♦s q✉❡ ♥ã♦ s❡❥❛♠ ❞❡ ✐♥t❡rs❡❝çã♦✳ ❖❜s❡r✈❡ ❛s ✜❣✉r❛s ❛❜❛✐①♦ ♣❛r❛ ❡ss❡ ❡♥t❡♥❞✐♠❡♥t♦✳

  F igura F igura ♣cq ♣dq

  ❖❜s❡r✈❡ q✉❡ ♥❛ ✜❣✉r❛ ♣cq ❛ r❡t❛ ❞❡ ♦r❞❡♠ n 1 ❝♦rt❛ ❛s ♦✉tr❛s r❡t❛s ♥❛ ✐♥t❡rs❡❝çã♦✱ ♦ q✉❡ ❞✐✈✐❞❡ ♦ ♣❧❛♥♦ ❡♠ s❡✐s ♣❛rt❡s✱ ❡♥q✉❛♥t♦ q✉❡ ♥❛ ✜❣✉r❛ ♣dq ❛ r❡t❛ ♥ã♦ ❝♦rt❛ ♥❛ ✐♥t❡rs❡❝çã♦ ❡ ❞✐✈✐❞❡ ♦ ♣❧❛♥♦ ❡♠ s❡t❡ ♣❛rt❡s✳

  ❆❧é♠ ❞✐ss♦✱ ❛ r❡t❛ ❞❡ ♦r❞❡♠ n 1 ❛♦ ❝♦rt❛r ❛ ♣r✐♠❡✐r❛ r❡t❛✱ ❞✐✈✐❞❡ ❡♠ ❞♦✐s ♣❡❞❛ç♦s ❛ ♣❛rt❡ q✉❡ s❡ ❡♥❝♦♥tr❛✈❛ ✐♥✐❝✐❛❧♠❡♥t❡✳ ❊♠ s❡❣✉✐❞❛✱ ❞✐✈✐❞❡ ❛ ♣❛rt❡ s❡❣✉✐♥t❡✱ ❛ss✐♠ q✉❡ t♦❝❛ ❛ s❡❣✉♥❞❛ r❡t❛ ❡ s✉❝❡ss✐✈❛♠❡♥t❡ ❡❧❛ ✈❛✐ ❞✐✈✐❞✐♥❞♦ ❡♠ ❞♦✐s ❡s♣❛ç♦s ❛s ♣❛rt❡s ❡♠ q✉❡ ♣❛ss❛ ❛♥t❡s ❞❡ ❡♥❝♦♥tr❛r ❝❛❞❛ ✉♠❛ ❞❛s n r❡t❛s✳ ❈♦♠ ✐ss♦ sã♦ ❣❡r❛❞❛s n 1 ♣❛rt❡s ❛ ♠❛✐s ❞♦ q✉❡ s❡ t✐♥❤❛✳ ❊✱ ♣♦rt❛♥t♦✱

  P ♣n 1q ✏ P ♣nq ♣n 1q n

  ☎ ♣n 1q

  1 ✏ ♣n 1q

  2 n ☎ ♣n 1q 2 ☎ ♣n 1q

  1 ✏

  2 ♣n 1q ☎ ♣n 2q 1. ✏

  ✶✳✶ Pr✐♥❝í♣✐♦ ❞❛ ■♥❞✉çã♦ ❋✐♥✐t❛ ✶✵

  ❆ ♣r♦♣r✐❡❞❛❞❡ P ♣n 1q é ✈❡r❞❛❞❡✐r❛✱ ❡✱ ♣♦rt❛♥t♦ P ♣nq t❛♠❜é♠ é ✈❡r❞❛❞❡✐r❛ ♣❛r❛ t♦❞♦ n P N✳ ❈♦♥❝❧✉✐✲s❡ q✉❡ ♦ ♠❛✐♦r ♥ú♠❡r♦ ❞❡ ♣❛rt❡s ♣♦ssí✈❡❧ q✉❡ ♣♦❞❡♠♦s ❞✐✈✐❞✐r ♦ ♣❧❛♥♦ n

  ☎ ♣n 1q

  1 ❝♦♠ n r❡t❛s é ❞❛❞♦ ♣♦r ✳

  2

  ✶✳✶ Pr✐♥❝í♣✐♦ ❞❛ ■♥❞✉çã♦ ❋✐♥✐t❛ ✶✶

  Pr♦❜❧❡♠❛s

  ✶✳ ❉❡♠♦♥str❛r ♣♦r ✐♥❞✉çã♦ q✉❡ ♣❛r❛ n ➙ 1 ♥❛t✉r❛❧ 2 2 2 ♣n 1q♣2n 1q n 2 . ✭❛✮ 1 ☎ ☎ ☎ n ✏ 3 3 3

  6 2

  2 ✭❜✮ 1 ☎ ☎ ☎ n ✏ ♣1 2 ☎ ☎ ☎ nq 2 2 n 2 n ☎ ♣n 1q n

  

✁1 ✁1

  3 . ✭❝✮ 1 ✁ 2 ✁ ☎ ☎ ☎ ♣✁1q ☎ n ✏ ♣✁1q ☎

  2 ✷✳ ❉❡♠♦♥str❛r q✉❡ 3

  ✭❛✮ n ✁ n é ✉♠ ♠ú❧t✐♣❧♦ ❞❡ ✻ ♣❛r❛ t♦❞♦ ♥❛t✉r❛❧ n✳ n ✭❜✮ 5 ✁ 1 é ♠ú❧t✐♣❧♦ ❞❡ ✷✹ ♣❛r❛ t♦❞♦ ♥ú♠❡r♦ ♥❛t✉r❛❧ n ♣❛r✳ n

  1 ✭❝✮ 2 é ♠ú❧t✐♣❧♦ ❞❡ ✸ ♣❛r❛ t♦❞♦ ♥❛t✉r❛❧ í♠♣❛r n✳ 2

  ✸✳ ❉❡♠♦♥str❡ q✉❡ ❛ s♦♠❛ ❞♦s n ♣r✐♠❡✐r♦s ♥ú♠❡r♦s í♠♣❛r❡s é n ✱ ♦✉ s❡❥❛ 2

  1 3 .

  ☎ ☎ ☎ ♣2n ✁ 1q ✏ n ✹✳ ❉❡♠♦♥str❡ q✉❡ ❛ s♦♠❛ ❞♦s n ♣r✐♠❡✐r♦s ♥ú♠❡r♦s ♣❛r❡s é n ☎ ♣n 1q✱ ♦✉ s❡❥❛

  2

  4 ☎ ☎ ☎ 2n ✏ n ☎ ♣n 1q.

  10 n n

  11

  11 ✺✳ ▼♦str❡✱ ✉s❛♥❞♦ ✐♥❞✉çã♦✱ q✉❡ 1 ☎ ☎ ☎ ☎ ☎ ☎ 1 ✏ ♣10 ✁ 1q ✁ ✱ ♦♥❞❡ n

  81

  9 r❡♣r❡s❡♥t❛ ❛ q✉❛♥t✐❞❛❞❡ ❞❡ ✧✉♥s✧ ♣r❡s❡♥t❡s ♥♦ ú❧t✐♠♦ t❡r♠♦ ❞❛ ❡①♣r❡ssã♦✳ n

  ♣n ✁ 3q ✻✳ ▼♦str❡ ♣♦r ✐♥❞✉çã♦ q✉❡ ✉♠ ♣♦❧í❣♦♥♦ ❝♦♥✈❡①♦ ❞❡ ♥ ❧❛❞♦s ♣♦ss✉✐ ❡①❛t❛♠❡♥t❡

  2 ❞✐❛❣♦♥❛✐s✳

  ✼✳ Pr♦❣r❡ssã♦ ❛r✐t♠ét✐❝❛ ✭P✳❆✳✮ é ✉♠❛ s❡q✉ê♥❝✐❛ ♥✉♠ér✐❝❛ ❡♠ q✉❡ ❝❛❞❛ t❡r♠♦✱ ❛ ♣❛rt✐r ❞♦ s❡❣✉♥❞♦✱ é ♦❜t✐❞♦ s♦♠❛♥❞♦✲s❡ ♦ t❡r♠♦ ❛♥t❡r✐♦r ❝♦♠ ✉♠❛ ❝♦♥st❛♥t❡✱ ❝❤❛♠❛❞❛ r❛③ã♦ ❞❛ P✳❆✳✱ ✐♥❞✐❝❛❞❛ ♣♦r r✳ ▼♦str❡✱ ♣♦r ✐♥❞✉çã♦✱ q✉❡ ♦ t❡r♠♦ ❣❡r❛❧ ❞❡ ✉♠❛ P✳❆✳ é ❞❛❞♦ ♣♦r a n 1

  ✏ a ♣n ✁ 1q ☎ r

  ✶✳✶ Pr✐♥❝í♣✐♦ ❞❛ ■♥❞✉çã♦ ❋✐♥✐t❛ ✶✷

  ✽✳ ❯s❛♥❞♦ ♦ Pr✐❝í♣✐♦ ❞❛ ■♥❞✉çã♦ ❋✐♥✐t❛✱ ♠♦str❡ q✉❡ ❛ s♦♠❛ ❞♦s n t❡r♠♦s ❞❡ ✉♠❛ P✳❆✳ é ❞❛❞❛ ♣♦r 1 a n

  ♣a q ☎ n S n

  ✏

  2 ✾✳ Pr♦❣r❡ssã♦ ❣❡♦♠étr✐❝❛ ✭P✳●✳✮ é ✉♠❛ s❡q✉ê♥❝✐❛ ♥✉♠ér✐❝❛ ❡♠ q✉❡ ❝❛❞❛ t❡r♠♦✱ ❛

  ♣❛rt✐r ❞♦ s❡❣✉♥❞♦✱ é ♦❜t✐❞♦ ♠✉❧t✐♣❧✐❝❛♥❞♦✲s❡ ♦ t❡r♠♦ ❛♥t❡r✐♦r ♣♦r ✉♠❛ ❝♦♥st❛♥t❡✱ ❝❤❛♠❛❞❛ r❛③ã♦ ❞❛ P✳●✳✱ ✐♥❞✐❝❛❞❛ ♣♦r q✳ ▼♦str❡✱ ♣♦r ✐♥❞✉çã♦✱ q✉❡ ♦ t❡r♠♦ ❣❡r❛❧ ❞❡ ✉♠❛ P✳●✳ é ❞❛❞♦ ♣♦r n

  ✁1

  a n 1 ✏ a ☎ q

  ✶✵✳ ▼♦str❡✱ ♣❡❧♦ Pr✐♥❝í♣✐♦ ❞❛ ■♥❞✉çã♦ ❋✐♥✐t❛✱ q✉❡ ❛ s♦♠❛ ❞♦s n t❡r♠♦s ❞❡ ✉♠❛ P✳●✳ é ❞❛❞❛ ♣♦r n a 1

  ♣q ✁ 1q S , n ✏ ♦♥❞❡ q ✘ 1. q

  ✁ 1 n ✶✶✳ ✭❆ ▼♦❡❞❛ ❋❛❧s❛ ❞♦ ❘❡✐✮ ❯♠❛ ❞❛s 3 ♠♦❡❞❛s ❞❡ ✉♠ ❘❡✐ é ❢❛❧s❛ ❡ ♣❡s❛ ♠❡♥♦s q✉❡ ❛s ✈❡r❞❛❞❡✐r❛s q✉❡ t❡♠ t♦❞❛s ♦ ♠❡s♠♦ ♣❡s♦✳ ❊ss❛ ❞✐❢❡r❡♥ç❛ ❞❡ ♣❡s♦ é q✉❛s❡

  ✐♠♣❡r❝❡♣tí✈❡❧ ❡ ♥ã♦ s❡ ♣♦❞❡ ❞❡t❡r♠✐♥❛r ❛ ♠♦❡❞❛ ❢❛❧s❛ s❡♠ ♦ ✉s♦ ❞❡ ✉♠❛ ❜❛❧❛♥ç❛ ❞❡ ❞♦✐s ♣r❛t♦s✳ ▼♦str❡ q✉❡ ❝♦♠ n ♣❡s❛❣❡♥s ♦ ❘❡✐ ♣♦❞❡ ❞❡s❝♦❜r✐r q✉❛❧ é ❛ ♠♦❡❞❛ ❢❛❧s❛✳

  ✶✳✷ Pr✐♥❝í♣✐♦ ❞❛ ❈❛s❛ ❞♦s P♦♠❜♦s ✶✸

  ✶✳✷ Pr✐♥❝í♣✐♦ ❞❛ ❈❛s❛ ❞♦s P♦♠❜♦s

  ◆❡st❛ s❡çã♦ ❡♥✉♥❝✐❛r❡♠♦s ✉♠ ♣r✐♥❝í♣✐♦✱ ❝✉❥❛ ❧ó❣✐❝❛ ♠❛t❡♠át✐❝❛ é ❞❡ ✉♠❛ s✐♠♣❧✐❝✐❞❛❞❡ ❡♥❝❛♥t❛❞♦r❛✱ ♠❛s ♣♦✉❝♦ ❛❜♦r❞❛❞♦ ♣❡❧♦s ♣r♦❢❡ss♦r❡s ❞♦ ❡♥s✐♥♦ ♠é❞✐♦✳ ❊st❡ ❜❡❧♦ ♣r✐♥❝í♣✐♦✱ ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ Pr✐♥❝í♣✐♦ ❞❛ ❈❛s❛ ❞♦s P♦♠❜♦s é ❞❡ ❣r❛♥❞❡ ✈❛❧✐❛ q✉❛♥❞♦ ❛t❛❝❛♠♦s ❡①❡♠✲ ♣❧♦s ♦♥❞❡ ♦ ✉s♦ ❞♦ r❛❝✐♦❝í♥✐♦ ❧ó❣✐❝♦ s❡ ❢❛③ ♥❡❝❡ssár✐♦ s❡♠ ❛ ✉t✐❧✐③❛çã♦ ❞❡ ❢ór♠✉❧❛s ♣❛r❛ r❡s♦❧✉çã♦ ❞♦s ♠❡s♠♦s✳

  1 Pr♦♣♦s✐çã♦ ✶✳✷✳✶✳ ✭P❈P ✲ ✈❡rsã♦ s✐♠♣❧❡s✮ ❙❡ ❞✐str✐❜✉í♠♦s N ♣♦♠❜♦s ❡♠ N ❝❛s❛s✱ ❡♥tã♦ ❛❧❣✉♠❛ ❞❛s ❝❛s❛s ❝♦♥té♠ ❞♦✐s ♦✉ ♠❛✐s ♣♦♠❜♦s✳ ❉❡♠♦♥str❛çã♦✿ ◗✉❛♥❞♦ ❞✐str✐❜✉í♠♦s ✉♠ ♣♦♠❜♦ ♣❛r❛ ❝❛❞❛ ✉♠❛ ❞❛s ❝❛s❛s ❡①✐st❡♥t❡s✱ ♥♦t❛♠♦s q✉❡ s♦❜r❛ ✉♠ ♣♦♠❜♦ ♣❛r❛ s❡r ❞✐str✐❜✉í❞♦ ❡♠ ✉♠❛ ❞❛s ❝❛s❛s q✉❡ ❥á ❡①✐st❡ ✉♠ ♣♦♠❜♦ ♥❡❧❛✳

  P♦r ♦✉tr♦ ❧❛❞♦✱ s✉♣♦♥❤❛♠♦s q✉❡ ❡♠ ❝❛❞❛ ❝❛s❛ ♥ã♦ ❡①✐st❡ ♠❛✐s ❞♦ q✉❡ ✉♠ ♣♦♠❜♦✱ ❡♥tã♦ q✉❛♥❞♦ ❝♦♥t❛♠♦s t♦❞♦s ♦s ♣♦♠❜♦s ❝♦♥t✐❞♦s ♥❛s N ❝❛s❛s✱ ♥ã♦ t❡r❡♠♦s ♠❛✐s ❞♦ q✉❡ N

  1 ♣♦♠❜♦s✳ ❖ q✉❡ ❝♦♥tr❛r✐❛ ❛ ❤✐♣ót❡s❡ ❞❡ t❡r♠♦s N ♣♦♠❜♦s✳ ▲♦❣♦ t❡r❡♠♦s ♣❡❧♦

  ♠❡♥♦s ✉♠❛ ❞❛s ❝❛s❛s ❝♦♠ ♠❛✐s ❞❡ ✉♠ ♣♦♠❜♦✳

  1 Pr♦♣♦s✐çã♦ ✶✳✷✳✷✳ ✭P❈P ✲ ✈❡rsã♦ ❣❡r❛❧✮ ❙❡ ❞✐str✐❜✉í♠♦s Nk ♣♦♠❜♦s ❡♠ N ❝❛s❛s✱ ❡♥tã♦ ❛❧❣✉♠❛ ❞❛s ❝❛s❛s ❝♦♥té♠ ♣❡❧♦ ♠❡♥♦s k 1 ♣♦♠❜♦s✳ ❉❡♠♦♥str❛çã♦✿ ❙✉♣♦♥❤❛ q✉❡ ❡♠ ❝❛❞❛ ❝❛s❛ ♥ã♦ ❡①✐st❡ ♠❛✐s ❞♦ q✉❡ k ♣♦♠❜♦s✱ ❡♥tã♦ ❝♦♥t❛♥❞♦ t♦❞♦s ♦s ♣♦♠❜♦s ❝♦♥t✐❞♦s ♥❛s N ❝❛s❛s ♥ã♦ t❡r❡♠♦s ♠❛✐s ❞♦ q✉❡ Nk ♣♦♠❜♦s✱ ❝♦♥tr❛❞✐③❡♥❞♦ ❛ ❤✐♣ót❡s❡ ❞❡ t❡r♠♦s Nk 1 ♣♦♠❜♦s ❞✐str✐❜✉í❞♦s ♥❛s N ❝❛s❛s✳

  ❙❡♥❞♦ k ✏ 1✱ t❡r❡♠♦s ❛ ✈❡rsã♦ s✐♠♣❧❡s ❞♦ Pr✐♥❝í♣✐♦ ❞❛ ❈❛s❛s ❞♦s P♦♠❜♦s✳ ❯♠❛ ✈❡rsã♦ ❛❧t❡r♥❛t✐✈❛ ♣❛r❛ ♦ Pr✐♥❝í♣✐♦ ❞❛ ❈❛s❛ ❞♦s P♦♠❜♦s q✉❡ ♣♦❞❡ s❡r ✉s❛❞❛ ❝♦♠♦

  ✉♠❛ ❛♣❧✐❝❛çã♦ é✿ ❙❡ ❛ s♦♠❛ ❞♦s n ♥ú♠❡r♦s ♥❛t✉r❛✐s é ✐❣✉❛❧ ❛ S✱ ❡♥tã♦ ❡①✐st❡ ♣❡❧♦ ♠❡♥♦s ✉♠ ❞❡❧❡s q✉❡

  S S

  ♥ã♦ é ♠❛✐♦r q✉❡ ✱ ❛ss✐♠ ❝♦♠♦ ❡①✐st❡ ♣❡❧♦ ♠❡♥♦s ✉♠ ❞❡❧❡s q✉❡ ♥ã♦ é ♠❡♥♦r q✉❡ ✳ n n

  ❖❜s❡r✈❡ ✉♠ ❜♦♠ ❡①❡♠♣❧♦ ❞♦ Pr✐♥❝í♣✐♦ ❞❛ ❈❛s❛ ❞♦s P♦♠❜♦s ❡♥✉♥❝✐❛❞♦ ♥❛ ♣r♦♣♦s✐çã♦ ✶✳✷✳✶✳ ❊①❡♠♣❧♦ ✶✳✷✳✶✳ ◆✉♠ sít✐♦ ❤á ✸✶✵ ❝❛❥✉❡✐r♦s✳ ➱ ❝♦♥❤❡❝✐❞♦ q✉❡ ✉♠ ❝❛❥✉❡✐r♦ ♥ã♦ ❝♦♥té♠ ♠❛✐s ❞♦ q✉❡ ✸✵✵ ❢r✉t♦s✳ Pr♦✈❡ q✉❡ ❡①✐st❡♠ ❞♦✐s ❝❛❥✉❡✐r♦s ♥❡ss❡ sít✐♦ q✉❡ tê♠ ❛ ♠❡s♠❛ q✉❛♥t✐❞❛❞❡ ❞❡ ❢r✉t♦s✳

  ✶✳✷ Pr✐♥❝í♣✐♦ ❞❛ ❈❛s❛ ❞♦s P♦♠❜♦s ✶✹

  ◆♦t❡ ❛ ❣r❛❝✐♦s✐❞❛❞❡ ❞❡ss❛ s♦❧✉çã♦✱ ♣❛r❛ ❡♥t❡♥❞❡r ❢❛❝✐❧♠❡♥t❡ ❡ss❡ ♣r✐♥❝í♣✐♦✳ ❖s ✸✶✵ ❝❛❥✉❡✐r♦s r❡♣r❡s❡♥t❛♠ ♦s ♣♦♠❜♦s ♥❡ss❡ ♣r✐♥❝í♣✐♦ ❡ ❛ q✉❛♥t✐❞❛❞❡ ❞❡ ❢r✉t♦s ❡♠

  ❝❛❞❛ ❝❛❥✉❡✐r♦ s❡rá r❡♣r❡s❡♥t❛❞❛ ♣❡❧❛ ❝❛s❛ ❞♦s ♣♦♠❜♦s✳ ❚❡r❡♠♦s ❝❛❥✉❡✐r♦ q✉❡ ♣♦❞❡ ♥ã♦ ❝♦❧♦❝❛r ❢r✉t♦s✱ ❛ss✐♠ ❝♦♠♦ ♣♦❞❡ ❝♦❧♦❝❛r ❛♣❡♥❛s ✉♠ ❢r✉t♦✱ ♦✉ ❞♦✐s✱ ❡ ❛ss✐♠ ♣♦r ❞✐❛♥t❡ ❛té ✉♠❛ q✉❛♥t✐❞❛❞❡ ♠á①✐♠❛ ❞❡ ✸✵✵ ❢r✉t♦s ♥♦ ♣é✳ ❊♥✉♠❡r❡♠♦s ❛s ❝❛s❛s ❞♦s ♣♦♠❜♦s ❞❡ ✵ ❛ ✸✵✵✱ ♦♥❞❡ ✵ r❡♣r❡s❡♥t❛ ♦ ❝❛❥✉❡✐r♦ q✉❡ ♥ã♦ ❝♦❧♦❝❛ ❢r✉t♦ ❡ ✸✵✵ r❡♣r❡s❡♥t❛ ♦ ❝❛❥✉❡✐r♦

  ♦ ♦

  q✉❡ ❝♦❧♦❝❛ ✸✵✵ ❢r✉t♦s ❡ ♦r❞❡♥❡♠♦s ♦s ♣♦♠❜♦s✱ q✉❡ r❡♣r❡s❡♥t❛ ♦s ❝❛❥✉❡✐r♦s✱ ❞❡ ✶ ❛ ✸✶✵ ✳

  ♦ ♦

  ❙✉♣♦♥❤❛ q✉❡ ♦ ✶ ❝❛❥✉❡✐r♦ ♥ã♦ ♣♦♥❤❛ ❢r✉t♦✱ ♦ ✷ ♣♦♥❤❛ ❛♣❡♥❛s ✉♠ ❢r✉t♦✱ ❡ ❛ss✐♠ ♣♦r ❞✐❛♥t❡✳ ❖❜s❡r✈❡ ♦ ❡sq✉❡♠❛ ❛ s❡❣✉✐r✳

  ◗✉❛♥t✐❞❛❞❡ ❞❡ ❢r✉t♦s ♥♦ ♣é ✭❝❛s❛ ❞♦s ♣♦♠❜♦s✮ ✵ ✶ ✷ ☎ ☎ ☎ ✸✵✵

  ♦ ♦ ♦ ♦

  1

  2 3 301 ❈❛❥✉❡✐r♦ ✭♣♦♠❜♦s✮ ☎ ☎ ☎

  ◆♦t❡ q✉❡✱ ❛✐♥❞❛ s♦❜r❛♠ ♥♦✈❡ ❝❛❥✉❡✐r♦s ♣❛r❛ ❝♦❧♦❝❛r ❡♠ ✉♠❛ ❞❛s ♣♦s✐çõ❡s ❥á ♦❝✉♣❛❞❛s ♣♦r ❛❧❣✉♠ ♦✉tr♦ ❝❛❥✉❡✐r♦✳ P♦rt❛♥t♦✱ t❡r❡♠♦s ❞♦✐s ❝❛❥✉❡✐r♦s ♦✉ ♠❛✐s q✉❡ t❡♥❤❛♠ ❛ ♠❡s♠❛ q✉❛♥t✐❞❛❞❡ ❞❡ ❢r✉t♦s✳

  ❖❜s❡r✈❡ ♦✉tr♦ ❡①❡♠♣❧♦✳ ❊①❡♠♣❧♦ ✶✳✷✳✷✳ ❊♠ ✉♠❛ r❡✉♥✐ã♦ ❤á ✷✵ ♣❡ss♦❛s✳ ▼♦str❡ q✉❡ ♥❡ss❛ r❡✉♥✐ã♦ ❡①✐st❡♠ ❞✉❛s ♣❡ss♦❛s q✉❡ ❝♦♥❤❡❝❡♠ ❡①❛t❛♠❡♥t❡ ♦ ♠❡s♠♦ ♥ú♠❡r♦ ❞❡ ♣❡ss♦❛s✳

  ◆❡st❡ ❝❛s♦✱ ❛s ✷✵ ♣❡ss♦❛s sã♦ r❡♣r❡s❡♥t❛❞♦s ♣❡❧♦s ♣♦♠❜♦s ❡ ❛s ❝❛s❛s ❞♦s ♣♦♠❜♦s✱ q✉❡ r❡♣r❡s❡♥t❛♠ ❛ q✉❛♥t✐❞❛❞❡ ❞❡ ♣❡ss♦❛s ❝♦♥❤❡❝✐❞❛s ♣♦r ❝❛❞❛ ♣❡ss♦❛✱ sã♦ ❡♥✉♠❡r❛❞❛s ❞❡ ✵ ❛ ✶✾✱ ♦♥❞❡ ③❡r♦ ✐♥❞✐❝❛ q✉❡ ❛ ♣❡ss♦❛ ♥ã♦ ❝♦♥❤❡❝❡ ♥✐♥❣✉é♠ ❡ ✶✾ ✐♥❞✐❝❛ q✉❡ ❛ ♣❡ss♦❛ ❝♦✲ ♥❤❡❝❡ t♦❞♦s ♦s ♦✉tr♦s ❞❛ r❡✉♥✐ã♦✳ ◆❡ss❛ r❡✉♥✐ã♦✱ ♥ã♦ ❡①✐st❡ ❛ ♣♦ss✐❜✐❧✐❞❛❞❡ ❞❡ ♦❝♦rr❡r✱ s✐♠✉❧t❛♥❡❛♠❡♥t❡✱ ♥ã♦ ❝♦♥❤❡❝❡r ♥✐♥❣✉é♠ ❡ ❝♦♥❤❡❝❡r ❛ t♦❞♦s✳ ▲♦❣♦✱ ✉♠❛ ❞❛s ❝❛s❛s✱ ✵ ♦✉ ✶✾✱ ♣❡r♠❛♥❡❝❡ ❞❡s♦❝✉♣❛❞❛✳ ❊♥✉♠❡r❛♥❞♦ ❞❡ ✵ ❛ ✶✾ ♦ ♥ú♠❡r♦ ❞❡ ♣❡ss♦❛s ❝♦♥❤❡❝✐❞❛s ❡ ♦r❞❡♥❛♥❞♦ ❛s ✷✵ ♣❡ss♦❛s ❞❛ r❡✉♥✐ã♦✱ t❡♠♦s✿

  ◗✉❛♥t✐❞❛❞❡ ❞❡ ♣❡ss♦❛s ❝♦♥❤❡❝✐❞❛s ✭❝❛s❛ ❞♦s ♣♦♠❜♦s✮ ✵ ✶ ✷ ☎ ☎ ☎ ✶✾

  ❛ ❛ ❛

  P❡ss♦❛s ✭♣♦♠❜♦s✮ ✶ ✷ ✸ ☎ ☎ ☎ ❳ ♦✉

  ◗✉❛♥t✐❞❛❞❡ ❞❡ ♣❡ss♦❛s ❝♦♥❤❡❝✐❞❛s ✭❝❛s❛ ❞♦s ♣♦♠❜♦s✮ ✵ ✶ ✷ ☎ ☎ ☎ ✶✾

  ❛ ❛ ❛

  P❡ss♦❛s ✭♣♦♠❜♦s✮ ❳ ✶ ✷ ☎ ☎ ☎ ✶✾ ▲♦❣♦✱ ❛ ♣ró①✐♠❛ ♣❡ss♦❛ ❞❡✈❡rá s❡r ❛❧♦❝❛❞❛ ❡♠ ✉♠❛ ❞❛s ❝❛s❛s ❥á ❡①✐st❡♥t❡s ❡✱ ❛ss✐♠✱ t❡r❡♠♦s ❞✉❛s ♣❡ss♦❛s ❝♦♠ ♦ ♠❡s♠♦ ♥ú♠❡r♦ ❞❡ ❝♦♥❤❡❝✐❞♦s ♥❡ss❛ r❡✉♥✐ã♦✳

  ✶✳✷ Pr✐♥❝í♣✐♦ ❞❛ ❈❛s❛ ❞♦s P♦♠❜♦s ✶✺

  P❡r❝❡❜❛ q✉❡ ✉s❛♠♦s ❛r❣✉♠❡♥t♦s ♣❛r❛ ❛ r❡s♦❧✉çã♦ ❞❡ss❡ ❡①❡♠♣❧♦✱ s❡♠ ❛ ♥❡❝❡ss✐❞❛❞❡ ❞❡ ❢ór♠✉❧❛s✳ ■ss♦ ♥♦s ♦❜r✐❣❛ ❛ ✉s❛r ♦ r❛❝✐♦❝í♥✐♦ ❞❡ ♠❛♥❡✐r❛ ♠❛✐s ❡❧❡✈❛❞❛✱ ♦ q✉❡ é ✐♠♣♦rt❛♥t❡ ♣❛r❛ ♦ ♥♦ss♦ ❝❛♠✐♥❤❛r ♣r♦✜ss✐♦♥❛❧✳ ❉❡ss❛ ❢♦r♠❛ ♣♦❞❡r❡♠♦s ❡♥❝❛r❛r ♦s ♣r♦❜❧❡♠❛s ❝♦♠ ♠❛✐s s❡r✐❡❞❛❞❡ ❡ tr❛♥q✉✐❧✐❞❛❞❡ ♣❛r❛ s♦❧✉❝✐♦♥á✲❧♦s✳ ❊①❡♠♣❧♦ ✶✳✷✳✸✳ ◆✉♠ ❝♦❧é❣✐♦ ❝♦♠ ✶✻ s❛❧❛s sã♦ ❞✐str✐❜✉í❞❛s ❝❛♥❡t❛s ♥❛s ❝♦r❡s ♣r❡t❛✱ ❛③✉❧ ❡ ✈❡r♠❡❧❤❛ ♣❛r❛ r❡❛❧✐③❛r ✉♠❛ ♣r♦✈❛ ❞❡ ❝♦♥❝✉rs♦✳ ❙❡ ❝❛❞❛ s❛❧❛ r❡❝❡❜❡ ❝❛♥❡t❛s ❞❛ ♠❡s♠❛ ❝♦r✱ ❡♥tã♦ ♣r♦✈❡ q✉❡ ❡①✐st❡♠ ♣❡❧♦ ♠❡♥♦s s❡✐s s❛❧❛s q✉❡ r❡❝❡❜❡r❛♠ ❝❛♥❡t❛s ❞❡ ♠❡s♠❛ ❝♦r✳

  ❉❡ss❛ ❢♦r♠❛✱ ❛s ✶✻ s❛❧❛s s❡rã♦ r❡♣r❡s❡♥t❛❞❛s ♣❡❧♦s ♣♦♠❜♦s ❡ ❛s ❝♦r❡s ❞❛s ❝❛♥❡t❛s s❡rã♦ r❡♣r❡s❡♥t❛❞❛s ♣❡❧❛s ❝❛s❛s✳ ❊♥✉♠❡r❡♠♦s ❛s s❛❧❛s ❞❡ ✶ ❛ ✶✻✳ ❙✉♣♦♥❤❛ q✉❡ ❛s s❛❧❛s ❞❡ ✶ ❛ ✺ r❡❝❡❜❡r❛♠ ❝❛♥❡t❛s ❞❡ ❝♦r ♣r❡t❛✱ ❛s s❛❧❛s ❞❡ ✻ ❛ ✶✵ r❡❝❡❜❡r❛♠ ❝❛♥❡t❛s ❞❡ ❝♦r ❛③✉❧✱ ❡ ❛s s❛❧❛s ❞❡ ✶✶ ❛ ✶✺ r❡❝❡❜❡r❛♠ ❝❛♥❡t❛s ❞❡ ❝♦r ✈❡r♠❡❧❤❛✱ r❡st❛♥❞♦ ✉♠❛ s❛❧❛ q✉❡ r❡❝❡❜❡✉ ❝❛♥❡t❛s ❞❡ ✉♠❛ ❞❡ss❛s ❝♦r❡s ❥á ❞✐str✐❜✉í❞❛s✳ ▲♦❣♦✱ t❡r❡♠♦s s❡✐s s❛❧❛s q✉❡ r❡❝❡❜❡r❛♠ ❝❛♥❡t❛s ❞❡ ♠❡s♠❛ ❝♦r✳

  P❡r❝❡❜❛✱ ♥♦✈❛♠❡♥t❡✱ q✉❡ ♥ã♦ ✉s❛♠♦s ❢ór♠✉❧❛s ♣❛r❛ s♦❧✉❝✐♦♥❛r ♦ ❡①❡♠♣❧♦✱ ❛♣❡♥❛s ✉♠❛ s✐♠♣❧❡s ♦r❣❛♥✐③❛çã♦ ❞♦ r❛❝✐♦❝í♥✐♦✳

  ✶✳✷ Pr✐♥❝í♣✐♦ ❞❛ ❈❛s❛ ❞♦s P♦♠❜♦s ✶✻

Pr♦❜❧❡♠❛s

  ✶✳ ◗✉❛❧ é ♦ ♥ú♠❡r♦ ♠í♥✐♠♦ ❞❡ ♣❡ss♦❛s q✉❡ ❞❡✈❡♠♦s r❡✉♥✐r ♣❛r❛ q✉❡ t❡♥❤❛♠♦s ❝❡rt❡③❛ ❞❡ q✉❡ ❡♥tr❡ ❡❧❛s ❤á ♣❡❧♦ ♠❡♥♦s ❞✉❛s q✉❡ ❢❛③❡♠ ❛♥✐✈❡rsár✐♦ ♥♦ ♠❡s♠♦ ♠ês❄

  ✷✳ ▼♦str❡ q✉❡✱ ❡♠ ✉♠ ❣r✉♣♦ ❞❡ ✺✵ ♣❡ss♦❛s✱ ❤á s❡♠♣r❡ ♣❡❧♦ ♠❡♥♦s ❝✐♥❝♦ q✉❡ ♥❛s❝❡r❛♠ ♥♦ ♠❡s♠♦ ♠ês✳

  ✸✳ ❊♠ ✉♠❛ r❡✉♥✐ã♦ ❤á n ♣❡ss♦❛s✳ ▼♦str❡ q✉❡ ❡①✐st❡♠ ❞✉❛s ♣❡ss♦❛s q✉❡ ❝♦♥❤❡❝❡♠ ❡①❛t❛♠❡♥t❡ ♦ ♠❡s♠♦ ♥ú♠❡r♦ ❞❡ ♦✉tr♦s ♣❛rt✐❝✐♣❛♥t❡s ❛❞♠✐t✐♠♦s q✉❡ ❝♦♥❤❡❝❡r s❡❥❛ ✉♠❛ r❡❧❛çã♦ s✐♠étr✐❝❛✱ ♦✉ s❡❥❛✱ s❡ a ❝♦♥❤❡❝❡ b✱ ❡♥tã♦ b ❝♦♥❤❡❝❡ a✳

  ✶

  ✹✳ ✭Pr✐♥❝í♣✐♦ ❞❛s ●❛✈❡t❛s ❞❡ ❉✐r✐❝❤❧❡t ✮ ❙❡ n 1 ♦✉ ♠❛✐s ♦❜❥❡t♦s sã♦ ❝♦❧♦❝❛❞♦s ❡♠ n ♦✉ ♠❡♥♦s ❣❛✈❡t❛s✱ ❡♥tã♦✱ ♣❡❧♦ ♠❡♥♦s ✉♠❛ ❣❛✈❡t❛ r❡❝❡❜❡ ♠❛✐s ❞❡ ✉♠ ♦❜❥❡t♦✳

  ✺✳ ❈✐♥❝♦ ♣♦♥t♦s sã♦ t♦♠❛❞♦s s♦❜r❡ ❛ s✉♣❡r❢í❡ ❞❡ ✉♠ q✉❛❞r❛❞♦ ❞❡ ❧❛❞♦ 2✳ ▼♦str❡ q✉❡ ❄

  2 ❤á ❞♦✐s ❞❡ss❡s ♣♦♥t♦s t❛✐s q✉❡ ❛ ❞✐stâ♥❝✐❛ ❡♥tr❡ ❡❧❡s é ♠❡♥♦r q✉❡✱ ♦✉ ✐❣✉❛❧ ❛✱ ✳

  ✻✳ ▼♦str❡ q✉❡ ❡♠ q✉❛❧q✉❡r ❝♦♥❥✉♥t♦ ❞❡ ♦✐t♦ ✐♥t❡✐r♦s ❤á s❡♠♣r❡ ❞♦✐s ❞❡❧❡s ❝✉❥❛ ❞✐❢❡r❡♥ç❛ é ✉♠ ♠ú❧t✐♣❧♦ ❞❡ s❡t❡✳

  ✼✳ ❈♦♥s✐❞❡r❡♠♦s ✉♠ ❝♦♥❥✉♥t♦ ❢♦r♠❛❞♦ ♣♦r ✶✵ ♥ú♠❡r♦s ♥❛t✉r❛✐s ❞✐❢❡r❡♥t❡s✳ ❙❡ ❝❛❧❝✉✲ ❧❛r♠♦s t♦❞❛s ❛s ❞✐❢❡r❡♥ç❛s ❡♥tr❡ ❡ss❡s ♥ú♠❡r♦s✱ ♣❡❧♦ ♠❡♥♦s ✉♠❛ ❞❡ss❛s ❞✐❢❡r❡♥ç❛s é ✉♠ ♠ú❧t✐♣❧♦ ❞❡ ✾❄

  ✽✳ ❊♠ ✉♠ ❝❛♠♣❡♦♥❛t♦✱ ❝❛❞❛ ❞♦✐s t✐♠❡s ❥♦❣❛♠ ❡♥tr❡ s✐ ✉♠❛ ú♥✐❝❛ ✈❡③✳ ▼♦str❡ q✉❡✱ ❡♠ q✉❛❧q✉❡r ♠♦♠❡♥t♦✱ ❤á s❡♠♣r❡ ❞♦✐s t✐♠❡s q✉❡ ❞✐s♣✉t❛r❛♠ ♦ ♠❡s♠♦ ♥ú♠❡r♦ ❞❡ ♣❛rt✐❞❛s✳

  ✾✳ ❉♦ ❝♦♥❥✉♥t♦ A ✏ t1, 2, ☎ ☎ ☎ , 99, 100✉✱ ❡s❝♦❧❤❡♠♦s ❛♦ ❛❝❛s♦ ✺✶ ♥ú♠❡r♦s✳ ❉❡♠♦♥str❛r q✉❡ ❡♥tr❡ ♦s ♥ú♠❡r♦s ❡s❝♦❧❤✐❞♦s s❡♠♣r❡ ❡①✐st❡♠ ❞♦✐s q✉❡ sã♦ ❝♦♥s❡❝✉t✐✈♦s✳ ✶✵✳ ❊s❝♦❧❤❡♠✲s❡ ✼ ♣♦♥t♦s ♥♦ ✐♥t❡r✐♦r ❞❡ ✉♠ r❡tâ♥❣✉❧♦ ❞❡ ❞✐♠❡♥sõ❡s 2 ✂ 3✳ ❉❡♠♦♥str❛r q✉❡ s❡♠♣r❡ é ♣♦ssí✈❡❧ ❡♥❝♦♥tr❛r ❞♦✐s ♣♦♥t♦s t❛❧ q✉❡ s✉❛ ❞✐stâ♥❝✐❛ é ♠❡♥♦r ♦✉ ✐❣✉❛❧

  ❄

  2 ❛ ✳

  ✶✶✳ ❉❛❞❛s s❡✐s ♣❡ss♦❛s ♥✉♠❛ ❢❡st❛✱ ❞❡♠♦♥str❛r q✉❡ ♥❡❝❡ss❛r✐❛♠❡♥t❡ ❡①✐st❡♠ três ♣❡ss♦❛s q✉❡ s❡ ❝♦♥❤❡❝❡♠ ♠✉t✉❛♠❡♥t❡ ♦✉ três ♣❡ss♦❛s q✉❡ ♥ã♦ s❡ ❝♦♥❤❡❝❡♠ ♠✉t✉❛♠❡♥t❡✳ ❙✉♣♦♥❤❛ q✉❡ ❛ r❡❧❛çã♦ ❞❡ ❝♦♥❤❡❝❡r é s✐♠étr✐❝❛✳

  P❡t❡r ●✉st❛✈ ▲❡❥❡✉♥❡ ❉✐r✐❝❤❧❡t ✭✶✽✵✺ ✲ ✶✽✺✾✮✱ ♠❛t❡♠át✐❝♦ ❛❧❡♠ã♦ ❛ q✉❡♠ s❡ ❛tr✐❜✉✐ ❛ ♠♦❞❡r♥❛ ❞❡✜♥✐çã♦ ❢♦r♠❛❧ ❞❡ ❢✉♥çã♦✳

  ✶✳✷ Pr✐♥❝í♣✐♦ ❞❛ ❈❛s❛ ❞♦s P♦♠❜♦s ✶✼

  ✶✷✳ ✭■▼❖✶✾✽✺✮✳ ❉❛❞♦ ✉♠ ❝♦♥❥✉♥t♦ M ❝♦♠ ✶✾✽✺ ✐♥t❡✐r♦s ♣♦s✐t✐✈♦s ❞✐st✐♥t♦s✱ ♥❡♥❤✉♠ ❞♦s q✉❛✐s t❡♠ ❞✐✈✐s♦r❡s ♣r✐♠♦s ♠❛✐♦r❡s ❞♦ q✉❡ ✷✸✱ ♠♦str❡ q✉❡ ❤á ✹ ❡❧❡♠❡♥t♦s ❡♠ M

  ❝✉❥♦ ♣r♦❞✉t♦ é ✉♠❛ q✉❛rt❛ ♣♦tê♥❝✐❛✳ ✶✸✳ ❊♠ ✉♠❛ ♣❡sq✉✐s❛ ❞❡ r✉❛✱ ❝❛❞❛ ❡♥tr❡✈✐st❛❞♦ r❡s♣♦♥❞❡✉ ❛ q✉❛tr♦ ♣❡r❣✉♥t❛s✱ ♣♦❞❡♥❞♦ s✉❛ r❡s♣♦st❛ s❡r s✐♠ ♦✉ ♥ã♦✱ ♣❛r❛ ❝❛❞❛ ✉♠❛ ❞❛s ♣❡r❣✉♥t❛s✳ ◗✉❛❧ ♦ ♥ú♠❡r♦ ♠í♥✐♠♦

  ❞❡ ❡♥tr❡✈✐st❛❞♦s ♣❛r❛ ❣❛r❛♥t✐r♠♦s q✉❡ ❞✉❛s ♣❡ss♦❛s r❡s♣♦♥❞❛♠ ✐❣✉❛❧♠❡♥t❡ ❛ t♦❞❛s ❛s ♣❡r❣✉♥t❛s❄

  ✶✹✳ ❙❡ n ❡ m sã♦ ♥ú♠❡r♦s ♥❛t✉r❛✐s✱ ❡♥tã♦ ♦ ❝♦♥❥✉♥t♦ A ✏ tm 1, m 2, ☎ ☎ ☎ , m n✉ ♣♦ss✉✐ ❛❧❣✉♠ ❞✐✈✐s♦r ❞❡ n✳

  ✶✺✳ ❉❡♠♦♥str❛r q✉❡ t♦❞♦ ✐♥t❡✐r♦ t❡♠ ✉♠ ♠ú❧t✐♣❧♦ ❝✉❥❛ r❡♣r❡s❡♥t❛çã♦ ❞❡❝✐♠❛❧ ❝♦♠❡ç❛ ❝♦♠ ♦ ❜❧♦❝♦ ❞❡ ❞í❣✐t♦s 1234567890✳

  ✶✻✳ ❉❛❞♦ ✉♠ ♥ú♠❡r♦ ✐♥t❡✐r♦ ♣♦s✐t✐✈♦ n✱ ♠♦str❡ q✉❡ ❡①✐st❡ ✉♠ ♠ú❧t✐♣❧♦ ❞❡ n q✉❡ s❡ ❡s❝r❡✈❡ ❝♦♠ ♦s ❛❧❣❛r✐s♠♦s ✵ ❡ ✶ ❛♣❡♥❛s✳ ✭P♦r ❡①❡♠♣❧♦✱ s❡ n ✏ 3✱ t❡♠♦s ✶✶✶ ♦✉ ✶✳✶✵✶✱ ❡t❝✳✮

  ✶✼✳ Pr♦✈❡ q✉❡ ❡♥tr❡ n 1 ❡❧❡♠❡♥t♦s ❡s❝♦❧❤✐❞♦s ♥♦ ❝♦♥❥✉♥t♦ t1, 2, 3, ☎ ☎ ☎ , 2n✉ ❡①✐st❡♠ ❞♦✐s q✉❡ sã♦ ♣r✐♠♦s r❡❧❛t✐✈♦s✳

  ❈❛♣ít✉❧♦

  2

❆❧❣✉♥s ❆s♣❡❝t♦s ❞❛ ❉✐✈✐s✐❜✐❧✐❞❛❞❡ ♥❛

➪❧❣❡❜r❛ ❞♦s ◆ú♠❡r♦s ■♥t❡✐r♦s

  ◆❡st❡ ❝❛♣ít✉❧♦✱ ❡st✉❞❛r❡♠♦s ♦ ❝♦♥❝❡✐t♦ ❞❡ ❞✐✈✐s✐❜✐❧✐❞❛❞❡ ♥♦s ❝♦♥❥✉♥t♦s ❞♦s ✐♥t❡✐r♦s✳ ❙❡♠ ❞✐♠✐♥✉✐r ❛ ✐♠♣♦rtâ♥❝✐❛ ❞❛s ♦♣❡r❛çõ❡s ❞❡ ❛❞✐çã♦✱ s✉❜tr❛çã♦ ❡ ♠✉❧t✐♣❧✐❝❛çã♦✱ t❡♠♦s ♥❛ ❞✐✈✐sã♦ ❞❡ ♥ú♠❡r♦s ✐♥t❡✐r♦s ✉♠❛ ❢♦♥t❡ ✐♥❡s❣♦tá✈❡❧ ❞❡ ♣r♦❜❧❡♠❛s ✐♥t❡r❡ss❛♥t❡s ❞❡♥tr♦ ❞❛ ▼❛t❡♠át✐❝❛✳ ❆♣❡s❛r ❞❛ s✐♠♣❧✐❝✐❞❛❞❡ ❞♦ ❝♦♥❝❡✐t♦✱ ❛ ❝♦♥str✉çã♦ ❞♦ ❢♦r♠❛❧✐s♠♦ ♠❛t❡♠át✐❝♦ ❞❡st❡ ❝♦♥❝❡✐t♦ é ❛ ❝♦♥❞✐çã♦ ♥❡❝❡ssár✐❛ ♣❛r❛ q✉❡ ♦ ♣r♦❢❡ss♦r ♣♦ss❛ ❞❡s❡♥✈♦❧✈❡r ❛ t❡♦r✐❛ ❞❡ ❢♦r♠❛ ♥❛t✉r❛❧ ❡ ❝♦♥s✐st❡♥t❡✳

  ✷✳✶ ❉✐✈✐s✐❜✐❧✐❞❛❞❡

  ❉❡✜♥✐çã♦ ✷✳✶✳✶✳ ❉❛❞♦s a, b P Z✱ ❝♦♠ b ✘ 0✱ ❞✐③❡♠♦s q✉❡ b ❞✐✈✐❞❡ a✱ ❡ ❡s❝r❡✈❡♠♦s b ⑤ a✱ s❡ ❡①✐st✐r c P Z t❛❧ q✉❡ a ✏ bc✳ ❈❛s♦ b ♥ã♦ ❞✐✈✐❞❛ a✱ ❡s❝r❡✈❡♠♦s b ∤ a✳ ❉❛❞♦ ✉♠ ♥ú♠❡r♦ ✐♥t❡✐r♦ b ✘ 0✱ s❡ b ❞✐✈✐❞✐r a✱ ❞✐③❡♠♦s q✉❡ b é ✉♠ ❞✐✈✐s♦r ❞❡ a✱ q✉❡ a

  é ❞✐✈✐sí✈❡❧ ♣♦r b ♦✉ ❛✐♥❞❛ q✉❡ a é ✉♠ ♠ú❧t✐♣❧♦ ❞❡ b✳ ❙❡ b ⑤ a ❡ b → 0✱ ❡♥tã♦ b é ✉♠ ❞✐✈✐s♦r ♣♦s✐t✐✈♦ ❞❡ a✳

  ❆ ♣r♦♣♦s✐çã♦ q✉❡ s❡❣✉❡ ❡st❛❜❡❧❡❝❡ ♣r♦♣r✐❡❞❛❞❡s ❜ás✐❝❛s ❞❛ r❡❧❛çã♦ ❞❡ ❞✐✈✐s✐❜✐❧✐❞❛❞❡✳ ❙❛❧✐❡♥t❛♠♦s q✉❡ ❡st❛s ♣r♦♣r✐❡❞❛❞❡s sã♦ ♠✉✐t♦ r❡❧❡✈❛♥t❡s ♥♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❛ t❡♦r✐❛ ❡✱ ♣♦rt❛♥t♦✱ ❞❡✈❡♠♦s ❞❛r ❛ ❡❧❛s ❛ ❛t❡♥çã♦ ♥❡❝❡ssár✐❛ ♣❛r❛ ✉♠ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❛ ♠❡s♠❛✳ Pr♦♣♦s✐çã♦ ✷✳✶✳✶✳ ❙❡❥❛♠ a, b ❡ c ✐♥t❡✐r♦s ♥ã♦ ♥✉❧♦s ❡ x, y ✐♥t❡✐r♦s q✉❛✐sq✉❡r✳

  ✭❛✮ ❙❡ b ⑤ a ❡ a ⑤ b✱ ❡♥tã♦ a ✏ ✟b✳ ✭❞✮ ❙❡ b ⑤ a✱ ❡♥tã♦ ⑤b⑤ ↕ ⑤a⑤✳ ✭❜✮ ❙❡ c ⑤ b ❡ b ⑤ a✱ ❡♥tã♦ c ⑤ a✳ ✭❡✮ ❙❡ c ⑤ b✱ ❡♥tã♦ c ⑤ ab✳

  ✷✳✷ ❉✐✈✐sã♦ ❊✉❝❧✐❞✐❛♥❛ ✶✾

  ❉❡♠♦♥str❛çã♦✿ ✭❛✮ ❙❡ m ❡ n sã♦ ✐♥t❡✐r♦s t❛✐s q✉❡ a ✏ bm ❡ b ✏ an✱ ❡♥tã♦ a ✏ ♣anq ☎ m ✏ a ☎ ♣mnq ❡✱ ❞❛í✱

  ★ m ✏ 1 ❡ n ✏ 1 m , m, n

  ☎ n ✏ 1 ñ P Z. m

  ✏ ✁1 ❡ n ✏ ✁1 ❙❡❣✉❡ q✉❡ a ✏ ✟b✳

  ✭❜✮ ❙❡ b ✏ cn ❡ a ✏ bm✱ ❝♦♠ m, n P Z✱ ❡♥tã♦ a ✏ ♣cnq ☎ m ✏ c ☎ ♣mnq✱ ❝♦♠ mn P Z✳ ▲♦❣♦✱ c ⑤ a✳

  ✭❝✮ ❙❡ a ✏ cm ❡ b ✏ cn✱ ❝♦♠ m, n P Z✱ ❡♥tã♦ ax by ✏ cmx cny ✏ ♣mx nyq ☎ c✱ ❝♦♠ x ❡ y P Z✳ ▲♦❣♦✱ c ⑤ ♣ax byq✳

  ✭❞✮ ❙❡ a ✏ bm✱ ❝♦♠ m P Z✳ ❚♦♠❛♥❞♦ ♦s ♠ó❞✉❧♦s✱ t❡♠♦s q✉❡ ⑤a⑤ ✏ ⑤b⑤⑤m⑤✳ ❈♦♠♦ a ✘ 0✱ t❡♠♦s q✉❡ m ✘ 0✱ ❧♦❣♦ 1 ↕ ⑤m⑤ ❡✱ ❝♦♥s❡q✉❡♥t❡♠❡♥t❡✱ ⑤b⑤ ↕ ⑤b⑤⑤m⑤ ✏ ⑤a⑤✳ ✭❡✮ ❙❡ b ✏ cn✱ ❝♦♠ n P Z✱ ❡♥tã♦ ab ✏ acn✱ ❧♦❣♦ ab ✏ ♣anq ☎ c ❡ ❝♦♥s❡q✉❡♥t❡♠❡♥t❡ c ⑤ ab✳

  ✭❢✮ ❙❡ a ✏ bm✱ ❝♦♠ m P Z✱ ❡♥tã♦ ac ✏ cbm ❡ ❝♦♥s❡q✉❡♥t❡♠❡♥t❡ bc ⑤ ac✳

  ✷✳✷ ❉✐✈✐sã♦ ❊✉❝❧✐❞✐❛♥❛

  ▼❡s♠♦ q✉❛♥❞♦ ✉♠ ♥ú♠❡r♦ b ✐♥t❡✐r♦ ♥ã♦ ♥✉❧♦ ♥ã♦ ❞✐✈✐❞❡ ✉♠ ♥ú♠❡r♦ ✐♥t❡✐r♦ a✱ é ♣♦ssí✈❡❧ ❡❢❡t✉❛r ❛ ❞✐✈✐sã♦ ❞❡ a ♣♦r b ❝♦♠ r❡st♦✳ ❚❡♦r❡♠❛ ✷✳✷✳✶✳ ❙❡❥❛♠ a ❡ b ❞♦✐s ♥ú♠❡r♦s ✐♥t❡✐r♦s ❝♦♠ b ✘ 0✳ ❊①✐st❡♠ ❞♦✐s ú♥✐❝♦s ♥ú♠❡r♦s ✐♥t❡✐r♦s q ❡ r t❛✐s q✉❡ a

  ✏ b ☎ q r, ❝♦♠ 0 ↕ r ➔ ⑤b⑤ ❉❡♠♦♥str❛çã♦✿

  ❊①✐stê♥❝✐❛✿ a

  ❙✉♣♦♥❤❛ b → 0✱ ❡ s❡❥❛ q ♦ ♠❛✐♦r ✐♥t❡✐r♦ t❛❧ q✉❡ q ↕ ✳ ❊♥tã♦✱ ♣❡❧❛ ♦❜s❡r✈❛çã♦ ❛❝✐♠❛✱ b a q

  ↕ ➔ q 1✱ ♦✉ s❡❥❛✱ b ☎ q ↕ a ➔ b ☎ ♣q 1q✳ ❙♦♠❛♥❞♦ ✁b ☎ q ❛ ❡st❛ ❞❡s✐❣✉❛❧❞❛❞❡✱ ♦❜t❡♠♦s b r q✉❡ 0 ↕ a ✁ b ☎ q ➔ b✳ P♦rt❛♥t♦✱ ❞❡✜♥✐♥❞♦ r ✏ a ✁ b ☎ q✱ s❡❣✉❡ q✉❡ a ✏ b ☎ q ✱ ❝♦♠

  ↕ r ➔ ⑤b⑤✱ ♣♦✐s b → 0✳

  ✷✳✸ ▼á①✐♠♦ ❉✐✈✐s♦r ❈♦♠✉♠ ❡ ▼í♥✐♠♦ ▼ú❧t✐♣❧♦ ❈♦♠✉♠ ✷✵ ◆♦ ❝❛s♦ ❞❡ b ➔ 0✱ ❡♥tã♦ ✁b → 0✳ P♦rt❛♥t♦✱ ♣❡❧♦ ❝❛s♦ ❛❝✐♠❛✱ ❡①✐st❡♠ q✱ r P Z t❛✐s q✉❡ a

  ✏ ♣✁bq ☎ q r✱ ❝♦♠ 0 ↕ r ➔ ✁b✳ ❉❛í a ✏ b ☎ ♣✁qq r✱ ❝♦♠ 0 ↕ r ➔ ✁b ✏ ⑤b⑤✳ Pr♦✈❛♥❞♦✱ ❞❡ss❛ ❢♦r♠❛✱ ❛ ♣r✐♠❡✐r❛ ♣❛rt❡ ❞♦ t❡♦r❡♠❛✳

  ❱❡r✐✜❝❛r❡♠♦s✱ ❛❣♦r❛✱ ❛ ✉♥✐❝✐❞❛❞❡✿

  ✶ ✶ ✶ ✶ ✶

  r ❙✉♣♦♥❤❛ q✉❡ a ✏ b ☎ q r ✏ b ☎ q ✱ ♦♥❞❡ q✱ q ✱ r✱ r P Z ❡ 0 ↕ r ➔ ⑤b⑤ ❡ 0 ↕ r ➔ ⑤b⑤✳

  P❡❧❛ ✐❣✉❛❧❞❛❞❡s✱ t❡♠♦s q✉❡

  ✶ ✶ ✶ ✶

  b ☎ q ✁ b ☎ q ✏ r ✁ r ñ b ☎ ♣q ✁ q q ✏ r ✁ r. ❊♥tã♦

  ✶

  ⑤r ✁ r⑤ ➔ ⑤b⑤ ❡

  ✶ ✶

  b ☎ ♣q ✁ q q ✏ r ✁ r.

  ✶ ✶

  ◆♦t❡ q✉❡ q ✘ q ✱ ❡♥tã♦ ⑤q ✁ q ⑤ ➙ 1✱ ❞❡ ♠♦❞♦ q✉❡

  ✶ ✶ ⑤b⑤ ↕ ⑤b⑤ ☎ ⑤q ✁ q ⑤ ✏ ⑤r ✁ r ⑤ ➔ ⑤b⑤. ✶ ✶

  ❯♠❛ ❝♦♥tr❛❞✐çã♦✳ ▲♦❣♦ q ✏ q ❡✱ ❞❛í✱ r ✏ r ✳ P❛r❛ ♦ t❡♦r❡♠❛ ✷✳✷✳✶✱ é ✐♠♣♦rt❛♥t❡ t♦♠❛r ♥♦t❛ ❞❡ ✉♠❛ ♦❜s❡r✈❛çã♦✳

  ❨ ❪ a a ❉❛❞♦ P R✱ s✉❛ ♣❛rt❡ ✐♥t❡✐r❛ P R é ❞❡✜♥✐❞❛ ♣♦r b b

  ❨ ❪ a a ✏ maxtn P Z; n ↕ ✉. b b

  ❙❡ b → 0✱ ❡♥tã♦ ❨ ❪ ❨ ❪ a a q

  ✏ ❡ r ✏ a ✁ ☎ b✳ b b ❉❡ ♦✉tr♦ ♠♦❞♦✱ ♣❛r❛ n P Z✱ t❡♠♦s

  ❨ ❪ a a ✏ n ô n ↕ ➔ n 1. b b

  ❚❛✐s ✐♥t❡✐r♦s q ❡ r sã♦✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ♦ q✉♦❝✐❡♥t❡ ❡ ♦ r❡st♦ ❞❛ ❞✐✈✐sã♦ ❞❡ a ♣♦r b✳

  

✷✳✸ ▼á①✐♠♦ ❉✐✈✐s♦r ❈♦♠✉♠ ❡ ▼í♥✐♠♦ ▼ú❧t✐♣❧♦ ❈♦♠✉♠

  ◆❡st❛ s❡çã♦✱ tr❛t❛r❡♠♦s ❞❛s ❞❡✜♥✐çõ❡s ❞❡ ▼á①✐♠♦ ❉✐✈✐s♦r ❈♦♠✉♠ ❡ ▼í♥✐♠♦ ▼ú❧t✐♣❧♦

  ✷✳✸ ▼á①✐♠♦ ❉✐✈✐s♦r ❈♦♠✉♠ ❡ ▼í♥✐♠♦ ▼ú❧t✐♣❧♦ ❈♦♠✉♠ ✷✶ ❋❛❧❛r❡♠♦s ❞❛ ❞❡❝♦♠♣♦s✐çã♦ ❞❡ ✉♠ ♥ú♠❡r♦ ✐♥t❡✐r♦ ❝♦♠♦ ♣r♦❞✉t♦ ❞❡ ♥ú♠❡r♦s ♣r✐♠♦s✳

  ✷✳✸✳✶ ▼á①✐♠♦ ❉✐✈✐s♦r ❈♦♠✉♠

  ❙❡❥❛♠ ❞❛❞♦s ❞♦✐s ✐♥t❡✐r♦s a ❡ b✱ ❞✐st✐♥t♦s ♦✉ ♥ã♦✳ ❯♠ ♥ú♠❡r♦ ✐♥t❡✐r♦ d s❡rá ❞✐t♦ ❞✐✈✐s♦r ❝♦♠✉♠ ❞❡ a ❡ b s❡ d ⑤ a ❡ d ⑤ b✳ ❉❡✜♥✐çã♦ ✷✳✸✳✶✳ ❉❛❞♦s ❞♦✐s ♥ú♠❡r♦s ✐♥t❡✐r♦s a ❡ b✱ ❞✐③❡♠♦s q✉❡ ✉♠ ♥ú♠❡r♦ ✐♥t❡✐r♦ d

  → 0 é ♦ ♠á①✐♠♦ ❞✐✈✐s♦r ❝♦♠✉♠ ❞❡ a ❡ b s❡ d é ✉♠ ❞✐✈✐s♦r ❝♦♠✉♠ ❞❡ a ❡ b ❡ d é ❞✐✈✐sí✈❡❧ ♣♦r t♦❞♦ ❞✐✈✐s♦r ❝♦♠✉♠ ❞❡ a ❡ b✳

  ❖❜s❡r✈❡ q✉❡ ❛ ❞❡✜♥✐çã♦ ❛❝✐♠❛ ♥♦s ❞✐③ q✉❡ s❡ ❡①✐st❡ ✉♠ ♥ú♠❡r♦ c q✉❡ é ❞✐✈✐s♦r ❝♦♠✉♠ ❞❡ a ❡ b✱ ❡♥tã♦ c ❞✐✈✐❞❡ d✳

  ❖❜s❡r✈❡ q✉❡ ❞❛❞♦s a✱ b P Z✱ s❡ ❡①✐st✐r ♦ ♠á①✐♠♦ ❞✐✈✐s♦r ❝♦♠✉♠ ❞❡ a ❡ b✱ ❞❡♥♦t❛❞♦ ♣♦r mdc ♣a, bq✱ ❡♥tã♦✿ mdc

  ♣a, bq ✏ mdc♣✁a, bq ✏ mdc♣a, ✁bq ✏ mdc♣✁a, ✁bq. ❉❡ss❛ ❢♦r♠❛✱ ♣❛r❛ ❡❢❡✐t♦ ❞❡ ❝á❧❝✉❧♦ ❞♦ ♠á①✐♠♦ ❞✐✈✐s♦r ❝♦♠✉♠ ❞❡ ❞♦✐s ♥ú♠❡r♦s ✐♥t❡✐r♦s✱

  ♣♦❞❡♠♦s s❡♠♣r❡ s✉♣ô✲❧♦s ♥ã♦ ♥❡❣❛t✐✈♦s✳ ▲❡♠❛ ✷✳✸✳✶✳ ❙❡❥❛♠ a✱ b✱ n P Z✳ ❙❡ ❡①✐st❡ mdc♣a, b ✁ naq✱ ❡♥tã♦✱ mdc♣a, bq ❡①✐st❡ ❡ mdc

  ♣a, bq ✏ mdc♣a, b ✁ naq ❉❡♠♦♥str❛çã♦✿ ❙❡❥❛ d ✏ mdc♣a, b ✁ naq✳ ❈♦♠♦ d ⑤ a ❡ d ⑤ ♣b ✁ naq✱ s❡❣✉❡ q✉❡ d ⑤ n ☎ a✱ ❡♥tã♦ d ⑤ b✳ ▲♦❣♦ d é ✉♠ ❞✐✈✐s♦r ❝♦♠✉♠ ❡♥tr❡ a ❡ b✳ ❙✉♣♦♥❞♦ ❛❣♦r❛ q✉❡ c s❡❥❛ ✉♠ ❞✐✈✐s♦r ❝♦♠✉♠ ❞❡ a ❡ b✳ ▲♦❣♦✱ c é ✉♠ ❞✐✈✐s♦r ❝♦♠✉♠ ❞❡ a ❡ b✁na ❡✱ ♣♦rt❛♥t♦✱ c ⑤ d✳ ❈♦♠♣❧❡t❛♥❞♦ ❛ ❞❡♠♦♥str❛çã♦ ❞♦ ❧❡♠❛✳

  ✷✳✸✳✷ Pr♦♣r✐❡❞❛❞❡s ❞♦ ♠❞❝

  ◆❡st❛ s✉❜s❡çã♦ ❞❡♠♦♥str❛r❡♠♦s ♣r♦♣r✐❡❞❛❞❡s ❞♦ ♠á①✐♠♦ ❞✐✈✐s♦r ❝♦♠✉♠ q✉❡ sã♦ r❡✲ ❧❡✈❛♥t❡s ❛♦ q✉❡ ♥♦s ♣r♦♣♦♠♦s ❛ ❢❛③❡r✳ ❉❛❞♦s d P Z, d ✘ 0✱ ❞❡♥♦t❛r❡♠♦s ♦ ❝♦♥❥✉♥t♦ ty ✏ d ☎ n, n P Z✉ ♣♦r dZ✳ ❚❡♦r❡♠❛ ✷✳✸✳✶✳ ❙❡❥❛♠ a✱ b P Z✱ ♥ã♦ ❛♠❜♦s ♥✉❧♦s✳ ❙❡ d ✏ min♣txa yb; x, y P Z✉❳Nq✱ ❡♥tã♦ d é ♦ ♠á①✐♠♦ ❞✐✈✐s♦r ❝♦♠✉♠ ❞❡ a ❡ b ❡ txa by; x, y P Z✉ ✏ dZ :✏ tx; x ✏ ld; l P Z✉.

  ✷✳✸ ▼á①✐♠♦ ❉✐✈✐s♦r ❈♦♠✉♠ ❡ ▼í♥✐♠♦ ▼ú❧t✐♣❧♦ ❈♦♠✉♠ ✷✷ ❉❡♠♦♥str❛çã♦✿ ❙✉♣♦♥❤❛ q✉❡ c é ✉♠ ❞✐✈✐s♦r ❝♦♠✉♠ ❞❡ a ❡ b✳ ❈♦♥s❡q✉❡♥t❡♠❡♥t❡✱ c ❞✐✈✐❞❡ t♦❞♦s ♦s ♥ú♠❡r♦s ♥❛t✉r❛✐s ❞❛ ❢♦r♠❛ xa yb✳ P♦rt❛♥t♦✱ c ❞✐✈✐❞❡ t♦❞♦s ♦s ❡❧❡♠❡♥t♦s ❞❡ txa yb; x, y P Z✉ ❡✱ ❞❡ss❛ ❢♦r♠❛✱ c ⑤ d✳

  ❚♦♠❛♥❞♦ z P txa yb; x, y P Z✉ ❳ N✱ s✉♣♦♥❤❛✱ ♣♦r ❛❜s✉r❞♦✱ q✉❡ d ∤ z✳ P♦rt❛♥t♦✱ ♣❡❧❛ ❞✐✈✐sã♦ ❡✉❝❧✐❞✐❛♥❛✱ z

  ✏ d ☎ q r, ❝♦♠ 0 ➔ r ➔ d. ✭✷✳✶✮ ❈♦♠♦ z, d P txa yb; x, y P Z✉ ❳ N t❡♠✲s❡ q✉❡ z ✏ xa yb ❡ d ✏ ma nb✱ ♣❛r❛

  ❛❧❣✉♥s x, y, m, n P Z✱ s❡❣✉❡✲s❡ ❞❛ ❡q✉❛çã♦ ✷✳✶ q✉❡ xa yb ✏ ♣ma nbq ☎ q r✳ ❉♦♥❞❡ ❝♦♥❝❧✉í♠♦s q✉❡ r

  ✏ ♣x ✁ qmq ☎ a ♣y ✁ qnq ☎ b ❡✱ ♣♦rt❛♥t♦✱ r P txa yb; x, y P Z✉❳N✳ ❯♠ ❛❜s✉r❞♦✱ ♣♦✐s d ✏ mintxa yb; x, y P Z✉❳N ❡ r ➔ d✳ ❊♠ ♣❛rt✐❝✉❧❛r✱ d ⑤ a ❡ d ⑤ b✳ ❆ss✐♠✱ ♣r♦✈❛♠♦s q✉❡ d é ♦ ♠á①✐♠♦ ❞✐✈✐s♦r ❝♦♠✉♠ ❞❡ a ❡ b✳

  ❆❣♦r❛✱ ♣r♦✈❛r❡♠♦s q✉❡ txa yb; x, y P Z✉ ✏ dZ✳ ❉❛❞♦ q✉❡ t♦❞♦ ❡❧❡♠❡♥t♦ ❞♦ ❝♦♥❥✉♥t♦ txa yb; x, y P Z✉ é ❞✐✈✐sí✈❡❧ ♣♦r d✱ t❡♠♦s q✉❡ txa yb; x, y P Z✉ ⑨ dZ✳ P♦r ♦✉tr♦ ❧❛❞♦✱ ♣❛r❛ t♦❞♦ w P dZ✱ t❡♠♦s q✉❡ w

  ✏ ld ✏ l ☎ ♣ma nbq ✏ ♣lmq ☎ a ♣lnq ☎ b ✏ xa yb, x ✏ lm ❡ y ✏ ln ❡✱ ♣♦rt❛♥t♦✱ w P txa yb; x, y P Z✉✳ ▲♦❣♦✱ dZ ⑨ txa yb; x, y P Z✉✳ ❉❡ss❛ ❢♦r♠❛✱ ❝♦♥❝❧✉í♠♦s q✉❡ txa yb; x, y P Z✉ ✏ dZ✳

  ❯♠❛ ❝♦♥s❡q✉ê♥❝✐❛ ✐♥t✉✐t✐✈❛ ❞♦ t❡♦r❡♠❛ ❛♥t❡r✐♦r é ♦ ❝♦r♦❧ár✐♦ ❛❜❛✐①♦✳ ❈♦r♦❧ár✐♦ ✷✳✸✳✶✳ ◗✉❛✐sq✉❡r q✉❡ s❡❥❛♠ a, b P Z✱ ♥ã♦ ❛♠❜♦s ♥✉❧♦s✱ ❡ n P N✱ t❡♠✲s❡ q✉❡ mdc

  ♣na, nbq ✏ n ☎ mdc♣a, bq. ❉❡♠♦♥str❛çã♦✿ ◆♦t❡ q✉❡ tn☎xa n☎yb; x, y P Z✉ ✏ tn☎♣xa ybq; x, y P Z✉ ✏ tw ✏ n☎z; z P txa yb; x, y P Z✉✉.

  ❆❣♦r❛ ♦ r❡s✉❧t❛❞♦ s❡❣✉❡✲s❡ ❞♦ t❡♦r❡♠❛ ✷✳✸✳✶ ❡ ❞♦ ❢❛t♦ ❞❡ q✉❡ min ♣txa yb; x, y P Z✉ ❳ Nq ✏ n ☎ mintxa yb; x, y P Z ❳ N✉.

  ✷✳✸ ▼á①✐♠♦ ❉✐✈✐s♦r ❈♦♠✉♠ ❡ ▼í♥✐♠♦ ▼ú❧t✐♣❧♦ ❈♦♠✉♠ ✷✸ ❈♦r♦❧ár✐♦ ✷✳✸✳✷✳ ❉❛❞♦s a, b P Z✱ ♥ã♦ ❛♠❜♦s ♥✉❧♦s✱ t❡♠✲s❡ q✉❡

  ✂ ✡ a b mdc , ✏ 1. mdc mdc

  ♣a, bq ♣a, bq ❉❡♠♦♥str❛çã♦✿ P❡❧♦ ❝♦r♦❧ár✐♦ ✷✳✸✳✶✱ t❡♠♦s q✉❡

  ✂ ✡ ✁ ✠ a b a b mdc , mdc , mdc

  ♣a, bq ☎ mdc mdc mdc ✏ mdc ♣a, bq ♣a, bq

  ♣a,bq ♣a,bq

  mdc mdc ♣a, bq ♣a, bq

  ✏ mdc♣a, bq, ♦ q✉❡ ♣r♦✈❛ ♦ r❡s✉❧t❛❞♦✳

  ❆ s❡❣✉✐r ❡♥✉♥❝✐❛r❡♠♦s ✉♠❛ ♣r♦♣♦s✐çã♦✱ ❝✉❥♦ r❡s✉❧t❞❛♦ é ❞❡ s✉♠❛ ✐♠♣♦rtâ♥❝✐❛ ♣❛r❛ ♥♦ss❛s ♣r❡t❡♥sõ❡s✳ ❉❡✜♥✐çã♦ ✷✳✸✳✷✳ ❉♦✐s ♥ú♠❡r♦s ✐♥t❡✐r♦s s❡rã♦ ❞✐t♦s ♣r✐♠♦s ❡♥tr❡ s✐✱ ♦✉ ❝♦♣r✐♠♦s✱ s❡ ♦ ú♥✐❝♦ ❞✐✈✐s♦r ♣♦s✐t✐✈♦ ❞❡ ❛♠❜♦s é ✶✳ ❞❡ ✈✐st❛ t❡ór✐❝♦ ❡ q✉❡ ❢❛❝✐❧✐t❛ ♦ r❡❝♦♥❤❡❝✐♠❡♥t♦ ❞❡ ✉♠ ♥ú♠❡r♦ ♣r✐♠♦✳ Pr♦♣♦s✐çã♦ ✷✳✸✳✶✳ ❉♦✐s ♥ú♠❡r♦s ✐♥t❡✐r♦s a ❡ b sã♦ ♣r✐♠♦s ❡♥tr❡ s✐ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ ❡①✐st❡♠ ♥ú♠❡r♦s ✐♥t❡✐r♦s m ❡ n t❛✐s q✉❡ ma nb ✏ 1✳ ❉❡♠♦♥str❛çã♦✿ ❙✉♣♦♥❤❛ q✉❡ a ❡ b sã♦ ♣r✐♠♦s ❡♥tr❡ s✐✳ ▲♦❣♦ mdc♣a, bq ✏ 1✳ ❊♥tã♦✱ ♣❡❧♦ t❡♦r❡♠❛ ✷✳✸✳✶✱ t❡♠♦s q✉❡ ❡①✐st❡♠ ♥ú♠❡r♦s ✐♥t❡✐r♦s m ❡ n t❛✐s q✉❡ ma nb ✏ mdc♣a, bq ✏ 1✳

  ❘❡❝✐♣r♦❝❛♠❡♥t❡✱ s✉♣♦♥❤❛ q✉❡ ❡①✐st❛♠ ♥ú♠❡r♦s ✐♥t❡✐r♦s m ❡ n t❛✐s q✉❡ ma nb ✏ 1✳ ❙❡ d ✏ mdc♣a, bq✱ t❡♠♦s q✉❡ d ⑤ ♣ma nbq✱ ♦ q✉❡ ♠♦str❛ q✉❡ d ⑤ 1✱ ❡✱ ♣♦rt❛♥t♦✱ d ✏ 1✳ ❚❡♦r❡♠❛ ✷✳✸✳✷✳ ✭▲❡✐ ❞❡ ●❛✉ss✮ ❙❡❥❛♠ a, b ❡ c ♥ú♠❡r♦s ✐♥t❡✐r♦s✳ ❙❡ a ⑤ bc ❡ mdc♣a, bq ✏ 1✱ ❡♥tã♦ a ⑤ c✳ ❉❡♠♦♥str❛çã♦✿ ❙❡ a ⑤ bc✱ ❡♥tã♦ ❡①✐st❡ e P Z t❛❧ q✉❡ b ☎ c ✏ a ☎ e✳ P♦r ♦✉tr♦ ❧❛❞♦✱ s❡ mdc

  ♣a, bq ✏ 1✱ ❡♥tã♦✱ ♣❡❧❛ ♣r♦♣♦s✐çã♦ ✷✳✸✳✶✱ t❡♠♦s q✉❡ ❡①✐st❡♠ m, n P Z t❛✐s q✉❡

  ✷✳✸ ▼á①✐♠♦ ❉✐✈✐s♦r ❈♦♠✉♠ ❡ ▼í♥✐♠♦ ▼ú❧t✐♣❧♦ ❈♦♠✉♠ ✷✹ ma nb ✏ 1. ✭✷✳✷✮

  ▼✉❧t✐♣❧✐❝❛♥❞♦ ♣♦r c ❛♠❜♦s ♦s ♠❡♠❜r♦s ❞❛ ❡q✉❛çã♦ ✷✳✷✱ ♦❜t❡♠♦s✿ ma ☎ c nb ☎ c ✏ c. ❉❡ss❛ ❢♦r♠❛✱ s✉❜st✐✉í♥❞♦ b ☎ c ♣♦r a ☎ e ♥❡st❛ ú❧t✐♠❛ ✐❣✉❛❧❞❛❞❡✱ t❡♠♦s c

  ✏ mac nae ✏ a ☎ ♣mc neq. ❈♦♥s❡q✉❡♥t❡♠❡♥t❡✱ a ⑤ c✳

  ❈♦r♦❧ár✐♦ ✷✳✸✳✸✳ ❉❛❞♦s a, b, c P Z ❝♦♠ b ❡ c ♥ã♦ ❛♠❜♦s ♥✉❧♦s✱ t❡♠♦s q✉❡ bc b ⑤ a e c ⑤ a ô ⑤ a. mdc

  ♣b, cq ❉❡♠♦♥str❛çã♦✿ ❈♦♠♦ b ⑤ a ❡ c ⑤ a✱ t❡♠♦s q✉❡ a ✏ nb ✏ mc ♣❛r❛ ❛❧❣✉♠ n, m P Z✳ ▲♦❣♦✱ n m

  ☎ b ☎ c .

  ✏ ✭✷✳✸✮ mdc mdc ♣b, cq ♣b, cq

  ✂ ✡ ,

  P♦r ♦✉tr♦ ❧❛❞♦✱ ♣❡❧♦ ❝♦r♦❧ár✐♦ ✷✳✸✳✷✱ t❡♠♦s q✉❡ mdc ✏ 1✱ ❡ mdc mdc ♣b, cq ♣b, cq b b

  ☎ c ❝❧❛r❛♠❡♥t❡ ⑤ m ♥❛ ❡q✉❛çã♦ ✷✳✸✱ ♦ q✉❡ ✐♠♣❧✐❝❛ q✉❡ ⑤ m ☎ c✳ ❈♦♠♦ mdc mdc

  ♣b, cq ♣b, cq m

  ☎ c ✏ a✱ ❛ ✐♠♣❧✐❝❛çã♦ ❞✐r❡t❛ ✜❝❛ ♣r♦✈❛❞❛ ❡ ❛ r❡❝í♣r♦❝❛ é ✐♠❡❞✐❛t❛✳ ❖ ❝♦♥❝❡✐t♦ ❞❡ ♠á①✐♠♦ ❞✐✈✐s♦r ❝♦♠✉♠ ❡♥tr❡ ❞♦✐s ♥ú♠❡r♦s ♣♦❞❡ s❡r ❣❡♥❡r❛❧✐③❛❞♦

  ♣❛r❛ ✉♠❛ q✉❛♥t✐❞❛❞❡ ✜♥✐t❛ ❞❡ ♥ú♠❡r♦s✳ 1 , ..., a n ❉❡ ❢❛t♦✱ ✉♠ ♥ú♠❡r♦ ♥❛t✉r❛❧ d s❡rá ❞✐t♦ ♠❞❝ ❞❡ ✉♠❛ q✉❛♥t✐❞❛❞❡ ✜♥✐t❛ a ❞❡ 1 , ..., a n

  ♥ú♠❡r♦s ✐♥t❡✐r♦s✱ ♥ã♦ t♦❞♦s ♥✉❧♦s✱ s❡ d é ✉♠ ❞✐✈✐s♦r ❝♦♠✉♠ ❞❡ a ❡ s❡ c é ✉♠ ❞✐✈✐s♦r 1 , ..., a n ❝♦♠✉♠ ❞❡ a ✱ ❡♥tã♦ c ⑤ d✳

  ✷✳✸✳✸ ▼í♥✐♠♦ ▼ú❧t✐♣❧♦ ❈♦♠✉♠

  ❉✐r❡♠♦s q✉❡ ✉♠ ♥ú♠❡r♦ ✐♥t❡✐r♦ é ✉♠ ♠ú❧t✐♣❧♦ ❝♦♠✉♠ ❞❡ ❞♦✐s ♥ú♠❡r♦s ✐♥t❡✐r♦s ❞❛❞♦s s❡ ❡❧❡ é s✐♠✉❧t❛♥❡❛♠❡♥t❡ ♠ú❧t✐♣❧♦ ❞❡ ❛♠❜♦s ♦s ♥ú♠❡r♦s✳

  ✷✳✸ ▼á①✐♠♦ ❉✐✈✐s♦r ❈♦♠✉♠ ❡ ▼í♥✐♠♦ ▼ú❧t✐♣❧♦ ❈♦♠✉♠ ✷✺ ❉❡✜♥✐çã♦ ✷✳✸✳✸✳ ❉✐r❡♠♦s q✉❡ ✉♠ ♥ú♠❡r♦ ✐♥t❡✐r♦ m ➙ 0 é ✉♠ ♠í♥✐♠♦ ♠ú❧t✐♣❧♦ ❝♦♠✉♠ ❞♦s ♥ú♠❡r♦s ✐♥t❡✐r♦s a ❡ b✱ s❡ m é ✉♠ ♠ú❧t✐♣❧♦ ❝♦♠✉♠ ❞❡ a ❡ b✱ ❡ s❡ c é ✉♠ ♠ú❧t✐♣❧♦ ❝♦♠✉♠ ❞❡ a ❡ b✱ ❡♥tã♦ m ⑤ c✳

  ❈♦♠♦ ♦❝♦rr❡ ❝♦♠ ♦ ♠❞❝✱ ♦ ♠í♥✐♠♦ ♠ú❧t✐♣❧♦ ❝♦♠✉♠ ❞❡ a ❡ b✱ s❡ ❡①✐st❡✱ ❞❡♥♦t❛❞♦ ♣♦r mmc ♣a, bq✱ s❛t✐s❢❛③ ❛ r❡❧❛çã♦ ❛❜❛✐①♦✿ mmc

  ♣✁a, bq ✏ mmc♣a, ✁bq ✏ mmc♣✁a, ✁bq ✏ mmc♣a, bq. ❆ss✐♠✱ ♣❛r❛ ❡❢❡✐t♦ ❞❡ ❝á❧❝✉❧♦ ❞♦ ♠í♥✐♠♦ ♠ú❧t✐♣❧♦ ❝♦♠✉♠ ❞❡ ❞♦✐s ♥ú♠❡r♦s✱ ♣♦❞❡♠♦s s❡♠♣r❡ s✉♣ô✲❧♦s ♥ã♦ ♥❡❣❛t✐✈♦s✳

  Pr♦♣♦s✐çã♦ ✷✳✸✳✷✳ ❉❛❞♦s ❞♦✐s ♥ú♠❡r♦s ✐♥t❡✐r♦s a ❡ b✱ t❡♠♦s q✉❡ mdc♣a, bq ❡①✐st❡ ❡ mmc ♣a, bq ☎ mdc♣a, bq ✏ ⑤a⑤ ☎ ⑤b⑤. ❉❡♠♦♥str❛çã♦✿ ❙❡ a ✏ 0 ♦✉ b ✏ 0✱ ❛ ✐❣✉❛❧❞❛❞❡ ❛❝✐♠❛ é tr✐✈✐❛❧♠❡♥t❡ s❛t✐s❢❡✐t❛✳ ❙❡❥❛ a

  ☎ b m ✏ ❡✱ s❡♠ ♣❡r❞❛ ❞❡ ❣❡♥❡r❛❧✐❞❛❞❡✱ s✉♣♦♥❤❛ a, b P N✳ ❈♦♠♦ mdc

  ♣a, bq a b ☎ b ☎ a m ,

  ✏ ✏ mdc mdc ♣a, bq ♣a, bq t❡♠♦s q✉❡ a ⑤ m ❡ b ⑤ m✳ P♦rt❛♥t♦✱ m é ✉♠ ♠ú❧t✐♣❧♦ ❝♦♠✉♠ ❞❡ a ❡ b✳ ❙❡❥❛ c ✉♠ ♠ú❧t✐♣❧♦

  ✶

  b ❝♦♠✉♠ ❞❡ a ❡ b❀ ❧♦❣♦✱ c ✏ na ✏ n ✳ ❙❡❣✉❡ ❞❛í q✉❡ a b

  ✶ n .

  ☎ ✏ n ☎ mdc mdc ♣a, bq ♣a, bq a b

  P❡❧♦ ❝♦r♦❧ár✐♦ ✷✳✸✳✷✱ ❡ sã♦ ♣r✐♠♦s ❡♥tr❡ s✐✳ ❙❡❣✉❡✲s❡✱ ❡♥tã♦✱ q✉❡ mdc mdc ♣a, bq ♣a, bq a a

  ☎ b

  ✶ ✶

  b ❞✐✈✐❞❡ n ✱ ❡✱ ♣♦rt❛♥t♦✱ m ✏ ❞✐✈✐❞❡ n q✉❡✱ é ✐❣✉❛❧ ❛ c✳ ▲♦❣♦✱ mdc mdc

  ♣a, bq ♣a, bq m ✏ mmc ♣a, bq ❡ ❞❛í s❡❣✉❡ ❛ r❡❧❛çã♦✳

  ❱❛❧❡ s❛❧✐❡♥t❛r ✉♠ r❡s✉❧t❛❞♦ q✉❡ s❡❣✉❡ ✐♠❡❞✐❛t❛♠❡♥t❡ ❞❛ ♣r♦♣♦s✐çã♦ ✷✳✸✳✷✿ ❈♦r♦❧ár✐♦ ✷✳✸✳✹✳ ❙❡ a ❡ b sã♦ ♥ú♠❡r♦s ✐♥t❡✐r♦s ♣r✐♠♦s ❡♥tr❡ s✐✱ ❡♥tã♦ mmc♣a, bq ✏ ⑤a☎b⑤.

  ❆ss✐♠ ❝♦♠♦ ✜③❡♠♦s ❝♦♠ ♦ ♠á①✐♠♦ ❞✐✈✐s♦r ❝♦♠✉♠✱ ♣♦❞❡♠♦s ❣❡♥❡r❛❧✐③❛r ♦ ❝♦♥❝❡✐t♦ ❞❡ ♠í♥✐♠♦ ♠ú❧t✐♣❧♦ ❝♦♠✉♠✳ ❉❡ ❢❛t♦✱ ❞❛❞♦ ✉♠ ♥ú♠❡r♦ ♥❛t✉r❛❧ m ❡ ♥ú♠❡r♦s ✐♥t❡✐r♦s a 1 , a 2 , n

  ✷✳✸ ▼á①✐♠♦ ❉✐✈✐s♦r ❈♦♠✉♠ ❡ ▼í♥✐♠♦ ▼ú❧t✐♣❧♦ ❈♦♠✉♠ ✷✻

  ✶ 1 , a 2 , n

  ♣❛r❛ t♦❞♦ ♠ú❧t✐♣❧♦ ❝♦♠✉♠ m ❞♦s ♥ú♠❡r♦s a ☎ ☎ ☎ , a ✱ t❡♠✲s❡ q✉❡ m ⑤ m ✳ 1 , n Pr♦♣♦s✐çã♦ ✷✳✸✳✸✳ ❙❡❥❛♠ a ☎ ☎ ☎ , a ♥ú♠❡r♦s ✐♥t❡✐r♦s ♥ã♦ ♥✉❧♦s✳ ❊♥tã♦ ❡①✐st❡ ♦ ♥ú♠❡r♦ mmc 1 , n

  ♣a ☎ ☎ ☎ , a q ❡ mmc 1 , , a n n n n n 1 , , mmc , a ♣a ☎ ☎ ☎ , a ✁1 q ✏ mmc♣a ☎ ☎ ☎ , a ✁2 ♣a ✁1 qq. ❱❡❥❛♠♦s ✉♠ ❡①❡♠♣❧♦ q✉❡ ✐❧✉str❛ ❛ ♣r♦♣♦s✐çã♦ ✷✳✸✳✸✳

  ❊①❡♠♣❧♦ ✷✳✸✳✶✳ ❉❡t❡r♠✐♥❡ ♦ ♠í♥✐♠♦ ♠ú❧t✐♣❧♦ ❝♦♠✉♠ ❡♥tr❡ 4, 6 e 12✳ ❖❜s❡r✈❡ ✉♠❛ s♦❧✉çã♦✳ ❉❡♥♦t❡♠♦s m♣4q, m♣6q ❡ m♣12q ❝♦♠♦ ♦s ♠ú❧t✐♣❧♦s ❞❡ ✹✱ ✻ ❡ ✶✷ r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ❖s

  ♠ú❧t✐♣❧♦s ❞❡ss❡s ♥ú♠❡r♦s sã♦ m ♣4q ✏ t0, 4, 8, ✶✷, 16, 20, ✷✹, 28, 32, ✸✻, 40, 44, ✹✽, 52, 56, ✻✵, ☎ ☎ ☎ ✉ m ♣6q ✏ t0, 6, ✶✷, 18, ✷✹, 30, ✸✻, 42, ✹✽, 54, ✻✵, ☎ ☎ ☎ ✉ m ♣12q ✏ t0, ✶✷, ✷✹, ✸✻, ✹✽, ✻✵, ☎ ☎ ☎ ✉

  P❡r❝❡❜❛ q✉❡ ♦ ♠❡♥♦r ♠ú❧t✐♣❧♦ ❝♦♠✉♠ ❡♥tr❡ ❡ss❡s ♥ú♠❡r♦s é ♦ ✶✷✳ ▲♦❣♦✱ ♦ mmc♣4, 6, 12q é ♦ ♥ú♠❡r♦ ✶✷✳

  ❱❡❥❛ ♦✉tr❛ s♦❧✉çã♦ ♣❡❧♦ ♣r♦❝❡ss♦ ❞❛ ❞❡❝♦♠♣♦s✐çã♦ s✐♠✉❧tâ♥❡❛✳ ❈♦❧♦❝❛♠♦s ♦s ♥ú♠❡r♦s ✹✱ ✻ ❡ ✶✷ ❡♠ ✉♠❛ ❧✐♥❤❛ ❡ ❢❛③❡♠♦s ❛ ❞✐✈✐sã♦ ❞❡ss❡s ♥ú♠❡r♦s ♣❡❧♦

  ❢❛t♦r ♣r✐♠♦ ❝♦❧♦❝❛❞♦ ♥❛ ú❧t✐♠❛ ❝♦✉❧♥❛✳ ✹ ✻ ✶✷ ✷ ✷ ✸ ✻

  ❈♦❧♦❝❛♠♦s ❛❜❛✐①♦ ❞❡ss❡s ♥ú♠❡r♦s ♦ r❡s✉❧t❛❞♦ ❞❛ ❞✐✈✐sã♦ ❞❡❧❡s ♣❡❧♦ ♥ú♠❡r♦ ❝♦❧♦❝❛❞♦ ♥❛ ú❧t✐♠❛ ❝♦❧✉♥❛✳ ❙❡ ♥♦ r❡s✉❧t❛❞♦ ❞❡st❛ ❞✐✈✐sã♦ ❛✐♥❞❛✱ ❛ss✐♠✱ ❛♣❛r❡❝❡r ♥ú♠❡r♦s q✉❡ sã♦ ❞✐✈✐sí✈❡✐s ♣❡❧♦ ♠❡s♠♦ ❢❛t♦r ♣r✐♠♦✱ r❡❛❧✐③❛r❡♠♦s ❛ ❞✐✈✐sã♦ ❞❡st❡s r❡s✉❧t❛❞♦s ♣❡❧♦ ♠❡s♠♦ ❢❛t♦r ♣r✐♠♦✳ ◗✉❛♥❞♦ ♥ã♦ ❡①✐st❡ ♠❛✐s ♥❡♥❤✉♠ ♥ú♠❡r♦ ♥♦ r❡s✉❧t❛❞♦ ❞❛ ❞✐✈✐sã♦ ♠❡♥❝✐♦♥❛❞❛ ❛♥t❡r✐♦r♠❡♥t❡ q✉❡ s❡❥❛ ❞✐✈✐sí✈❡❧ ♣♦r ❡st❡ ❢❛t♦r ♣r✐♠♦✱ r❡♣❡t✐r❡♠♦s ♦ ♣r♦❝❡ss♦ ❛♥t❡r✐♦r ❝♦♠ ♦ ♣ró①✐♠♦ ❢❛t♦r ♣r✐♠♦✳

  ✹ ✻ ✶✷ ✷ ✷ ✸ ✻ ✷ ✶ ✸ ✸ ✸ ✶ ✶ ✶

  ✷✳✹ ◆ú♠❡r♦s Pr✐♠♦s ✷✼

  P❛r❛ ❞❡t❡r♠✐♥❛r ♦ ♠í♥✐♠♦ ♠ú❧t✐♣❧♦ ❝♦♠✉♠ ❜❛st❛ ♠✉❧t✐♣❧✐❝❛r♠♦s t♦❞♦s ♦s ❢❛t♦r❡s q✉❡ s❡ ❡♥❝♦♥tr❛♠ ♥❛ ú❧t✐♠❛ ❝♦❧✉♥❛✱ ♦✉ s❡❥❛✱ 2 ☎ 2 ☎ 3 ✏ 12✳ P♦rt❛♥t♦✱ mmc♣4, 6, 12q ✏ 12✳ ❆❣♦r❛✱ ✉s❛♥❞♦ ❛ ♣r♦♣♦s✐çã♦ ✷✳✸✳✸✱ ❞❛r❡♠♦s ✉♠❛ s♦❧✉çã♦ ♠❛✐s s✐♠♣❧✐✜❝❛❞❛✳ mmc

  ♣4, 6, 12q ✏ mmc♣4, mmc♣6, 12qq ✏ mmc♣4, 12q ✏ 12.

  ✷✳✹ ◆ú♠❡r♦s Pr✐♠♦s

  ◆❡st❛ s❡çã♦ ❞❡✜♥✐r❡♠♦s ♦ ❝♦♥❝❡✐t♦ ❞❡ ♥ú♠❡r♦ ♣r✐♠♦✱ ✉♠ ❞♦s ❝♦♥❝❡✐t♦s ♠❛✐s ✐♠♣♦rt❛♥t❡ ❞❛ ❚❡♦r✐❛ ❞♦s ◆ú♠❡r♦s✳ ❆ss✐♠ ❝♦♠♦ ❞❡s❡♥✈♦❧✈❡r❡♠♦s ❛❧❣✉♥s ❛s♣❡❝t♦s t❡ór✐❝♦s r❡❧❛t✐✈♦s ❛ ❡ss❡ ❝♦♥❝❡✐t♦ ❝❛♣❛③ ❞❡ ❞❛r s♦❧✐❞❡③ ❡ ❝♦♥s✐stê♥❝✐❛ ❛ ❡ss❛ t❡♦r✐❛ ❡ ❡♥✉♥❝✐❛r❡♠♦s ♦ ❚❡♦r❡♠❛ ❋✉♥❞❛♠❡♥t❛❧ ❞❛ ❆r✐t♠ét✐❝❛✳ ❉❡✜♥✐çã♦ ✷✳✹✳✶✳ ❯♠ ♥ú♠❡r♦ ♥❛t✉r❛❧ p, p → 1✱ é ❝❤❛♠❛❞♦ ❞❡ ♥ú♠❡r♦ ♣r✐♠♦ s❡ ♦s ú♥✐❝♦s ❞✐✈✐s♦r❡s ♣♦s✐t✐✈♦s ❞❡ p é ♦ ♣ró♣r✐♦ p ❡ ♦ ♥ú♠❡r♦ ✶✳

  ❯♠❛ ❝♦♥s❡q✉ê♥❝✐❛ ✐♠❡❞✐❛t❛ ❞❛ ❞❡✜♥✐çã♦ ❛❝✐♠❛ é ❛ s❡❣✉✐♥t❡ ❛✜r♠❛çã♦✿ ❞❛❞♦s ❞♦✐s ♥ú♠❡r♦s ♣r✐♠♦s p ❡ q ❡ ✉♠ ♥ú♠❡r♦ ✐♥t❡✐r♦ a q✉❛❧q✉❡r✱ t❡♠♦s

  ✭✐✮ ❙❡ p ⑤ q✱ ❡♥tã♦ p ✏ q✳ ✭✐✐✮ ❙❡ p ∤ a✱ ❡♥tã♦ mdc♣p, aq ✏ 1✳

  ❉❡♠♦♥str❛çã♦✿ ✭✐✮ ❈♦♠♦ p ⑤ q ❡ s❡♥❞♦ q ♣r✐♠♦✱ t❡♠♦s q✉❡ p ✏ 1 ♦✉ p ✏ q✳ ❙❡♥❞♦ p ♣r✐♠♦✱ t❡♠✲s❡ q✉❡ p

  → 1✱ ♦ q✉❡ ❛❝❛rr❡t❛ p ✏ q✳ ✭✐✐✮ ❙❡ mdc♣p, aq ✏ d✱ t❡♠♦s q✉❡ d ⑤ p ❡ d ⑤ a✳ P♦rt❛♥t♦✱ d ✏ p ♦✉ d ✏ 1✳ ▼❛s d ✘ p✱

  ♣♦✐s p ∤ a ❡✱ ❝♦♥s❡q✉❡♥t❡♠❡♥t❡✱ d ✏ 1✳ ❯♠ ♥ú♠❡r♦ ✐♥t❡✐r♦ ♠❛✐♦r q✉❡ 1 ❡ q✉❡ ♥ã♦ é ♣r✐♠♦ s❡rá ❝❤❛♠❛❞♦ ❞❡ ♥ú♠❡r♦ ❝♦♠♣♦st♦✳

  Pr♦♣♦s✐çã♦ ✷✳✹✳✶✳ ✭▲❡♠❛ ❞❡ ❊✉❝❧✐❞❡s✮ ❙❡❥❛♠ a, b, p P Z✱ ❝♦♠ p ♣r✐♠♦✳ ❙❡ p ⑤ a ☎ b✱

  ✷✳✹ ◆ú♠❡r♦s Pr✐♠♦s ✷✽

  ❉❡♠♦♥str❛çã♦✿ ❉✐✈✐❞✐r❡♠♦s ❛ ❞❡♠♦♥str❛çã♦ ❡♠ ❞✉❛s s✐t✉❛çõ❡s✿ ♣p ⑤ a ☎ bq ♦✉ ♣p ∤ a ❡ p ∤ b q✳

  ❙❡ p ⑤ a ☎ b ❡ p ∤ a✱ ❡♥tã♦ p ⑤ b✳ ❈♦♠♦ p ⑤ a ☎ b✱ ❡♥tã♦ ❡①✐st❡ ✉♠ ♥ú♠❡r♦ c t❛❧ q✉❡ a ☎ b ✏ p ☎ c✳ ▼❛s p ∤ a✱ ❧♦❣♦ mdc♣p, aq ✏ 1 ❡ ♣♦rt❛♥t♦ p ⑤ b✳

  P♦r ♦✉tr♦ ❧❛❞♦✱ ❙❡ p ⑤ a ☎ b ❡ p ∤ b✱ ❡♥tã♦ p ⑤ a✳ ❈♦♠♦ p ⑤ a ☎ b✱ ❡♥tã♦ ❡①✐st❡ ✉♠ ♥ú♠❡r♦ e t❛❧ q✉❡ a ☎ b ✏ p ☎ e✳ ▼❛s p ∤ b✱ ❧♦❣♦ mdc♣p, bq ✏ 1 ❡ ♣♦rt❛♥t♦ p ⑤ a✳ 1 , n n 1 2 n

  ❈♦r♦❧ár✐♦ ✷✳✹✳✶✳ ❙❡❥❛♠ p, p ☎ ☎ ☎ , p ♥ú♠❡r♦s ♣r✐♠♦s✳ ❙❡ p ⑤ p ☎ p ☎ ☎ ☎ p ✱ ❡♥tã♦ p i ✏ p ♣❛r❛ ❛❧❣✉♠ i ✏ 1, ☎ ☎ ☎ , n✳

  ❋✐♥❛❧✐③❛r❡♠♦s ❛ s❡çã♦ ❝♦♠ ♦ ❚❡♦r❡♠❛ ❋✉♥❞❛♠❡♥t❛❧ ❞❛ ❆r✐t♠ét✐❝❛✳ ❯♠ t❡♦r❡♠❛ ❞❡ ❝♦♥t❡ú❞♦ r❡❧❡✈❛♥t❡ ♥♦ q✉❡ ❞✐③ r❡s♣❡✐t♦ ❛♦ ❡st✉❞♦ ❞♦s ♥ú♠❡r♦s ♣r✐♠♦s✳ ❚❡♦r❡♠❛ ✷✳✹✳✶✳ ✭❚❡♦r❡♠❛ ❋✉♥❞❛♠❡♥t❛❧ ❞❛ ❆r✐t♠ét✐❝❛✮ ❚♦❞♦ ♥ú♠❡r♦ ♥❛t✉r❛❧ ♠❛✐♦r ❞♦ q✉❡ ✶ ♦✉ é ♣r✐♠♦ ♦✉ s❡ ❡s❝r❡✈❡ ❞❡ ♠♦❞♦ ú♥✐❝♦ ❝♦♠♦ ✉♠ ♣r♦❞✉t♦ ❞❡ ♥ú♠❡r♦s ♣r✐♠♦s✱ ❛ ♠❡♥♦s ❞❛ ♦r❞❡♠ ❞♦s ❢❛t♦r❡s✳ ❉❡♠♦♥str❛çã♦✿ ❯s❛r❡♠♦s ♦ Pr✐♥❝í♣✐♦ ❞❛ ■♥❞✉çã♦✳

  ❙❡ n ✏ 2✱ ♦ r❡s✉❧t❛❞♦ é ❢❛❝✐❧♠❡♥t❡ ✈❡r✐✜❝❛❞♦✱ ♣♦✐s ✷ é ♣r✐♠♦✳ ❙✉♣♦♥❤❛ q✉❡ ❛ ❛✜r♠❛çã♦ s❡❥❛ ✈❡r❞❛❞❡✐r❛ ♣❛r❛ t♦❞♦ 2 ↕ n ➔ N✱ ♦✉ s❡❥❛✱ ♦✉ n é ♣r✐♠♦ i r i 2

  ♦✉ n ✏ p ☎ p ☎ ☎ ☎ p ✱ ♦♥❞❡ p é ♣r✐♠♦ ❡ i P t1, 2, ☎ ☎ ☎ , r✉✳ p, q P N t❛❧ q✉❡ N ✏ p ☎ q✱ ❝♦♠ 1 ➔ p ➔ N ❡ 1 ➔ q ➔ N✳ P♦rt❛♥t♦✱ ♣❡❧❛ ❤✐♣ót❡s❡ 1 , p 2 , r 1 , q 2 , s

  ❞❡ ✐♥❞✉çã♦ t❡♠ q✉❡ ❡①✐st❡♠ ♥ú♠❡r♦s ♣r✐♠♦s p ☎ ☎ ☎ , p ❡ q ☎ ☎ ☎ , q t❛✐s q✉❡ p 1 2 r 1 2 s ✏ p ☎ p ☎ ☎ ☎ p ❡ q ✏ q ☎ q ☎ ☎ ☎ q ✳ P♦rt❛♥t♦✱ 1 2 1 2 N r s ,

  ✏ p ☎ q ✏ p ☎ p ☎ ☎ ☎ p ☎ q ☎ q ☎ ☎ ☎ q ♦✉ s❡❥❛✱ N s❡ ❞❡❝♦♠♣♦♥❤❛ ❝♦♠♦ ♣r♦❞✉t♦ ❞❡ ♥ú♠❡r♦s ♣r✐♠♦s✳

  ▼♦str❛r❡♠♦s ❛❣♦r❛ ❛ ✉♥✐❝✐❞❛❞❡ ❞❛ ❞❡❝♦♠♣♦s✐çã♦ ♣♦r ✐♥❞✉çã♦✳ P❛r❛ n ✏ 2✱ é ✈❡r❞❛❞❡✳ ❙✉♣♦♥❤❛ ✈á❧✐❞❛ ♣❛r❛ t♦❞♦ m✱ ❝♦♠ 2 ↕ m ➔ n✳ n 1 2 r s 1 2

  ✏ p ☎ p ☎ ☎ ☎ p ✏ q ☎ q ☎ ☎ ☎ q ✭✷✳✹✮ i i , q , 1 1 1 2 s ♦♥❞❡ p ↕ i ↕ r ❡ 1 ↕ i ↕ s sã♦ ♥ú♠❡r♦s ♣r✐♠♦s✳ ❙❡❣✉❡ ❞❡ ✷✳✹ q✉❡ p ⑤ q ☎ q ☎ ☎ ☎ q ✳ 1 j

  ❉♦♥❞❡ ❝♦♥❝❧✉í♠♦s q✉❡ p ⑤ q ♣❛r❛ ❛❧❣✉♠ j P t1, 2, ☎ ☎ ☎ , s✉✳ ❙❡♠ ♣❡r❞❛ ❞❡ ❣❡♥❡r❛❧✐❞❛❞❡✱ 1 1 r❡♦r❞❡♥❛♥❞♦ s❡ ♥❡❝❡ssár✐♦✱ t❡♠✲s❡ q✉❡ p ✏ q ✳ P♦rt❛♥t♦ p 2 3 r 2 3 s .

  ✷✳✹ ◆ú♠❡r♦s Pr✐♠♦s ✷✾

  ❈♦♠♦ 2 3 2 3 p r s ☎ p ☎ ☎ ☎ p ✏ q ☎ q ☎ ☎ ☎ q

  é ♠❡♥♦r q✉❡ n✱ t❡♠✲s❡ ♣❡❧❛ ❤✐♣ót❡s❡ ❞❡ ✐♥❞✉çã♦✱ q✉❡ r ✏ s✳ ◆♦✈❛♠❡♥t❡✱ ✐♥✈♦❝❛♥❞♦ ♦ i j ❝♦r♦❧ár✐♦ ✷✳✹✳✶✱ ❝♦♥❝❧✉í♠♦s q✉❡ p ❡ q sã♦ ✐❣✉❛✐s ❛♦s ♣❛r❡s ❡✱ ♣♦rt❛♥t♦✱ r❡♦r❞❡♥❛♥❞♦ s❡ 2 2 , . r s

  ♥❡❝❡ssár✐♦✱ q✉❡ p ✏ q ☎ ☎ ☎ , p ✏ q ❈♦♥✈é♠ ♦❜s❡r✈❛r q✉❡ ❡stá✈❛♠♦s ✉s❛♥❞♦ ♦ Pr✐♥❝í♣✐♦ ❞❛ ■♥❞✉çã♦ ❋✐♥✐t❛ ❡♠ ✉♠ ❝♦♥t❡ú❞♦

  ♠❛t❡♠át✐❝♦ ❞❛ á❧❣❡❜r❛ ✭❞✐✈✐s✐❜✐❧✐❞❛❞❡✮✳ ■ss♦ ♥♦s ♠♦str❛ q✉❡ ♦s ❝♦♥t❡ú❞♦s ♠❛t❡♠át✐❝♦s ♥ã♦ sã♦ ❝♦♠♣❧❡t❛♠❡♥t❡ ✐♥❞❡♣❡♥❞❡♥t❡s✱ q✉❛s❡ s❡♠♣r❡ ❛s ❢❡rr❛♠❡♥t❛s ♠❛t❡♠át✐❝❛s ✉t✐❧✐✲ ③❛❞❛s ♣❡r♠❡✐❛♠ ♦s ✈ár✐♦s r❛♠♦s ❞❛ ▼❛t❡♠át✐❝❛✳ ❖ Pr✐♥❝í♣✐♦ ❞❛ ■♥❞✉çã♦ ❋✐♥✐t❛ é ✉♠❛ ❞❡ss❛s ❢❡rr❛♠❡♥t❛s✱ ✉♠ ✐♥str✉♠❡♥t♦ ❜❛st❛♥t❡ ❡✜❝❛③ ♥❛ ❞❡♠♦♥str❛çã♦ ❞❡ ✈ár✐❛s ❛✜r♠❛çõ❡s ♠❛t❡♠át✐❝❛s ❡♠ ❞✐✈❡rs❛s ár❡❛s✳ 1 r

  ❚❡♦r❡♠❛ ✷✳✹✳✷✳ ❉❛❞♦ ✉♠ ♥ú♠❡r♦ ✐♥t❡✐r♦ n ✘ 0, 1, ✁1✱ ❡①✐st❡♠ ♣r✐♠♦s p ➔ ☎ ☎ ☎ ➔ p ❡ α 1 , r

  ☎ ☎ ☎ , α P N✱ ✉♥✐✈♦❝❛♠❡♥t❡ ❞❡t❡r♠✐♥❛❞♦s✱ t❛✐s q✉❡ α α α 1 2 r n . 1 2 r 1 , ✏ ✟p ☎ p ☎ ☎ ☎ p r 1 α α 1 2 2 α r Pr♦♣♦s✐çã♦ ✷✳✹✳✷✳ ❙❡❥❛♠ p ☎ ☎ ☎ , p ♥ú♠❡r♦s ♣r✐♠♦s ❡ n ✏ ✟p ☎p ☎ ☎ ☎ p r ✉♠ ♥ú♠❡r♦

  ✶

  ♥❛t✉r❛❧ ❡s❝r✐t♦ ♥❡st❛ ❢♦r♠❛✳ ❙❡ n é ✉♠ ❞✐✈✐s♦r ♣♦s✐t✐✈♦ ❞❡ n✱ ❡♥tã♦ β β β r

  ✶ 1 2

  n , 1 2 i i ✏ p ☎ p ☎ ☎ ☎ p r ♦♥❞❡ 0 ↕ β ↕ α ✱ ♣❛r❛ i ✏ 1, ☎ ☎ ☎ , r✳ β

  ✶

  ❉❡♠♦♥str❛çã♦✿ ❙❡❥❛ n ✉♠ ❞✐✈✐s♦r ❞❡ n ❡ s❡❥❛ p ❛ ♣♦tê♥❝✐❛ ❞❡ ✉♠ ♣r✐♠♦ p q✉❡ ❢❛③ β β

  ✶

  ♣❛rt❡ ❞❛ ❞❡❝♦♠♣♦sçã♦ ❞❡ n ❡♠ ❢❛t♦r❡s ♣r✐♠♦s✳ ❈♦♠♦ p ⑤ n✱ s❡❣✉❡ q✉❡ p ❞✐✈✐❞❡ ❛❧❣✉♠ α i p i ✱ ♣❛r❛ ❛❧❣✉♠ i P t1, ☎ ☎ ☎ , r✉✳ ❈♦♥s❡q✉❡♥t❡♠❡♥t❡✱ 0 ↕ β ↕ α ✳ P♦r ♦✉tr♦ ❧❛❞♦✱ ♣♦r s❡r α j i , j i

  ♣r✐♠♦ ❝♦♠ ♦s ❞❡♠❛✐s p j ✘ i, j P t1, 2, ☎ ☎ ☎ , r✉✱ s❡❣✉❡✲s❡ q✉❡ p ✏ p ✳ 1 α α 1 2 2 α n 1 β β 1 2 i 2 β n ❚❡♦r❡♠❛ ✷✳✹✳✸✳ ❙❡❥❛♠ a ✏ ✟p ☎ p ☎ ☎ ☎ p n ❡ b ✏ ✟p ☎ p ☎ ☎ ☎ p n ✱ ❝♦♠ p ♣r✐♠♦✱

  1 ↕ i ↕ n✳

  γ , β , β i i i i i i ✏ mintα ✉ e δ ✏ maxtα ✉, i ✏ 1, ☎ ☎ ☎ , n, t❡♠✲s❡ q✉❡ γ γ δ δ 1 2 γ n δ n 1 2 mdc e mmc . 1 2 1 2

  ♣a, bq ✏ p ☎ p ☎ ☎ ☎ p n ♣a, bq ✏ p ☎ p ☎ ☎ ☎ p n

  ✷✳✺ ❆r✐t♠ét✐❝❛ ❞♦s ❘❡st♦s ❡ ❙✉❛s ❆♣❧✐❝❛çõ❡s 1 γ γ 1 2 n 2 n γ ✸✵ ❉❡♠♦♥str❛çã♦✿ ◆♦t❡ q✉❡ ♦ ♥ú♠❡r♦ p ☎ p ☎ ☎ ☎ p é ✉♠ ❞✐✈✐s♦r ❝♦♠✉♠ ❞❡ a ❡ b❀ ❧♦❣♦✱ ε ε 1 2 r ε γ γ 1 2 n γ c i i , β i 1 2 r 1 2 n

  ✏ ✟p ☎ p ☎ ☎ ☎ p ✱ ♦♥❞❡ ε ↕ mintα ✉ ❡✱ ♣♦rt❛♥t♦✱ c ⑤ p ☎ p ☎ ☎ ☎ p ✳ 1 δ δ 1 2 n 2 δ n ◆♦t❡ t❛♠❜é♠ q✉❡ ♦ ♥ú♠❡r♦ p ☎ p ☎ ☎ ☎ p é ✉♠ ♠ú❧t✐♣❧♦ ❝♦♠✉♠ ❞❡ a ❡ b✳ ❙❡❥❛ m ̟ ̟ 1 2 ̟ r 1 2 i i i r , β

  ✉♠ ♠ú❧t✐♣❧♦ ❝♦♠✉♠ ❞❡ a ❡ b❀ ❧♦❣♦✱ m ✏ ✟p ☎ p ☎ ☎ ☎ p ✱ ♦♥❞❡ ̟ ➙ maxtα ✉ ❡✱ δ δ 1 1 2 2 δ n ♣♦rt❛♥t♦✱ m ✏ p ☎ p ☎ ☎ ☎ p n ☎ q✳

  ✷✳✺ ❆r✐t♠ét✐❝❛ ❞♦s ❘❡st♦s ❡ ❙✉❛s ❆♣❧✐❝❛çõ❡s

  ◆❡st❛ s❡çã♦ ✐♥tr♦❞✉③✐r❡♠♦s ♦ ❝♦♥❝❡✐t♦ ❞❡ ❝♦♥❣r✉ê♥❝✐❛ ♠♦❞✉❧❛r✳ ❊ss❡ ❝♦♥❝❡✐t♦ é ✉♠❛ ❢❡rr❛♠❡♥t❛ ♠✉✐t♦ út✐❧ ♣❛r❛ tr❛❜❛❧❤❛r♠♦s ❝♦♠ ❞✐✈✐s✐❜✐❧✐❞❛❞❡✳ ❆❝r❡s❝❡♥t❛r❡♠♦s t❛♠❜é♠✱ ❛❧❣✉♥s ❝♦♥❝❡✐t♦s ❞❡ ❞✐✈✐s✐❜✐❧✐❞❛❞❡✳ ❉❡✜♥✐çã♦ ✷✳✺✳✶✳ ❙❡ a, b e m sã♦ ♥ú♠❡r♦s ✐♥t❡✐r♦s✱ ❝♦♠ m → 0✱ ❞✐③❡♠♦s q✉❡ a é ❝♦♥❣r✉✲ ❡♥t❡ ❛ b ♠ó❞✉❧♦ m✱ s❡ m ⑤ ♣a ✁ bq✳ ◗✉❛♥❞♦ ♦s ✐♥t❡✐r♦s a ❡ b sã♦ ❝♦♥❣r✉❡♥t❡s ♠ó❞✉❧♦ m✱ ❡s❝r❡✈❡♠♦s a

  ✑ b ♣mod mq. ◗✉❛♥❞♦ ❛ r❡❧❛çã♦ a ✑ b ♣mod mq ❢♦r ❢❛❧s❛✱ ❞✐r❡♠♦s q✉❡ a ❡ b ♥ã♦ sã♦ ❝♦♥❣r✉❡♥t❡s✱ ♦✉ q✉❡ sã♦ ✐♥❝♦♥❣r✉❡♥t❡s✱ ♠ó❞✉❧♦ m✳ ❊s❝r❡✈❡r❡♠♦s✱ ♥❡ss❡ ❝❛s♦✱ a ✙ b ♣mod mq✳ ❯♠ ❡①❡♠♣❧♦ s✐♠♣❧❡s q✉❡ ♣♦❞❡♠♦s ❝✐t❛r é q✉❡ 18 ✑ 3 ♣mod 5q✱ ♣♦✐s 5 ⑤ ♣18 ✁ 3q✳ ❆ss✐♠

  ❝♦♠♦✱ ♣♦❞❡♠♦s ❢❛❧❛r q✉❡ 30 ✙ 25 ♣mod 4q✱ ♣♦✐s 4 ∤ ♣30 ✁ 25q✳ ❈♦♠♦ ♦ r❡st♦ ❞❛ ❞✐✈✐sã♦ ❞❡ ✉♠ ♥ú♠❡r♦ ✐♥t❡✐r♦ q✉❛❧q✉❡r ♣♦r 1 é s❡♠♣r❡ ♥✉❧♦✱ t❡♠♦s q✉❡ a ✑ b ♣mod 1q✱ q✉❛✐sq✉❡r q✉❡ s❡❥❛♠ a ❡ b P Z✳ ❙❛❧✐❡♥t❛♠♦s q✉❡ ❛ ❝♦♥❣r✉ê♥❝✐❛ ♠ó❞✉❧♦

  ✶ ♥ã♦ ❛❝r❡s❝❡♥t❛ s✉❜st❛♥❝✐❛❧♠❡♥t❡ ♥❛❞❛ ❛ ❛r✐t♠ét✐❝❛✳ ❊ ♣♦r ❡st❛ r❛③ã♦✱ ✐r❡♠♦s ❝♦♥s✐❞❡r❛r s❡♠♣r❡ m → 1✳ ❉❡❝♦rr❡ ✐♠❡❞✐❛t❛♠❡♥t❡ ❞❛ ❞❡✜♥✐çã♦✱ q✉❡ ❛ ❝♦♥❣r✉ê♥❝✐❛ ♠ó❞✉❧♦ m✱ ❝♦♠ m ✜①♦✱ é ✉♠❛ r❡❧❛çã♦ ❞❡ ❡q✉✐✈❛❧ê♥❝✐❛✳ ▼♦str❛r❡♠♦s ❡ss❡ ❢❛t♦ ❛ s❡❣✉✐r✳

  Pr♦♣♦s✐çã♦ ✷✳✺✳✶✳ ❙❡❥❛ m P N✳ P❛r❛ t♦❞♦s a, b, c P Z✱ t❡♠✲s❡ q✉❡ ✭✐✮ a ✑ a ♣mod mq✱

  ✭✐✐✮ ❙❡ a ✑ b ♣mod mq✱ ❡♥tã♦ b ✑ a ♣mod mq✱ ✭✐✐✐✮ ❙❡ a ✑ b ♣mod mq ❡ b ✑ c ♣mod mq✱ ❡♥tã♦ a ✑ c ♣mod mq✳

  ❉❡♠♦♥str❛çã♦✿ ✭✐✮ ❇❛st❛ ♦❜s❡r✈❛r q✉❡ m ⑤ a ✁ a ✏ 0✳

  ✷✳✺ ❆r✐t♠ét✐❝❛ ❞♦s ❘❡st♦s ❡ ❙✉❛s ❆♣❧✐❝❛çõ❡s ✸✶

  ✭✐✐✮ ❙❡ m ⑤ a ✁ b✱ ❡♥tã♦ m ⑤ ✁♣a ✁ bq✳ ❉♦♥❞❡✱ s❡❣✉❡ q✉❡ m ⑤ b ✁ a✳ ✭✐✐✐✮ ❙❡ m ⑤ a ✁ b ❡ m ⑤ b ✁ c✱ ❡♥tã♦ m ⑤ ♣a ✁ bq ♣b ✁ cq✳ ❙❡❣✉❡✱ ❞❛í✱ q✉❡ m ⑤ a ✁ c✳

  P❛r❛ ✈❡r✐✜❝❛r s❡ ❞♦✐s ♥ú♠❡r♦s sã♦ ❝♦♥❣r✉❡♥t❡s ♠ó❞✉❧♦ m✱ ♥ã♦ é ♥❡❝❡ssár✐♦ ❡❢❡t✉❛r ❛ ❞✐✈✐sã♦ ❡✉❝❧✐❞✐❛♥❛ ❞❡ ❛♠❜♦s ♣♦r m ♣❛r❛ ❞❡♣♦✐s ❝♦♠♣❛r❛r ♦s s❡✉s r❡st♦s✳ ➱ s✉✜❝✐❡♥t❡✱ ❛♣❡♥❛s✱ ❛♣❧✐❝❛r ♦ s❡❣✉✐♥t❡ r❡s✉❧t❛❞♦✿ Pr♦♣♦s✐çã♦ ✷✳✺✳✷✳ ❙✉♣♦♥❤❛ q✉❡ a, b, m P Z✱ ❝♦♠ m → 1✳ ❚❡♠✲s❡ q✉❡ a ✑ b ♣mod mq s❡ ❡ s♦♠❡♥t❡ s❡✱ m ⑤ ♣b ✁ aq✳

  ✶ ✶ ✶

  r ❉❡♠♦♥str❛çã♦✿ ❙❡❥❛♠ a ✏ m ☎ q r✱ ❝♦♠ 0 ↕ r ➔ m ❡ b ✏ m ☎ q ✱ ❝♦♠ 0 ↕ r ➔ m✱ ❛s ❞✐✈✐sõ❡s ❡✉❝❧✐❞✐❛♥❛s ❞❡ a ❡ b ♣♦r m✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ▲♦❣♦✱

  ✶ ✶

  b ✁ a ✏ m ☎ ♣q ✁ qq ♣r ✁ rq.

  ✶

  P♦rt❛♥t♦✱ a ✑ b ♣mod mq s❡✱ ❡ s♦♠❡♥t❡ s❡✱ r ✏ r ✱ ♦ q✉❡✱ ❡♠ ✈✐st❛ ❞❛ ✐❣✉❛❧❞❛❞❡ ❛❝✐♠❛✱

  ✶

  é ❡q✉✐✈❛❧❡♥t❡ ❛ ❞✐③❡r q✉❡ m ⑤ ♣b ✁ aq✱ ❥á q✉❡ ⑤r ✁ r ⑤ ➔ m✳ ❖ q✉❡ t♦r♥❛ út✐❧ ❡ ♣♦❞❡r♦s❛ ❛ ♥♦çã♦ ❞❡ ❝♦♥❣r✉ê♥❝✐❛ é ♦ ❢❛t♦ ❞❡ s❡r ✉♠❛ r❡❧❛çã♦ ❞❡

  ❡q✉✐✈❛❧ê♥❝✐❛ ❝♦♠♣❛tí✈❡❧ ❝♦♠ ❛s ♦♣❡r❛çõ❡s ❞❡ ❛❞✐çã♦ ❡ ♠✉❧t✐♣❧✐❝❛çã♦ ♥♦s ✐♥t❡✐r♦s✱ ❝♦♥❢♦r♠❡ ✈❡r❡♠♦s ♥❛ ♣r♦♣♦s✐çã♦ ❛ s❡❣✉✐r✳ Pr♦♣♦s✐çã♦ ✷✳✺✳✸✳ ❙❡❥❛♠ a, b, c, d, m P Z✱ ❝♦♠ m → 1✳

  ✭✐✮ ❙❡ a ✑ b ♣mod mq ❡ c ✑ d ♣mod mq✱ ❡♥tã♦ a c ✑ b d ♣mod mq✳ ✭✐✐✮ ❙❡ a ✑ b ♣mod mq ❡ ✑ d ♣mod mq✱ ❡♥tã♦ a ☎ c ✑ b ☎ d ♣mod mq✳

  ❉❡♠♦♥str❛çã♦✿ ❙✉♣♦♥❤❛♠♦s q✉❡ a ✑ b ♣mod mq ❡ c ✑ d ♣mod mq✳ ▲♦❣♦✱ t❡♠♦s q✉❡ m ⑤ ♣b ✁ aq ❡ m ⑤ ♣d ✁ cq✳

  P❛r❛ ♠♦str❛r ♣iq✱ ❜❛st❛ ♦❜s❡r✈❛r q✉❡ m ⑤ ♣b✁aq ♣d✁cq ❡✱ ♣♦rt❛♥t♦✱ m ⑤ ♣b dq✁♣a cq✳ Pr♦✈❛♥❞♦✱ ❞❡ss❛ ❢♦r♠❛ ♣iq✳

  ◆❛ ♣r♦✈❛ ❞❡ ♣iiq✱ ❜❛st❛ ♦❜s❡r✈❛r q✉❡ b ☎ d ✁ a ☎ c ✏ d ☎ ♣b ✁ aq a ☎ ♣d ✁ cq ❡ ❝♦♥❝❧✉✐r q✉❡ m ⑤ b ☎ d ✁ a ☎ c✳

  ❈♦r♦❧ár✐♦ ✷✳✺✳✶✳ P❛r❛ t♦❞♦s n P N, a, b P Z✱ s❡ a ✑ b ♣mod mq✱ ❡♥tã♦ t❡♠✲s❡ q✉❡ n n a

  ✷✳✺ ❆r✐t♠ét✐❝❛ ❞♦s ❘❡st♦s ❡ ❙✉❛s ❆♣❧✐❝❛çõ❡s ✸✷

  ❉❡♠♦♥str❛çã♦✿ ❋❛r❡♠♦s ❛ ❞❡♠♦♥str❛çã♦ ❞❡ss❡ ❝♦r♦❧ár✐♦ ✉s❛♥❞♦ ❛ ✐♥❞✉çã♦ s♦❜r❡ n✳ ◆♦t❡ q✉❡ tr✐✈✐❛❧♠❡♥t❡ ❛ r❡❧❛çã♦ é ✈á❧✐❞❛ ♣❛r❛ n ✏ 1✱ ♣♦✐s a n n ✑ b ♣mod mq. ❙✉♣♦♥❤❛✱ ♣♦r ✐♥❞✉çã♦✱ q✉❡ a ✑ b ♣mod mq ♣❛r❛ ❛❧❣✉♠ n P N✳ ❉❡✈❡♠♦s ♠♦str❛r q✉❡ ❛ ❝♦♥❣r✉ê♥❝✐❛ é ✈á❧✐❞❛ ♣❛r❛ n 1✱ ♦✉ s❡❥❛✱ n 1 n 1 a

  ✑ b ♣mod mq. n n ❚❡♠♦s q✉❡ a ✑ b ♣mod mq ❡ ♣❡❧❛ ❤✐♣ót❡s❡ ❞❡ ✐♥❞✉çã♦ a ✑ b ♣mod mq✳ P♦rt❛♥t♦✱

  ♣❡❧❛ ♣r♦♣♦s✐çã♦ ✷✳✺✳✸✱ t❡♠♦s q✉❡ n n a n 1 n 1 ☎ a ✑ b ☎ b ♣mod mq. n n ❉❛í✱ a ✑ b ♣mod mq✳ P♦rt❛♥t♦✱ ♣♦r ✐♥❞✉çã♦✱ ❝♦♥❝❧✉í♠♦s q✉❡ a ✑ b ♣mod mq✱

  ♣❛r❛ t♦❞♦ n P N✳

  ✶

  ❈♦♠ ❛ ♥♦t❛çã♦ ❞❡ ❝♦♥❣r✉ê♥❝✐❛s✱ ♦ P❡q✉❡♥♦ ❚❡♦r❡♠❛ ❞❡ ❋❡r♠❛t ❡♥✉♥❝✐❛✲s❡ ❝♦♠♦ s❡ s❡❣✉❡✿ ❙❡ p é ♣r✐♠♦ ❡ a P Z✱ ❡♥tã♦ p a

  ✑ a ♣mod pq. ❆❧é♠ ❞✐ss♦✱ s❡ p ∤ a✱ ❡♥tã♦ p

  ✁1

  a ✑ 1 ♣mod pq. Pr♦♣♦s✐çã♦ ✷✳✺✳✹✳ ❙❡❥❛♠ a, b, c, m P Z✱ ❝♦♠ m → 1✳ ❊♥tã♦✱ a c

  ✑ b c ♣mod mq ô a ✑ b ♣mod mq. ❉❡♠♦♥str❛çã♦✿ ❙❡ a ✑ b ♣mod mq✱ s❡❣✉❡ ✐♠❡❞✐❛t❛♠❡♥t❡ ❞❛ ♣r♦♣♦s✐çã♦ ✷✳✺✳✸✱ q✉❡ a c

  ✑ b c ♣mod mq✱ ♣♦✐s c ✑ c ♣mod mq✳ ❘❡❝✐♣r♦❝❛♠❡♥t❡✱ s❡ a c ✑ b c ♣mod mq✱ ❡♥tã♦ m ⑤ ♣b cq ✁ ♣a cq✱ ♦ q✉❡ ✐♠♣❧✐❝❛ q✉❡ m ⑤ ♣b ✁ aq ❡✱ ❝♦♥s❡q✉❡♥t❡♠❡♥t❡✱ a ✑ b ♣mod mq✳ ❆ ♣r♦♣♦s✐çã♦ ❛❝✐♠❛ ♥♦s ❞✐③ q✉❡✱ ♣❛r❛ ❛s ❝♦♥❣r✉ê♥❝✐❛s✱ ✈❛❧❡ ♦ ❝❛♥❝❡❧❛♠❡♥t♦ ❝♦♠ r❡❧❛çã♦

  à ❛❞✐çã♦✳ ❡♥tr❡t❛♥t♦✱ ♥ã♦ ✈❛❧❡✱ ❡♠ ❣❡r❛❧✱ ♦ ❝❛♥❝❡❧❛♠❡♥t♦ ♣❛r❛ ❛ ♠✉❧t✐♣❧✐❝❛çã♦✱ ❝♦♠♦

  P✐❡rr❡ ❞❡ ❋❡r♠❛t✱ ♥❛s❝✐❞♦ ♥❛ ♣r✐♠❡✐r❛ ❞❡ ❞é❝❛❞❛ ❞♦ sé❝✉❧♦ XV II✱ ❢♦✐ ✉♠ ♠❛t❡♠át✐❝♦ ❡ ❝✐❡♥t✐st❛ ❢r❛♥❝ês✳

  ✷✳✺ ❆r✐t♠ét✐❝❛ ❞♦s ❘❡st♦s ❡ ❙✉❛s ❆♣❧✐❝❛çõ❡s ✸✸ s❡ ♣♦❞❡ ✈❡r✐✜❝❛r ♥♦ ❡①❡♠♣❧♦ ❛ s❡❣✉✐r✳ ❈♦♠♦ 6 ☎ 9 ✁ 6 ☎ 5 ✏ 24 ❡ 8 ⑤ 24✱ t❡♠♦s q✉❡

  6 ☎ 9 ✑ 6 ☎ 5 ♣mod 8q✱ ❡✱ ♥♦ ❡♥t❛♥t♦✱ 9 ∤ 5 ♣mod 8q✳

  ◆♦ ❡♥t❛♥t♦✱ ❞❛r❡♠♦s ❛ s❡❣✉✐r ✉♠ r❡s✉❧t❛❞♦ r❡❧❛❝✐♦♥❛❞♦ ❝♦♠ ♦ ❝❛♥❝❡❧❛♠❡♥t♦ ♠✉❧t✐♣❧✐✲ ❝❛t✐✈♦✳ Pr♦♣♦s✐çã♦ ✷✳✺✳✺✳ ❙❡❥❛♠ a, b, c, m P Z✱ ❝♦♠ m → 1✳ ❚❡♠♦s q✉❡

  ✂ ✡ m a mod .

  ☎ c ✑ b ☎ c ♣mod mq ô a ✑ b mdc ♣c, mq m c

  ❉❡♠♦♥str❛çã♦✿ t❡♠♦s✱ ♣❡❧♦ ❝♦r♦❧ár✐♦ ✷✳✸✳✷✱ q✉❡ ❡ sã♦ ❝♦♣r✐♠♦s ❡ mdc mdc ♣c, mq ♣c, mq

  ♣♦r ❤✐♣ót❡s❡ q✉❡ a ☎ c ✑ b ☎ c ♣mod mq.

  P❡❧❛ ♣r♦♣♦s✐çã♦ ✷✳✺✳✶✱ ♣♦❞❡♠♦s ♦❜s❡r✈❛r q✉❡ m ⑤ ♣b ✁ aq ☎ c, s❡♥❞♦ q✉❡ ♣❡❧❛ ❞❡✜♥✐çã♦ ✷✳✶✳✶✱ t❡♠♦s q✉❡ m

  ✏ ♣b ✁ aq ☎ c ☎ k, ♣❛r❛ ❛❧❣✉♠ k P Z✳

  ❉✐✈✐❞✐♥❞♦ ❛♠❜♦s ♦s ♠❡♠❜r♦s ❞❛ ✐❣✉❛❧❞❛❞❡ ♣♦r mdc♣c, mq✱ ♦❜t❡♠♦s m ♣b ✁ aq ☎ c

  ✏ ☎ k, mdc mdc ♣c, mq ♣c, mq

  ♦ q✉❡ ✐♠♣❧✐❝❛ ❞✐③❡r q✉❡ m ♣b ✁ aq ☎ c . ⑤ mdc mdc

  ♣c, mq ♣c, mq m c ❈♦♠♦ ❥á ❢♦✐ ♦❜s❡r✈❛❞♦ ♥♦ ✐♥í❝✐♦ ❞❛ ❞❡♠♦♥str❛çã♦ q✉❡ ❡ sã♦ mdc mdc

  ♣c, mq ♣c, mq ❝♦♣r✐♠♦s✱ ♣❡❧♦ ❝♦r♦❧ár✐♦ ✷✳✸✳✷✱ s❡❣✉❡ q✉❡ m

  ⑤ ♣b ✁ aq. mdc

  ♣c, mq ❉♦♥❞❡✱

  ✂ ✡ m a mod .

  ✑ b mdc ♣c, mq

  ✷✳✺ ❆r✐t♠ét✐❝❛ ❞♦s ❘❡st♦s ❡ ❙✉❛s ❆♣❧✐❝❛çõ❡s ✸✹

  ❘❡❝✐♣r♦❝❛♠❡♥t❡✱ s❡ ✂ ✡ m a mod ,

  ✑ b mdc ♣c, mq t❡♠♦s ♣❡❧❛ ❞❡✜♥✐çã♦ ✷✳✶✳✶ ❡ ♣r♦♣♦s✐çã♦ ✷✳✺✳✶ q✉❡ m

  ⑤ ♣b ✁ aq, mdc ♣c, mq

  ♦✉ s❡❥❛✱ m c ✏ ♣b ✁ aq ☎ ☎ k, mdc mdc

  ♣c, mq ♣c, mq m c ♣❛r❛ ❛❧❣✉♠ k P Z✱ ♣♦✐s ❡ ✱ ♣❡❧♦ ❝♦r♦❧ár✐♦ ✷✳✸✳✷✱ sã♦ ♣r✐♠♦s ❡♥tr❡ s✐✳ mdc mdc

  ♣c, mq ♣c, mq ▲♦❣♦✱ m

  ✏ ♣b ✁ aq ☎ c ☎ k, ❡✱ ♣♦rt❛♥t♦✱ m

  ⑤ ♣b ✁ aq ☎ c. ❉♦♥❞❡✱ s❡❣✉❡ q✉❡ a

  ☎ c ✑ b ♣mod mq ❙❡❣✉❡✱ ❛❣♦r❛✱ ❛❧❣✉♥s ❝r✐tér✐♦s ❞❡ ❞✐✈✐s✐❜✐❧✐❞❛❞❡✳ Pr✐♠❡✐r♦✱ ✈❡r❡♠♦s ♦ ❝r✐tér✐♦ ❞❡ ❞✐✈✐s✐✲

  ❜✐❧✐❞❛❞❡ ♣♦r 5 ♦✉ 10✳ Pr♦♣♦s✐çã♦ ✷✳✺✳✻✳ ❙❡❥❛ a ✉♠ ♥ú♠❡r♦ ✐♥t❡✐r♦✱ ❝✉❥❛ r❡♣r❡s❡♥t❛çã♦ ♥♦ s✐st❡♠❛ ❞❡❝✐♠❛❧ n r n 1 r

  ✁1

  é r ☎ ☎ ☎ r ✳ ❯♠❛ ❝♦♥❞✐çã♦ ♥❡❝❡ssár✐❛ ❡ s✉✜❝✐❡♥t❡ ♣❛r❛ q✉❡ a s❡❥❛ ❞✐✈✐sí✈❡❧ ♣♦r 5 ✭r❡s♣❡❝t✐✈❛♠❡♥t❡ ♣♦r 10✮ é q✉❡ r s❡❥❛ 0 ♦✉ 5 ✭r❡s♣❡❝t✐✈❛♠❡♥t❡ 0✮✳ n 1

  ❉❡♠♦♥str❛çã♦✿ ❙❡♥❞♦ a ✏ 10☎♣r ☎ ☎ ☎ r q r ✱ t❡♠♦s q✉❡ a é ❞✐✈✐sí✈❡❧ ♣♦r 5 s❡✱ ❡ s♦♠❡♥t❡ s❡✱ r é ❞✐✈✐sí✈❡❧ ♣♦r 5✱ ❡✱ ♣♦rt❛♥t♦✱ r ✏ 0 ♦✉ r ✏ 5✳ P♦r ♦✉tr♦ ❧❛❞♦✱ a é ❞✐✈✐sí✈❡❧ ♣♦r 10 s❡✱ ❡ só s❡✱ r é ❞✐✈✐sí✈❡❧ ♣♦r 10✱ ♦ q✉❡ s♦♠❡♥t❡ ♦❝♦rr❡ q✉❛♥❞♦ r ✏ 0✳

  ❆❣♦r❛✱ ✈❡r❡♠♦s ♦ ❝r✐tér✐♦ ❞❡ ❞✐✈✐s✐❜✐❧✐❞❛❞❡ ♣♦r 3 ♦✉ 9✳ Pr♦♣♦s✐çã♦ ✷✳✺✳✼✳ ❙❡❥❛ a ✉♠ ♥ú♠❡r♦ ✐♥t❡✐r♦✱ ❝✉❥❛ r❡♣r❡s❡♥t❛çã♦ ♥♦ s✐st❡♠❛ ❞❡❝✐♠❛❧ n n r 1 r é r ✁1 ☎ ☎ ☎ r ✳ ❯♠❛ ❝♦♥❞✐çã♦ ♥❡❝❡ssár✐❛ ❡ s✉✜❝✐❡♥t❡ ♣❛r❛ q✉❡ a s❡❥❛ ❞✐✈✐sí✈❡❧ ♣♦r 3 n 1 r ✭r❡s♣❡❝t✐✈❛♠❡♥t❡ ♣♦r 9✮ é q✉❡ r ☎ ☎ ☎ r s❡❥❛ ❞✐✈✐sí✈❡❧ ♣♦r 3 ✭r❡s♣❡❝t✐✈❛♠❡♥t❡ ♣♦r

  9 ✮✳

  ✷✳✺ ❆r✐t♠ét✐❝❛ ❞♦s ❘❡st♦s ❡ ❙✉❛s ❆♣❧✐❝❛çõ❡s ✸✺

  ✁1 ☎ 10 2 k ✁1

  ✁1

  ✁ a 1 ☎ ☎ ☎ ✁ a 2 k

  r 2 k ♣mod 11q. P♦rt❛♥t♦✱ a é ❞✐✈✐sí✈❡❧ ♣♦r 11 s❡✱ ❡ s♦♠❡♥t❡ s❡✱ ♦ ♥ú♠❡r♦ a

  ✁1

  ✑ r ✁ r 1 r 2 ✁ r 3 ☎ ☎ ☎ ✁ r 2 k

  ♣mod 11q ❙♦♠❛♥❞♦ ♠❡♠❜r♦ ❛ ♠❡♠❜r♦✱ ❛s ❝♦♥❣r✉ê♥❝✐❛s ❛❝✐♠❛✱ t❡♠♦s q✉❡ a

  r 2 k ☎ 10 2 k ✑ r 2 k

  ✁1 ♣mod 11q

  ✑ ✁r 2 k

  ♣mod 11q ✳✳✳ ✳✳✳ r 2 k

  ❉❡♠♦♥str❛çã♦✿ ❚❡♠♦s q✉❡ a ✁ ♣r n ☎ ☎ ☎ r 1 r q ✏ r n ☎ 10 n ☎ ☎ ☎ r 1 ☎ 10 r ✁ ♣r n ☎ ☎ ☎ r 1 r q

  ☎ 10 ✑ ✁r ♣mod 11q r 2 ☎ 10 2 ✑ r 2

  ✑ r ♣mod 11q

  ✑ ✁1 ♣mod 11q✱ ❝♦♠ k P N✳ ❙❡❥❛ a ✏ r n ☎ ☎ ☎ r 1 r ✉♠ ♥ú♠❡r♦ ❡s❝r✐t♦ ♥♦ s✐st❡♠❛ ❞❡❝✐♠❛❧✳ ❚❡♠♦s✱ ❡♥tã♦ q✉❡ r

  ✁1

  ✳ ❯♠❛ ❝♦♥❞✐çã♦ ♥❡❝❡ssár✐❛ ❡ s✉✜❝✐❡♥t❡ ♣❛r❛ q✉❡ a s❡❥❛ ❞✐✈✐sí✈❡❧ ♣♦r 11 é q✉❡ r ✁ r 1 r 2 ✁ r 3 ☎ ☎ ☎ r n s❡❥❛ ❞✐✈✐sí✈❡❧ ♣♦r 11✳ ❉❡♠♦♥str❛çã♦✿ ❯s❛♥❞♦ ❛ ♥♦t❛çã♦ ❞❡ ❝♦♥❣r✉ê♥❝✐❛✱ é ❢á❝✐❧ ♦❜s❡r✈❛r q✉❡ 10 ✑ ✁1 ♣mod 11q✱ ❡ t❛♠❜é♠ q✉❡ 10 2 k ✑ 1 ♣mod 11q ❡ 10 2 k

  ✁1 ☎ ☎ ☎ r 1 r

  Pr♦♣♦s✐çã♦ ✷✳✺✳✽✳ ❙❡❥❛ a ✉♠ ♥ú♠❡r♦ ✐♥t❡✐r♦✱ ❝✉❥❛ r❡♣r❡s❡♥t❛çã♦ ♥♦ s✐st❡♠❛ ❞❡❝✐♠❛❧ é r n r n

  ♣❛r❛ ❛❧❣✉♠ q P Z✳ ❉❡ ♦♥❞❡ s❡❣✉❡ ♦ r❡s✉❧t❛❞♦✳ ❆❣♦r❛✱ ✈❡❥❛♠♦s ♦ ❝r✐tér✐♦ ❞❡ ❞✐✈✐s✐❜✐❧✐❞❛❞❡ ♣♦r 11✳

  ❈♦♠♦ ♦ t❡r♠♦ à ❞✐r❡✐t❛ ♥❛ ✐❣✉❛❧❞❛❞❡ ❛❝✐♠❛ é ❞✐✈✐sí✈❡❧ ♣♦r 9✱ t❡♠♦s q✉❡ a ✏ ♣r n ☎ ☎ ☎ r 1 r q 9q,

  ✏ r n ☎ ♣10 n ✁ 1q ☎ ☎ ☎ r 1 ☎ ♣10 ✁ 1q.

  a 2 k é ❞✐✈✐sí✈❡❧ ♣♦r ✶✶✳

  ✷✳✺ ❆r✐t♠ét✐❝❛ ❞♦s ❘❡st♦s ❡ ❙✉❛s ❆♣❧✐❝❛çõ❡s ✸✻

Pr♦❜❧❡♠❛s

  ✭❛✮ ✻✸✼ ❡ ✸ ✽✽✼ ✭❜✮ ✻✹✽ ❡ ✶ ✷✶✽

  2 99

  ✭❜✮ 2 100 é ❞✐✈✐❞✐❞♦ ♣♦r 3❀

  ✾✳ ❊♥❝♦♥tr❡ ♦ r❡st♦ q✉❡ é ❞❡✐①❛❞♦ q✉❛♥❞♦ ✭❛✮ 2001 ☎ 2002 ☎ 2003 ☎ 2004 2005 2 é ❞✐✈✐❞✐❞♦ ♣♦r 7❀

  4 10 ☎ ☎ ☎ 100 10 é ❞✐✈✐sí✈❡❧ ♣♦r 101✳

  3 10

  2 10

  ✽✳ ▼♦str❡ q✉❡ 1 10

  5 99

  4 99

  3 99

  ✼✳ ✭❍✉♥❣r✐❛✮ Pr♦✈❡ q✉❡ ✺ ❞✐✈✐❞❡ 1 99

  ✭❝✮ ✺✺✶ ❡ ✽✼✹ ✭❞✮ ✼ ✸✷✺ ❡ ✽ ✹✽✺✳

  ✶✳ P❛r❛ ❝❛❞❛ ♣❛r ❞❡ ♥ú♠❡r♦s ✐♥t❡✐r♦s a ❡ b ❞❛❞♦s ❛❜❛✐①♦✱ ❛❝❤❡ mdc ♣a, bq ❡ ❞❡t❡r♠✐♥❡ ♥ú♠❡r♦s ✐♥t❡✐r♦s m ❡ n t❛✐s q✉❡ mdc ♣a, bq ✏ ma nb✳

  ✻✳ ✭❍✉♥❣r✐❛✮ P❛r❛ n P N✱ ♣r♦✈❡ q✉❡ ♦ ♥ú♠❡r♦ 8 n ✁ 3 n ✁ 6 n

  9 ✳

  ✺✳ ▼♦str❡ q✉❡ ♦ ú❧t✐♠♦ ❛❧❣❛r✐s♠♦ ❞❡ ✉♠ q✉❛❞r❛❞♦ ♣❡r❢❡✐t♦ só ♣♦❞❡ s❡r 0✱ 1✱ 4✱ 5✱ 6 ♦✉

  ✹✳ ❉❛❞♦s ✐♥t❡✐r♦s a 1 ✱ a 2 ❡ b✱ s❡♥❞♦ b ♥ã♦ ♥✉❧♦✱ ♠♦str❡ q✉❡ b ⑤ ♣a 1 ✁ a 2 q s❡✱ ❡ só s❡✱ a 1 ❡ a 2 ❞❡✐①❛♠ r❡st♦s ✐❣✉❛✐s ♥❛ ❞✐✈✐sã♦ ♣♦r b✳

  ✸✳ ▼♦str❡ q✉❡ t♦❞♦ q✉❛❞r❛❞♦ ♣❡r❢❡✐t♦✿ ✭❛✮ ❞❡✐①❛ r❡st♦ 0 ♦✉ 1 q✉❛♥❞♦ ❞✐✈✐❞✐❞♦ ♣♦r 3❀ ✭❝✮ ❞❡✐①❛ r❡st♦ 0✱ 1 ♦✉ 4 q✉❛♥❞♦ ❞✐✈✐❞✐❞♦ ♣♦r 8✳

  ✭❝✮ mdc ♣2n 1, 9n 4q ✏ 1❀ ✭❞✮ mdc ♣n! 1, ♣n 1q! 1q ✏ 1✳

  ✭❜✮ mdc ♣n 1, n 2 n 1 q ✏ 1❀

  ✷✳ ❙❡❥❛ n P N✳ ▼♦str❡ q✉❡ ✭❛✮ mdc ♣n, 2n 1q ✏ 1❀

  1 n é ✉♠ ♠ú❧t✐♣❧♦ ❞❡ ✶✵✳

  ✷✳✺ ❆r✐t♠ét✐❝❛ ❞♦s ❘❡st♦s ❡ ❙✉❛s ❆♣❧✐❝❛çõ❡s 2468 ✸✼

  ✶✵✳ Pr♦✈❡ q✉❡ ♣999994q ✁ 1 é ❞✐✈✐sí✈❡❧ ♣♦r 333331✳ ✶✶✳ ❊♥❝♦♥tr❡ ♦ ú❧t✐♠♦ ❞í❣✐t♦ ❞♦s ♥ú♠❡r♦s 2015

  ✭❛✮ 1989 777 50

  2 ✭❜✮ 777 2 2 2

  3 ✭❝✮ 1 2 ☎ ☎ ☎ 2015 2014

  ✭❞✮ 2014 2015 ✭❡✮ 2015

  ✶✷✳ ❉❛❞♦s ♦✐t♦ ♥ú♠❡r♦s ✐♥t❡✐r♦s✱ ♠♦str❡ q✉❡ ❡①✐st❡♠ ❞♦✐s ❞❡❧❡s ❝✉❥❛ ❞✐❢❡r❡♥ç❛ é ❞✐✈✐sí✈❡❧ ♣♦r s❡t❡✳

  ✶✸✳ ❊♥❝♦♥tr❡ ♦ r❡st♦ ❞❛ ❞✐✈✐sã♦ ❞❡ 3 ✭❛✮ 1989 ☎ 1990 ☎ 1991 1992 ♣♦r 7❀ 100

  ✭❜✮ 9 ♣♦r 8❀ 2 2 2015

  ✭❝✮ 2014 ♣♦r 7✳ 5 4n

  ✶✹✳ Pr♦✈❡ q✉❡ n é ❞✐✈✐sí✈❡❧ ♣♦r 5 q✉❛❧q✉❡r q✉❡ s❡❥❛ ♦ ✐♥t❡✐r♦ n✳ 3 ✶✺✳ Pr♦✈❡ q✉❡ n ✁n é ❞✐✈✐sí✈❡❧ ♣♦r 24 q✉❛❧q✉❡r q✉❡ s❡❥❛ ♦ ✐♥t❡✐r♦ í♠♣❛r n✳ ❉✐❝❛✳ Pr♦✈❡ 3 q✉❡ ♦ ♥ú♠❡r♦ ❞❛❞♦ é ♠ú❧t✐♣❧♦ t❛♥t♦ ❞❡ 2 q✉❛♥t♦ ❞❡ 3✳ 2015

  ✶✻✳ ❊♥❝♦♥tr❡ ♦ r❡st♦ ❞❛ ❞✐✈✐sã♦ ❞♦ ♥ú♠❡r♦ 3 ♣♦r 7✳ 5555 2222 5555

  ✶✼✳ Pr♦✈❡ q✉❡ 2222 é ❞✐✈✐sí✈❡❧ ♣♦r 7✳ ✶✽✳ ❉❡t❡r♠✐♥❡ ♦ ✈❛❧♦r ❞❡✿ 2 2

  ✭❛✮ 25 ✁ 24 ❀ 2 2 ✭❜✮ 1975 ✁ 1974 ❀ 2 2

  ✭❝✮ 555555 ✁ 444444 ❀ 2 2 ✭❞✮ ♣n 1q ✁ n ✳ 3 2

  ✶✾✳ ❙❡❥❛ n ✏ 9867✳ ❙❡ ✈♦❝ê ❝❛❧❝✉❧❛ss❡ n ✁ n ✱ q✉❛❧ ♥ú♠❡r♦ ❡♥❝♦♥tr❛r✐❛ ♥♦ ❛❧❣❛r✐s♠♦ ❞❛s ✉♥✐❞❛❞❡s❄

  ✷✳✺ ❆r✐t♠ét✐❝❛ ❞♦s ❘❡st♦s ❡ ❙✉❛s ❆♣❧✐❝❛çõ❡s ✸✽

  ✷✵✳ ❊♥❝♦♥tr❡ ♦ ♣r♦❞✉t♦ ✂ ✡ ✂ ✡ ✂ ✡ ✂ ✡

  1

  1

  1

  1

  1

  1

  1 1 . ✁ ☎ ✁ ☎ ✁ ☎ ☎ ☎ ✁

  4

  9 2015 16 225

  1 ✷✶✳ ◗✉❛♥t♦s ③❡r♦s ❡①✐st❡♠ ♥♦ ✜♥❛❧ ❞♦ ♥ú♠❡r♦ 9 ❄ ✷✷✳ ❊①✐st❡ ✉♠ ♥ú♠❡r♦ ❞❡ 8 ❛❧❣❛r✐s♠♦s ❞❛ ❢♦r♠❛

  9999 ✝ ✝ ✝ ✝ q✉❡ é ✉♠ q✉❛❞r❛❞♦ ♣❡r❢❡✐t♦❄

  ✷✸✳ ◗✉❛❧ ♦ ♠❡♥♦r ♥ú♠❡r♦ ✐♥t❡✐r♦ ♣♦s✐t✐✈♦ n t❛❧ q✉❡ ❄

  ❄ n n ✁ ✁ 1 ➔ 0, 01?

  ✷✹✳ ▼♦str❡ q✉❡ s❡ ♦ ♣r♦❞✉t♦ N ✏ ♣n 6mq ☎ ♣2n 5mq ☎ ♣3n 4mq é ♠ú❧t✐♣❧♦ ❞❡ 7✱ ❝♦♠ 3 m ❡ n ♥ú♠❡r♦s ♥❛t✉r❛✐s✱ ❡♥tã♦ N é ♠ú❧t✐♣❧♦ ❞❡ 7 ✏ 343✳

  ✷✺✳ ❙❡❥❛ a ✉♠ ♥ú♠❡r♦ ✐♥t❡✐r♦ ♣♦s✐t✐✈♦ t❛❧ q✉❡ a é ♠ú❧t✐♣❧♦ ❞❡ ✺✱ a 1 é ♠ú❧t✐♣❧♦ ❞❡ ✼✱ a

  2 é ♠ú❧t✐♣❧♦ ❞❡ ✾ ❡ a 3 é ♠ú❧✐t♣❧♦ ❞❡ ✶✶✳ ❉❡t❡r♠✐♥❡ ♦ ♠❡♥♦r ✈❛❧♦r q✉❡ a ♣♦❞❡

  ❛ss✉♠✐r✳ ✷✻✳ ❉❡t❡r♠✐♥❡ t♦❞♦s ♦s ♥ú♠❡r♦s ♥❛t✉r❛✐s m ❡ n t❛✐s q✉❡ 8 11 m 2

  2

  2 2 . 444 333 ✏ n

  4 ✷✼✳ ❖ ♥ú♠❡r♦ 3 é ❞✐✈✐sí✈❡❧ ♣♦r ✺❄ 4

  ✷✽✳ ◗✉❛✐s ♦s ❝✉❜♦s ♣❡r❢❡✐t♦s q✉❡ ❞✐✈✐❞❡♠ 9 ❄ 3 3 3 b c ✷✾✳ ✭❊◆◗ ✷✵✶✺✳✶✮ ❙❡❥❛♠ a✱ b ❡ c ✐♥t❡✐r♦s t❛✐s q✉❡ a é ❞✐✈✐sí✈❡❧ ♣♦r ✾✳ ▼♦str❡ q✉❡ ♣❡❧♦ ♠❡♥♦s ✉♠ ❞♦s ✐♥t❡✐r♦s a✱ b ♦✉ c é ❞✐✈✐sí✈❡❧ ♣♦r ✸✳ ✸✵✳ ✭❊◆◗ ✷✵✶✹✳✶✮ ❖ ♠á①✐♠♦ ❞✐✈✐s♦r ❝♦♠✉♠ ❞❡ ❞♦✐s ✐♥t❡✐r♦s ♣♦s✐t✐✈♦s é ✷✵✳ P❛r❛ s❡

  ❝❤❡❣❛r ❛ ❡ss❡ r❡s✉❧t❛❞♦ ♣❡❧♦ ♣r♦❝❡ss♦ ❞❛s ❞✐✈✐sõ❡s s✉❝❡ss✐✈❛s✱ ♦s q✉♦❝✐❡♥t❡s ❡♥❝♦♥✲ tr❛❞♦s ❢♦r❛♠✱ ♣❡❧❛ ♦r❞❡♠✱ ✶✱ ✺✱ ✸✱ ✸✱ ✶ ❡ ✸✳ ❊♥❝♦♥tr❡ ♦s ❞♦✐s ♥ú♠❡r♦s✳

  ❈❛♣ít✉❧♦

  3

❆❧❣✉♥s ❚ó♣✐❝♦s ❞❡ ●❡♦♠❡tr✐❛

❊✉❝❧✐❞✐❛♥❛

  ✸✳✶ ❈♦♥❣r✉ê♥❝✐❛ ❞❡ ❚r✐â♥❣✉❧♦s

  ❱❡r❡♠♦s ❛❧❣✉♥s ❝❛s♦s ♦♥❞❡ s❡rã♦ ❛♥❛❧✐s❛❞♦s ❛s ❝♦♥❞✐çõ❡s ♥❡❝❡ssár✐❛s ❡ s✉✜❝✐❡♥t❡s ♣❛r❛ q✉❡ t❡♥❤❛♠♦s ❞♦✐s tr✐â♥❣✉❧♦s s❡♥❞♦ ❝♦♥s✐❞❡r❛❞♦s ✧✐❣✉❛✐s✧✳ ❱❡r❡♠♦s✱ t❛♠❜é♠✱ ❛❧❣✉♠❛s ❝♦♥s❡q✉ê♥❝✐❛s ✐♥t❡r❡ss❛♥t❡s ❞❡ss❡s ❝♦♥❥✉♥t♦s ❞❡ ❝♦♥❞✐çõ❡s✱ ❝♦♠♦ ❛ ❞❡s✐❣✉❛❧❞❛❞❡ tr✐❛♥❣✉❧❛r ❡ ♦s ✈ár✐♦s t✐♣♦s ❞❡ q✉❛❞r✐❧át❡r♦s ♥♦tá✈❡✐s ❬✼❪✳

  ❆ s❡❣✉✐r✱ ❞❡✜♥✐r❡♠♦s â♥❣✉❧♦✳ ■r❡♠♦s ✉t✐❧✐③❛r ❡ss❛ ❞❡✜♥✐çã♦ ❡♠ ❛❧❣✉♠❛s s✐t✉❛çõ❡s✳ OA OB

  ❉❡✜♥✐çã♦ ✸✳✶✳✶✳ ❉❛❞❛s✱ ♥♦ ♣❧❛♥♦✱ ❞✉❛s s❡♠✐rr❡t❛s ÝÝÑ ❡ ÝÝÑ ✱ ✉♠ â♥❣✉❧♦ ❞❡ ✈ért✐❝❡ O OA OB

  OA OB ❡ ❧❛❞♦s ÝÝÑ ❡ ÝÝÑ é ✉♠❛ ❞❛s ❞✉❛s r❡❣✐õ❡s ❞♦ ♣❧❛♥♦ ❧✐♠✐t❛❞❛s ♣❡❧❛s s❡♠✐rr❡t❛s ÝÝÑ ❡ ÝÝÑ ✳

  B B O O A A ❋✐❣✉r❛ ✸✳✶✿ â♥❣✉❧♦s ♥♦ ♣❧❛♥♦✳

  ❯♠ â♥❣✉❧♦ ♣♦❞❡ s❡r ❝♦♥✈❡①♦ ♦✉ ♥ã♦ ❝♦♥✈❡①♦✳ ◆❛ ✜❣✉r❛ ✸✳✶✱ ♦ â♥❣✉❧♦ ❞❛ ❡sq✉❡r❞❛ é OA

  ✸✳✶ ❈♦♥❣r✉ê♥❝✐❛ ❞❡ ❚r✐â♥❣✉❧♦s ✹✵

  OB AOB ❡ ÝÝÑ ❡s❝r❡✈❡♥❞♦ ③ ✳ ❖ ❝♦♥t❡①t♦ ❞❡✐①❛rá ❝❧❛r♦ s❡ ❡st❛♠♦s ♥♦s r❡❢❡r✐♥❞♦ ❛♦ â♥❣✉❧♦ ❝♦♥✈❡①♦ ♦✉ ❛♦ ♥ã♦ ❝♦♥✈❡①♦✳ ❉❡✜♥✐çã♦ ✸✳✶✳✷✳ ❉✐r❡♠♦s q✉❡ ❞♦✐s s❡❣♠❡♥t♦s ❆❇ ❡ ❈❉ sã♦ ❝♦♥❣r✉❡♥t❡s q✉❛♥❞♦ ❡ss❡s

  A s❡❣♠❡♥t♦s ♣♦ss✉❡♠ ❛ ♠❡s♠❛ ♠❡❞✐❞❛✱ ♦✉ s❡❥❛✱ AB ✏ CD✳ ❉✐r❡♠♦s q✉❡ ❞♦✐s â♥❣✉❧♦s ♣ ❡ ♣

  B sã♦ ❝♦♥❣r✉❡♥t❡s s❡ ❡❧❡s tê♠ ❛ ♠❡s♠❛ ♠❡❞✐❞❛✳ ❖❜s❡r✈❡ q✉❡✱ ❝♦♠ ❡st❛ ❞❡✜♥✐çã♦✱ ❛s ♣r♦♣r✐❡❞❛❞❡s ❞❛ ✐❣✉❛❧❞❛❞❡ ❞❡ ♥ú♠❡r♦s ♣❛ss❛ ❛ ✈❛❧❡r

  ♣❛r❛ ❛ ❝♦♥❣r✉ê♥❝✐❛ ❞❡ s❡❣♠❡♥t♦s ❡ ❞❡ â♥❣✉❧♦s✳ ❈♦♠♦ ❝♦♥s❡q✉ê♥❝✐❛✱ ✉♠ s❡❣♠❡♥t♦ é s❡♠♣r❡ ❝♦♥❣r✉❡♥t❡ ❛ ❡❧❡ ♠❡s♠♦ ❡ ❞♦✐s s❡❣♠❡♥t♦s✱ ❝♦♥❣r✉❡♥t❡s ❛ ✉♠ t❡r❝❡✐r♦✱ sã♦ ❝♦♥❣r✉❡♥t❡s ❡♥tr❡ s✐✳ ❖ ♠❡s♠♦ ✈❛❧❡♥❞♦ ♣❛r❛ â♥❣✉❧♦s✳

  ❯s❛r❡♠♦s ♦ sí♠❜♦❧♦ ❞❡ ✐❣✉❛❧❞❛❞❡ ♣❛r❛ r❡♣r❡s❡♥t❛r ❛ ❝♦♥❣r✉ê♥❝✐❛ ❡♥tr❡ ❞♦✐s ♦❜❥❡t♦s✳ A B

  ❆ss✐♠✱ AB ✏ CD ❞❡✈❡ s❡r ❧✐❞♦ ❝♦♠♦ AB é ❝♦♥❣r✉❡♥t❡ ❛ CD ❡ ♣ ✏ ♣ ❞❡✈❡ s❡r ❧✐❞♦ ❝♦♠♦ â♥❣✉❧♦ ❆ é ❝♦♥❣r✉❡♥t❡ ❛♦ â♥❣✉❧♦ ❇✳ ❉❡✜♥✐çã♦ ✸✳✶✳✸✳ ❉♦✐s tr✐â♥❣✉❧♦s sã♦ ❝♦♥❣r✉❡♥t❡s s❡ ❢♦r ♣♦ssí✈❡❧ ❡st❛❜❡❧❡❝❡r ✉♠❛ ❝♦r✲ r❡s♣♦♥❞ê♥❝✐❛ ❜✐✉♥í✈♦❝❛ ❡♥tr❡ s❡✉s ✈ért✐❝❡s✱ ❞❡ ♠♦❞♦ q✉❡ ❧❛❞♦s ❡ â♥❣✉❧♦s ❝♦rr❡s♣♦♥❞❡♥t❡s s❡❥❛♠ ❝♦♥❣r✉❡♥t❡s✳

  ❙✉♣♦♥❤❛ q✉❡ ♦s tr✐â♥❣✉❧♦s ABC ❡ DEF ❞❛ ✜❣✉r❛ ❛❜❛✐①♦ s❡❥❛♠ ❝♦♥❣r✉❡♥t❡s✳ A

  F E B C D ❙✉♣♦♥❤❛ q✉❡ ❛ ❝♦rr❡s♣♦♥❞ê♥❝✐❛ ❛❜❛✐①♦

  A Ø D

  B Ø E

  C Ø F

  ❞❡✜♥❡ ❛ ❝♦♥❣r✉ê♥❝✐❛ ❡♥tr❡ ♦s tr✐â♥❣✉❧♦s ABC ❡ DEF ✳ ❊♥tã♦ ✈❛❧❡♠✱ s✐♠✉❧tâ♥❡❛♠❡♥t❡✱ ❛s s❡❣✉✐♥t❡s r❡❧❛çõ❡s✿

  ✸✳✶ ❈♦♥❣r✉ê♥❝✐❛ ❞❡ ❚r✐â♥❣✉❧♦s ✹✶

  AB BC AC ✏ DE; ✏ EF ; ✏ DF ;

  ♣ ♣ ♣ ♣ ♣ ♣ C F ; A D; B E.

  ✏ ✏ ✏ ◆❡st❛ s❡çã♦✱ ❡♥✉♥❝✐❛r❡♠♦s ♦s ♣r✐♥❝✐♣❛✐s r❡s✉❧t❛❞♦s ❞❡ ❝♦♥❣r✉ê♥❝✐❛ ❞❡ tr✐â♥❣✉❧♦s ♥❡✲

  ❝❡ssár✐♦s ♣❛r❛ ❛t❛❝❛r ❞✐✈❡rs♦s ♣r♦❜❧❡♠❛s r❡❧❛❝✐♦♥❛❞♦s ❛ ❣❡♦♠❡tr✐❛ ❡✉❝❧✐❞✐❛♥❛✳ ❆①✐♦♠❛ ✸✳✶✳✶✳ ❙❡ ❞♦✐s ❧❛❞♦s ❞❡ ✉♠ tr✐â♥❣✉❧♦ ❡ ♦ â♥❣✉❧♦ ❢♦r♠❛❞♦ ♣♦r ❡ss❡s ❞♦✐s ❧❛❞♦s ❢♦r❡♠ r❡s♣❡❝t✐✈❛♠❡♥t❡ ✐❣✉❛✐s ❛ ❞♦✐s ❧❛❞♦s ❞❡ ♦✉tr♦ tr✐â♥❣✉❧♦ ❡ ❛♦ â♥❣✉❧♦ ❢♦r♠❛❞♦ ♣♦r ❡ss❡s ❞♦✐s ❧❛❞♦s✱ ❡♥tã♦ ♦s ❞♦✐s tr✐â♥❣✉❧♦s sã♦ ❝♦♥❣r✉❡♥t❡s✳

  ❊st❡ ❛①✐♦♠❛ é ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ ♦ ❝❛s♦ ❞❡ ❝♦♥❣r✉ê♥❝✐❛ ❧❛❞♦✱ â♥❣✉❧♦✱ ❧❛❞♦ ✭▲❆▲✮✳ ❊❧❡ é ❞❡ ❢á❝✐❧ ❝♦♠♣r❡❡♥sã♦ ❡ s✉❛ ❛❝❡✐t❛çã♦ s❡ ❞❛r ♣♦r ❛r❣✉♠❡♥t♦s ♣✉r❛♠❡♥t❡ ❣❡♦♠étr✐❝♦s✳ ❙❡✱ ❞❡ ❛❧❣✉♠❛ ❢♦r♠❛✱ ♠♦✈❡r♠♦s ❞❡ ❢♦r♠❛ rí❣✐❞❛ ❛ ✜❣✉r❛ ❞♦ tr✐â♥❣✉❧♦ ABC ❝♦♥s❡❣✉✐r❡♠♦s s♦❜r❡♣♦r ❛ ✜❣✉r❛ s♦❜r❡ ♦ ♦✉tr♦ tr✐â♥❣✉❧♦✳

  ✶

  C A

  ✶

  A B C

  ✶

  B ❙✐♠❜♦❧✐❝❛♠❡♥t❡✱ tr❛❞✉③✐r❡♠♦s ♦ ❝❛s♦ ❞❡ ❝♦♥❣r✉ê♥❝✐❛ ❡♠ q✉❡stã♦ ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✿

  ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶

  B C B C A A ❉❛❞♦s ❞♦✐s tr✐â♥❣✉❧♦s ABC ❡ A s❡ AB ✏ A ✱ AC ✏ A ❡ ♣ ✏ ♣ ✱ ❞✐③❡♠♦s q✉❡ ♦s tr✐â♥❣✉❧♦s ❡♠ q✉❡stã♦ sã♦ ❝♦♥❣r✉❡♥t❡s✳

  ❆ ♣❛rt✐r ❞❡ss❡ ❛①✐♦♠❛✱ ♣r♦✈❛r❡♠♦s ♦✉tr♦s ❝❛s♦s ❞❡ ❝♦♥❣r✉ê♥❝✐❛ ❞❡ tr✐â♥❣✉❧♦s✳ ❖s ❞♦✐s ♣ró①✐♠♦s r❡s✉❧t❛❞♦s tr❛t❛rã♦ ❞♦s ❝❛s♦s â♥❣✉❧♦✱ ❧❛❞♦✱ â♥❣✉❧♦ ✭❆▲❆✮ ❡ ❧❛❞♦✱ ❧❛❞♦✱ ❧❛❞♦ ✭▲▲▲✮✳ ❚❡♦r❡♠❛ ✸✳✶✳✶✳ ✭❈❛s♦ ❆▲❆✮ ❉❛❞♦s ❞♦✐s tr✐â♥❣✉❧♦s q✉❛✐sq✉❡r ABC ❡ DEF ✱ s❡ t❡♠♦s

  ♣ A

  D, AB B E ✏ ♣ ✏ DE ❡ ♣ ✏ ♣ ✱ ❡♥tã♦ ♦s tr✐â♥❣✉❧♦s sã♦ ❝♦♥❣r✉❡♥t❡s✳

  ❉❡♠♦♥str❛çã♦✿ ❈♦♥s✐❞❡r❡♠♦s ♦s tr✐â♥❣✉❧♦s ABC ❡ DEF ✱ s❛t✐s❢❛③❡♥❞♦ ❛s ❤✐♣ót❡s❡s ❞♦ A

  D, AB B E t❡♦r❡♠❛✱ ♦✉ s❡❥❛✱ ♣ ✏ ♣ ✏ DE ❡ ♣ ✏ ♣ ✳ ❆ té❝♥✐❝❛ ❡♠♣r❡❣❛❞❛ ❛q✉✐ é ✐❞ê♥t✐❝❛ ❛ ✉s❛❞❛ ♥❛ ♣r♦♣♦s✐çã♦ ❛♥t❡r✐♦r✳ ■r❡♠♦s ♣r♦✈❛r q✉❡ ❡ss❡s tr✐â♥❣✉❧♦s ❛❞♠✐t❡♠ ♦✉tr♦ ❧❛❞♦ ✐❞ê♥t✐❝♦✳ ❈♦♠♣❛r❛♥❞♦ AC ❝♦♠ DF ✱ t❡r❡♠♦s

  ✸✳✶ ❈♦♥❣r✉ê♥❝✐❛ ❞❡ ❚r✐â♥❣✉❧♦s ✹✷

  ❉♦♥❞❡ ❝♦♥❝❧✉í♠♦s q✉❡ ④ F

  ✏ ④ DEF

  ✶

  ✳ P♦r ♦✉tr♦ ❧❛❞♦✱ ♣

  E ✏ ④

  DEF

  

  ④ F

  ✶ EF .

  ✶

  ✏ ♣ C

  EF ✏ 0 ❡✱ ♣♦rt❛♥t♦✱ ✉♠❛ ❝♦♥tr❛❞✐çã♦✱ ♣♦✐s F

  ✶

  EF → 0✳ ▲♦❣♦✱

  ♥ã♦ ♣♦❞❡ ♦❝♦rr❡r ✭✐✐✮✳ ❙❡ ♦❝♦rr❡r ✭✐✐✐✮✱ t♦♠❡ F

  ✶

  ♥❛ s❡♠✐rr❡t❛ ÝÝÑ DF t❛❧ q✉❡ DF

  ✶

  ✏ AC✳ A B C D E F F

  ❡ ♣ B

  ✶

  A B C D E F ✭✐✮ AC ✏ DF ❀

  ✳ ❱❡❥❛ ✜❣✉r❛ ❛❜❛✐①♦✳ A B C D E F F

  ✭✐✐✮ AC ➔ DF ❀ ✭✐✐✐✮ AC → DF ✳

  ◆♦t❡ q✉❡✱ s❡ ♦❝♦rr❡ ✭✐✮✱ ♥❛❞❛ ♠❛✐s t❡♠♦s ❛ ❞❡♠♦♥str❛r✳ ❖ r❡s✉❧t❛❞♦ s❡❣✉❡✱ ❡♥tã♦✱ ❞♦ ❛①✐♦♠❛ ✸✳✶✳✶✱ ♦✉ s❡❥❛✱

  ∆ABC ✏ ∆DEF. ❙❡ ♦❝♦rr❡ ✭✐✐✮✱ t♦♠❡ ✉♠ ♣♦♥t♦ F

  ✶

  ♥❛ s❡♠✐rr❡t❛ ÝÝÑ DF t❛❧ q✉❡ DF

  ✶

  ✏ AC ❡ ❝♦♥s✐❞❡r❡ ♦ tr✐â♥❣✉❧♦ DEF

  ✶

  ✶

  F

  P♦rt❛♥t♦✱ ♣❡❧♦ ❛①✐♦♠❛ ✸✳✶✳✶✱ ♦❜t❡♠♦s q✉❡✿ ∆ABC

  ✏ ∆DEF

  ✶

  , ✉♠❛ ✈❡③ q✉❡ AB ✏ DE✱ ♣

  A ✏ ♣

  D ❡ AC ✏ DF

  

  ✳ ❉❡ss❛ ❢♦r♠❛✱ ①

  ✶

  ✸✳✶ ❈♦♥❣r✉ê♥❝✐❛ ❞❡ ❚r✐â♥❣✉❧♦s ✹✸

  ✶

  A D ❈♦♠♦ AB ✏ DE✱ ♣ ✏ ♣ ❡ AC ✏ DF ✱ ♣♦r ❝♦♥str✉çã♦✱ s❡❣✉❡✲s❡ ♣❡❧♦ ❛①✐♦♠❛ ✸✳✶✳✶✱ q✉❡✿

  ✶ ∆ABC .

  ✏ ∆DEF

  ✶ ✶

  B DEF B E E DEF ❈♦♥s❡q✉❡♥t❡♠❡♥t❡✱ ♣ ✏ ④ ✳ ❈♦♠♦✱ ♣♦r ❤✐♣♦t❡s❡✱ ♣ ✏ ♣ ✱ s❡❣✉❡✲s❡ q✉❡ ♣ ✏ ④ ✳

  ✶ ✶

  F EF F EF

  ❉♦♥❞❡ ❝♦♥❝❧✉í♠♦s q✉❡ ④ ✏ 0 ❡✱ ♣♦rt❛♥t♦✱ ✉♠❛ ❝♦♥tr❛❞✐çã♦✱ ❝♦♠ ♦ ❢❛t♦ ❞❡ ④ → 0✳ ❆ss✐♠✱ ✭✐✐✐✮ t❛♠❜é♠ ♥ã♦ ♣♦❞❡ ♦❝♦rr❡r✳ ❚❡♦r❡♠❛ ✸✳✶✳✷✳ ✭❈❛s♦ ▲▲▲✮ ❙❡ ❞♦✐s tr✐â♥❣✉❧♦s tê♠ três ❧❛❞♦s ❝♦rr❡s♣♦♥❞❡♥t❡s ❝♦♥❣r✉✲ ❡♥t❡s✱ ❡♥tã♦ ♦s tr✐â♥❣✉❧♦s sã♦ ❝♦♥❣r✉❡♥t❡s✳ ❉❡♠♦♥str❛çã♦✿ ❈♦♥s✐❞❡r❡ ♦s tr✐â♥❣✉❧♦s ∆ABC ❡ ∆EF G✱ t❛✐s q✉❡✱ t❡♠♦s ♦s s❡❣♠❡♥t♦s AB

  ✏ EF ✱ BC ✏ F G ❡ AC ✏ EG✳ A E B C F G BM BC

  ❚r❛❝❡ ❛ s❡♠✐rr❡t❛ ÝÝÑ ♥♦ s❡♠✐✲♣❧❛♥♦ ❞❡t❡r♠✐♥❛❞♦ ♣❡❧❛ r❡t❛ ÐÑ ❞❡ t❛❧ ❢♦r♠❛ q✉❡ ♦ BC BM F

  â♥❣✉❧♦ ❢♦r♠❛❞♦ ♣❡❧❛s s❡♠✐rr❡t❛s ÝÝÑ ❡ ÝÝÑ s❡❥❛ ✐❣✉❛❧ ❛♦ â♥❣✉❧♦ ♣ ✳ A C B F G M

  ✸✳✷ ❆❧❣✉♠❛s ❆♣❧✐❝❛çõ❡s ❞❡ ❈♦♥❣r✉ê♥❝✐❛ ✹✹

  ✏ ④ BDA

  ✸✳✷ ❆❧❣✉♠❛s ❆♣❧✐❝❛çõ❡s ❞❡ ❈♦♥❣r✉ê♥❝✐❛

  ❈♦♥s❡q✉❡♥t❡♠❡♥t❡✱ ♦❜t❡♠♦s q✉❡ ∆ABC ✏ ∆EF G.

  ✸✳✶ ✏ ∆EF G.

  ∆ABC ✏ ∆DBC

  D ✱ AB ✏ BD ❡ AC ✏ DC✳ ❊♥tã♦✱

  A ✏ ♣

  ✏ ♣ D. P♦rt❛♥t♦✱ ∆ABC ✏ ∆DBC✱ ✉♠❛ ✈❡③ q✉❡ ♣

  ♣ A

  D, ♦✉ s❡❥❛✱

  ✏ ♣

  ④ ADC

  BAD ④ DAC

  ◆❛ s❡♠✐rr❡t❛ ÝÝÑ BM

  A ✏ ④

  ✳ ❉♦♥❞❡✱ ❝♦♥❝❧✉í♠♦s q✉❡ ♣

  ✏ ④ DAC

  ❡ ④ ADC

  ✏ ④ BDA

  ❡ ACD sã♦ ✐sós❝❡❧❡s✳ ▲♦❣♦✱ ④ BAD

  ❖❜s❡r✈❡ q✉❡ ❞❡ ✸✳✶ t❡♠✲s❡ ♥❡❝❡ss❛r✐❛♠❡♥t❡ q✉❡ DC ✏ EG✳ P♦rt❛♥t♦✱ ♦s tr✐â♥❣✉❧♦s ABD

  F ✱ BD ✏ EF ❡ BC ✏ F G✳

  CBD ✏ ♣

  ✏ ∆EF G ✭✸✳✶✮ ✉♠❛ ✈❡③ q✉❡ ④

  ❉❡ss❛ ❢♦r♠❛✱ t❡♠♦s q✉❡ ∆DBC

  ♠❛rq✉❡ ✉♠ ♣♦♥t♦ D t❛❧ q✉❡ BD ✏ EF ✳ A B C D E F G M

  ◆❡st❛ s❡çã♦ ♥♦s ❞❡t❡r❡♠♦s ❡♠ ❛❧❣✉♠❛s ❛♣❧✐❝❛çõ❡s ❞❡ ♥♦ss♦ ✐♥t❡r❡ss❡✱ ♦♥❞❡ ❛ ❝♦♥❣r✉ê♥✲ ❝✐❛ ❞❡ tr✐â♥❣✉❧♦s é ❛ ❢❡rr❛♠❡♥t❛ ✉t✐❧✐③❛❞❛ ♥❛ s♦❧✉çã♦ ❞❡st❛s ❛♣❧✐❝❛çõ❡s✳

  ✸✳✷ ❆❧❣✉♠❛s ❆♣❧✐❝❛çõ❡s ❞❡ ❈♦♥❣r✉ê♥❝✐❛ ✹✺

  AOB AOB OC ❉❡✜♥✐çã♦ ✸✳✷✳✶✳ ❉❛❞♦ ✉♠ â♥❣✉❧♦ ③ ✱ ❛ ❜✐ss❡tr✐③ ❞❡ ③ é ❛ s❡♠✐rr❡t❛ ÝÝÑ q✉❡ ♦

  OC AOB ❞✐✈✐❞❡ ❡♠ ❞♦✐s â♥❣✉❧♦s ✐❣✉❛✐s✳ ◆❡st❡ ❝❛s♦✱ ❞✐③❡♠♦s ❛✐♥❞❛ q✉❡ ÝÝÑ ❜✐ss❡❝t❛ ③ ✳

  B C O A ❋✐❣✉r❛ ✸✳✷✿ ❜✐ss❡tr✐③ ❖❈

  Pr♦♣♦s✐çã♦ ✸✳✷✳✶✳ ❙❡ ABC é ✉♠ tr✐â♥❣✉❧♦ ✐sós❝❡❧❡s ❡♠ ❜❛s❡ BC✱ ❡♥tã♦ ♦s â♥❣✉❧♦s ❞❛ B C

  ❜❛s❡ sã♦ ✐❣✉❛✐s✱ ♦✉ s❡❥❛✱ ♣ ✏ ♣ ✳ ❉❡♠♦♥str❛çã♦✿ ◆♦ tr✐â♥❣✉❧♦ ABC t❡♠✲s❡ q✉❡ ♦ ❧❛❞♦ BC é ❜❛s❡ ❞♦ tr✐â♥❣✉❧♦ ✐sós❝❡❧❡s ❡ AB ✏ AC✳ ❙❡❥❛ M ♦ ♣♦♥t♦ ♠é❞✐♦ ❞❡ ❜❛s❡ BC✳

  A C B M ❈♦♥s✐❞❡r❡ ❡♥tã♦✱ ♦s tr✐â♥❣✉❧♦s AMB ❡ AMC✳ ❈♦♠♦ AM é ❧❛❞♦ ❝♦♠✉♠ ❞♦s ❞♦✐s tr✐â♥❣✉❧♦s✱ AB ✏ AC ❡ BM ✏ CM✱ ♦ ❚❡♦r❡♠❛ ✸✳✶✳✷ ♥♦s ❣❛r❛♥t❡ q✉❡ ♦s tr✐â♥❣✉❧♦s AMB

  B C ❡ AMC sã♦ ❝♦♥❣r✉❡♥t❡s ✳ ❉♦♥❞❡ ❝♦♥❝❧✉í♠♦s q✉❡ ♣ ✏ ♣ ✳ Pr♦♣♦s✐çã♦ ✸✳✷✳✷✳ ❉❛❞♦ ✉♠ tr✐â♥❣✉❧♦ ABC✱ ❛ s♦♠❛ ❞♦s â♥❣✉❧♦s ✐♥t❡r♥♦s ❞❡ss❡ tr✐â♥❣✉❧♦

  ✆ ✆

  ♣ ♣ A B C

  é 180 ✱ ♦✉ s❡❥❛✱ ♣ ✏ 180 ✳

  XY ❉❡♠♦♥str❛çã♦✿ ❉❛❞♦ ✉♠ tr✐â♥❣✉❧♦ ABC✱ s❡❥❛ ÐÑ ❛ r❡t❛ ♣❛ss❛♥❞♦ ♣❡❧♦ ✈ért✐❝❡ A ❡ ♣❛r❛❧❡❧❛ à r❡t❛ q✉❡ ❝♦♥t❡♠ ♦ ❧❛❞♦ BC✳

  ✸✳✷ ❆❧❣✉♠❛s ❆♣❧✐❝❛çõ❡s ❞❡ ❈♦♥❣r✉ê♥❝✐❛ ✹✻

  A Y

  X C B BC

  XY C B k ÐÑ ❉❡st❛ ❢♦r♠❛✱ ❝♦♠♦ ÐÑ ✱ t❡♠♦s q✉❡ ♦s â♥❣✉❧♦s ♣ ❡ ♣ sã♦✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱

  Y AC

  XAB ✐❣✉❛✐s ❛♦s â♥❣✉❧♦s ③ ❡ ④ ✱ ✉♠❛ ✈❡③ q✉❡ sã♦ ❛❧t❡r♥♦s ✐♥t❡r♥♦s✳ ❆ss✐♠ t❡r❡♠♦s ❛ s✐t✉❛çã♦ r❡♣r❡s❡♥t❛❞❛ ♥❛ ✜❣✉r❛ ❛❜❛✐①♦✳

  A Y

  X C B

  XAY P♦r ♦✉tr♦ ❧❛❞♦ ♦ â♥❣✉❧♦ ④ é ✉♠ â♥❣✉❧♦ r❛s♦ ❡✱ ♣♦rt❛♥t♦✱

  ✆

  ④ ♣ ③ XAB A Y AC .

  ✏ 180 ❆ss✐♠ s❡♥❞♦✱ t❡♠♦s

  ✆

  ♣ ♣ ♣ A B C .

  ✏ 180 ❈♦r♦❧ár✐♦ ✸✳✷✳✶✳ ❙❡ ✉♠ tr✐â♥❣✉❧♦ é ❡q✉✐❧át❡r♦✱ ❡♥tã♦ ♦s â♥❣✉❧♦s ✐♥t❡r♥♦s ❞❡ss❡ tr✐â♥❣✉❧♦

  ✆

  sã♦ t♦❞♦s ✐❣✉❛✐s ❛ 60 ✳ ❆♥t❡s ❞❡ ✐♥✐❝✐❛r♠♦s ♦ ♣ró①✐♠♦ ❝♦r♦❧ár✐♦✱ ❢❛③✲s❡ ♥❡❝❡ssár✐♦ ❞❡✜♥✐r♠♦s ♦ â♥❣✉❧♦ ❡①t❡r♥♦

  ❞❡ ✉♠ tr✐â♥❣✉❧♦✳ ❉❡✜♥✐çã♦ ✸✳✷✳✷✳ ➶♥❣✉❧♦ ❡①t❡r♥♦ ❞❡ ✉♠ tr✐â♥❣✉❧♦ é ❛q✉❡❧❡ q✉❡ é ❛❞❥❛❝❡♥t❡ ❡ s✉♣❧❡♠❡♥t❛r

  ACX ❛ ✉♠ ❞♦s s❡✉s â♥❣✉❧♦s ✐♥t❡r♥♦s✳ ◆❛ ✜❣✉r❛ ✸✳✸✱ ♦ â♥❣✉❧♦ ④ é ✉♠ â♥❣✉❧♦ ❡①t❡r♥♦ ❞♦ tr✐â♥❣✉❧♦ ABC✳ ❈♦r♦❧ár✐♦ ✸✳✷✳✷✳ ❊♠ t♦❞♦ tr✐â♥❣✉❧♦✱ ❛ ♠❡❞✐❞❛ ❞❡ ✉♠ â♥❣✉❧♦ ❡①t❡r♥♦ é ✐❣✉❛❧ ❛ s♦♠❛ ❞❛s

  ✸✳✷ ❆❧❣✉♠❛s ❆♣❧✐❝❛çõ❡s ❞❡ ❈♦♥❣r✉ê♥❝✐❛ ✹✼

  A B C

  X ❋✐❣✉r❛ ✸✳✸✿ ➶♥❣✉❧♦ ❡①t❡r♥♦ ❞❡ ✉♠ tr✐â♥❣✉❧♦✳

  ❉❡♠♦♥str❛çã♦✿ ❇❛st❛ ♦❜s❡r✈❛r ❛ ✜❣✉r❛ ✸✳✸✱ ❡ q✉❡ ❛ ♣r♦♣♦s✐çã♦ ✸✳✷✳✷ ♥♦s ❣❛r❛♥t❡ q✉❡

  ✆

  ˆ ④

  ACX C A B ✏ 180 ✁ ˆ ✏ ˆ ✳

  ❈♦r♦❧ár✐♦ ✸✳✷✳✸✳ ❙❡ ❞♦✐s â♥❣✉❧♦s ❞❡ ✉♠ tr✐â♥❣✉❧♦ ❡ ♦ ❧❛❞♦ ♦♣♦st♦ ❛ ✉♠ ❞❡ss❡s â♥❣✉❧♦s ❢♦r❡♠ r❡s♣❡❝t✐✈❛♠❡♥t❡ ✐❣✉❛✐s ❛ ❞♦✐s â♥❣✉❧♦s ❞❡ ♦✉tr♦ tr✐â♥❣✉❧♦ ❡ ❛♦ ❧❛❞♦ ♦♣♦st♦ ❛♦ â♥❣✉❧♦ ❝♦rr❡s♣♦♥❞❡♥t❡ ♥❡ss❡ ♦✉tr♦ tr✐â♥❣✉❧♦✱ ❡♥tã♦ ♦s ❞♦✐s tr✐â♥❣✉❧♦s sã♦ ❝♦♥❣r✉❡♥t❡s✳

  A D B E ❉❡♠♦♥str❛çã♦✿ ❈♦♥s✐❞❡r❡ ♦s tr✐â♥❣✉❧♦s ABC ❡ DEF ♦♥❞❡ ♣ ✏ ♣ ✱ ♣ ✏ ♣ ❡ BC ✏ EF ✳

  F A B D E P❡❧❛ ♣r♦♣♦s✐çã♦ ✸✳✷✳✷✱ t❡♠♦s q✉❡

  ✆ ✆

  ♣ ♣ ♣ A B C C A B.

  ✏ 180 ñ ♣ ✏ 180 ✁ ♣ ✁ ♣ ✭✸✳✷✮ A D B E

  P♦r ❤✐♣ót❡s❡✱ t❡♠♦s q✉❡ ♣ ✏ ♣ ❡ ♣ ✏ ♣ ✳ ❯s❛♥❞♦ ❡ss❡ ❢❛t♦ ♥❛ ❡q✉❛çã♦ ✸✳✷✱ ♦❜t❡♠♦s

  ✆

  C D E. q✉❡ ♣ ✏ 180 ✁ ♣ ✁ ♣ C F .

  ❉❡ss❛ ❢♦r♠❛✱ ♦❜t❡♠♦s q✉❡ ♣ ✏ ♣ A E C F

  P♦rt❛♥t♦✱ t❡♠♦s q✉❡ ♣ ✏ ♣ ✱ ♣ ✏ ♣ ❡ BC ✏ EF ✳ ❆ss✐♠✱ ♦ t❡♦r❡♠❛ ✸✳✶✳✶ ♥♦s ❣❛r❛♥t❡ q✉❡ ♦s tr✐â♥❣✉❧♦s ABC ❡ DEF sã♦ ❝♦♥❣r✉❡♥t❡s

  ✸✳✸ ❆ ❉❡s✐❣✉❛❧❞❛❞❡ ❚r✐❛♥❣✉❧❛r ✹✽

  ✸✳✸ ❆ ❉❡s✐❣✉❛❧❞❛❞❡ ❚r✐❛♥❣✉❧❛r

  ◆❡st❛ s❡çã♦ ✐r❡♠♦s ♠♦str❛r q✉❡✱ ❡♠ t♦❞♦ tr✐â♥❣✉❧♦✱ ♦s ❝♦♠♣r✐♠❡♥t♦s ❞♦s ❧❛❞♦s ❣✉❛r✲ ❞❛♠ ✉♠❛ ❝❡rt❛ r❡❧❛çã♦✱ ❝♦♥❤❡❝✐❞❛ ❝♦♠♦ ❞❡s✐❣✉❛❧❞❛❞❡ tr✐❛♥❣✉❧❛r✳ ❈♦♠❡ç❛r❡♠♦s ❡st❛✲ ❜❡❧❡❝❡♥❞♦ ✉♠❛ r❡❧❛çã♦ ❡♥tr❡ ♦s ❝♦♠♣r✐♠❡♥t♦s ❞♦s ❧❛❞♦s ❡ ❛s ♠❡❞✐❞❛s ❞♦s â♥❣✉❧♦s ❛ ❡❧❡s ♦♣♦st♦s✳

  B C Pr♦♣♦s✐çã♦ ✸✳✸✳✶✳ ❙❡ ABC é ✉♠ tr✐â♥❣✉❧♦ t❛❧ q✉❡ ♣ → ♣ ✱ ❡♥tã♦ AC → AB✳

  B C BX ❉❡♠♦♥str❛çã♦✿ ❈♦♠♦ ♣ ✁ ♣ → 0✱ ♣♦❞❡♠♦s tr❛ç❛r ❛ s❡♠✐rr❡t❛ ÝÝÑ ✱ ✐♥t❡rs❡❝t❛♥❞♦ ♦

  ✁ ✠

  1 ♣

  CBX B C ✐♥t❡r✐♦r ❞❡ ABC ❡ t❛❧ q✉❡ ④ ✏ ✁ ♣ ✳

  2 A

  X P B C AP B ❖❜s❡r✈❡ q✉❡ ♦ â♥❣✉❧♦ ③ é ✉♠ â♥❣✉❧♦ ❡①t❡r♥♦ ❞♦ tr✐â♥❣✉❧♦ BP C✳ ❙❡♥❞♦ ❛ss✐♠✱ ♦

  ❝♦r♦❧ár✐♦ ✸✳✷✳✷ ♥♦s ❣❛r❛♥t❡ q✉❡ ✁ ✠ ✁ ✠

  1

  1 ③ ♣ ♣ ♣ ♣ ♣ AP B CBP C B C C B C .

  ✏ ④ ✏ ✁ ♣ ✏

  2

  2 ▼❛s✱ ❝♦♠♦

  ✁ ✠ ✁ ✠

  1

  1 ③ ♣ ♣ ♣

  ABP B B C B C , ✏ ♣ ✁ ✁ ♣ ✏

  2

  2 s❡❣✉❡ q✉❡ ♦ tr✐â♥❣✉❧♦ ABP é ✐sós❝❡❧❡s ❞❡ ❜❛s❡ BP ✳ P♦rt❛♥t♦✱ AB ✏ AP ➔ AC.

  ✆

  A ❈♦r♦❧ár✐♦ ✸✳✸✳✶✳ ❙❡ ABC é ✉♠ tr✐â♥❣✉❧♦ t❛❧ q✉❡ ♣ ➙ 90 ✱ ❡♥tã♦ BC é s❡✉ ♠❛✐♦r ❧❛❞♦✳ ❊♠ ♣❛rt✐❝✉❧❛r✱ ♥✉♠ tr✐â♥❣✉❧♦ r❡tâ♥❣✉❧♦ ❛ ❤✐♣♦t❡♥✉s❛ é ♦ ♠❛✐♦r ❧❛❞♦✳

  ✸✳✸ ❆ ❉❡s✐❣✉❛❧❞❛❞❡ ❚r✐❛♥❣✉❧❛r ✹✾

  C A B

  ✆

  A A ❉❡♠♦♥str❛çã♦✿ ❇❛st❛ ♦❜s❡r✈❛r q✉❡✱ s❡ ♣ ➙ 90 ✱ ❡♥tã♦ ♣ é ♦ ♠❛✐♦r â♥❣✉❧♦ ❞♦ tr✐â♥❣✉❧♦ ABC

  ✱ ❞❡ ♠♦❞♦ q✉❡ BC é✱ ♣❡❧❛ ♣r♦♣♦s✐çã♦ ✸✳✸✳✶✱ ♦ ♠❛✐♦r ❧❛❞♦✳ Pr♦♣♦s✐çã♦ ✸✳✸✳✷✳ ✭❉❡s✐❣✉❛❧❞❛❞❡ tr✐❛♥❣✉❧❛r✮ ❊♠ t♦❞♦ tr✐â♥❣✉❧♦✱ ❝❛❞❛ ❧❛❞♦ t❡♠ ✉♠ ❝♦♠♣r✐♠❡♥t♦ ♠❡♥♦r q✉❡ ❛ s♦♠❛ ❞♦s ❝♦♠♣r✐♠❡♥t♦s ❞♦s ♦✉tr♦s ❞♦✐s ❧❛❞♦s✳ ❉❡♠♦♥str❛çã♦✿ ❈♦♥s✐❞❡r❡ ✉♠ tr✐â♥❣✉❧♦ ABC✱ t❛❧ q✉❡ AB ✏ c✱ AC ✏ b ❡ BC ✏ a✳

  A c b a

  B C ▼♦str❛r❡♠♦s q✉❡ a ➔ b c✱ s❡♥❞♦ ❛ ❞❡♠♦♥str❛çã♦ ❞❛s ❞❡♠❛✐s ❞❡s✐❣✉❛❧❞❛❞❡s t♦t❛❧♠❡♥t❡

  ❛♥á❧♦❣❛✳ ❊s❝♦❧❤❛ ✉♠ ♣♦♥t♦ D ♥❛ s❡♠✐r❡t❛ CA t❛❧ q✉❡ CD → CA ❡ AD ✏ AB✳ ❖❜s❡r✈❡ D ABD q✉❡ ♣♦r ❝♦♥str✉çã♦ ♦ tr✐â♥❣✉❧♦ ABD é ✐sós❝❡❧❡s ❡✱ ♣♦rt❛♥t♦✱ ♦s â♥❣✉❧♦s ♣ ❡ ④ sã♦

  ❝♦♥❣r✉❡♥t❡s✳ D A c b a

  B C ❋✐❣✉r❛ ✸✳✹✿ ❉❡s✐❣✉❛❧❞❛❞❡ tr✐❛♥❣✉❧❛r

  P❡r❝❡❜❛ q✉❡✱ ♣♦r ❝♦♥str✉çã♦✱ CD ✏ AC AD ✏ b c✳ ◆♦ tr✐â♥❣✉❧♦ BCD ♥❛ ✜❣✉r❛ ④ ♣

  CBD CBA ABD CBA D D ABD ✸✳✹✱ ♦❜s❡r✈❛♠♦s q✉❡ ④ ✏ ③ ✏ ③ ✱ ♣♦✐s ♣ ✏ ④ ✳ P♦rt❛♥t♦

  ④ ABD ADB CBD

  ✏ ④ ➔ ④ ✳ ❱❡❥❛♠♦s ✉♠ ❡①❡♠♣❧♦ ♣❛r❛ ✐❧✉str❛r ❛ ❞❡s✐❣✉❛❧❞❛❞❡ tr✐❛♥❣✉❧❛r✳

  ✸✳✸ ❆ ❉❡s✐❣✉❛❧❞❛❞❡ ❚r✐❛♥❣✉❧❛r ✺✵ ❛✮ P B P C ➔ AB AC.

  ❜✮ P A P B P C ➔ AB AC BC.

  ❆♥❛❧✐s❡ ✉♠❛ s♦❧✉çã♦ ❛✮ ❈♦♥s✐❞❡r❡ P ✉♠ ♣♦♥t♦ q✉❛❧q✉❡r ♥♦ ✐♥t❡r✐♦r ❞♦ tr✐â♥❣✉❧♦ ABC✳

  A P B C BP Pr♦❧♦♥❣✉❡ ❛ s❡♠✐rr❡t❛ ÝÝÑ ❛té q✉❡ ❛ ♠❡s♠❛ ❡♥❝♦♥tr❡ ♦ ❧❛❞♦ AC ♥♦ ♣♦♥t♦ Q✳

  A Q P B C ❆♣❧✐❝❛♥❞♦ ❛ ❞❡s✐❣✉❛❧❞❛❞❡ tr✐❛♥❣✉❧❛r ♥♦ tr✐â♥❣✉❧♦ CP Q✱ ♦❜t❡♠♦s ❛ ❞❡s✐❣✉❛❧❞❛❞❡

  P C ➔ P Q QC. ❙♦♠❛♥❞♦✲s❡ P B ❛ ❛♠❜♦s ♠❡♠❜r♦s ❞❡ss❛ ❞❡s✐❣✉❛❧❞❛❞❡ t❡r❡♠♦s ♦ s❡❣✉✐♥t❡

  P B P C ➔ P B ♣P Q QCq ✏ BQ QC ✭✸✳✸✮

  ❯s❛♥❞♦ ❛ ❞❡s✐❣✉❛❧❞❛❞❡ tr✐❛♥❣✉❧❛r ♥♦ tr✐â♥❣✉❧♦ ABQ ♦❜t❡♠♦s q✉❡ BQ ➔ AB AQ✳ ❊ss❛ ✐♥❢♦r♠❛çã♦ ❥✉♥t❛♠❡♥t❡ ❝♦♠ ❛ ❡q✉❛çã♦ ✸✳✸ ♣r♦❞✉③ ❛ s❡❣✉✐♥t❡ ❞❡s✐❣✉❛❧❞❛❞❡

  P B P C ➔ AB AQ QC. P♦r ♦✉tr♦ ❧❛❞♦✱ AQ QC ✏ AC ❡✱ ♣♦rt❛♥t♦

  P B P C ➔ AB AC.

  ✸✳✹ ◗✉❛❞r✐❧át❡r♦s ◆♦tá✈❡✐s ✺✶

  ❜✮ ❱✐♠♦s ♥♦ ✐t❡♠ ❛♥t❡r✐♦r q✉❡ P B P C ➔ AB AC✳ ❉❡ ♠❛♥❡✐r❛ ✐♥t❡✐r❛♠❡♥t❡ ❛♥á❧♦❣❛ ♦❜t❡♠♦s ❛s s❡❣✉✐♥t❡s ❞❡s✐❣✉❛❧❞❛❞❡s✿

  P A P B ➔ AC BC

  ❡ P A P C

  ➔ AB BC ❙♦♠❛♥❞♦✲s❡ ♦r❞❡♥❛❞❛♠❡♥t❡ ❡ss❛s três ❞❡s✐❣✉❛❧❞❛❞❡s✱ ♦❜t❡♠♦s✿

  P A P B P A P C P B P C ➔ AC BC AB BC AB AC. ❉♦♥❞❡ s❡❣✉❡✲s❡ q✉❡✿

  2 ☎ ♣P A P B P Cq ➔ 2 ☎ ♣AB AC BCq ,

  ♦✉ s❡❥❛✱ ♣P A P B P Cq ➔ ♣AB AC BCq .

  ✸✳✹ ◗✉❛❞r✐❧át❡r♦s ◆♦tá✈❡✐s

  ◆❡st❛ s❡çã♦ ✐r❡♠♦s ❡st✉❞❛r ❛❧❣✉♥s ❞♦s q✉❛❞r✐❧át❡r♦s ♥♦tá✈❡✐s✳ ▼♦str❛r❡♠♦s ❛ ❞❡✜♥✐çã♦ ❞❡ ♣❛r❛❧❡❧♦❣r❛♠♦ ❡ ❛❧❣✉♠❛s ♣r♦♣♦s✐çõ❡s ❡♥✈♦❧✈❡♥❞♦ ❡ss❡ q✉❛❞r✐❧át❡r♦✳ ❉❡♣♦✐s ❞❡✜♥✐r❡♠♦s tr❛♣é③✐♦✱ r❡tâ♥❣✉❧♦ ❡ ❧♦s❛♥❣♦✳ ❊ss❡ é ✉♠ tó♣✐❝♦ ❜❡♠ ✐♥t❡r❡ss❛♥t❡ ♣❡❧❛ ♥❡❝❡ss✐❞❛❞❡ ❞❡ ❞❡✲ ✜♥✐r♠♦s ♣r❡❝✐s❛♠❡♥t❡ ❡ss❡s q✉❛❞r✐❧át❡r♦s✱ ♣❛r❛ ♥ã♦ ❝♦♥❢✉♥❞✐r♠♦s ❛s ❝❛r❛❝t❡ríst✐❝❛s ❡♥tr❡ ❡❧❡s✳

  ❉❡ ❛❝♦r❞♦ ❝♦♠ ❛ ♣r♦♣♦st❛ q✉❡ ❡stá s❡♥❞♦ ❢♦r♥❡❝✐❞❛ ♣❛r❛ ♦ ❧❡✐t♦r✱ ❞❡✜♥✐r❡♠♦s ❝♦♥❥✉♥t♦ ❝♦♥✈❡①♦✱ ♣♦❧í❣♦♥♦ ❡ ♣♦❧í❣♦♥♦ ❝♦♥✈❡①♦ ♣❛r❛ ✉♠ ♠❡❧❤♦r ❡♥t❡♥❞✐♠❡♥t♦ ❞♦s q✉❛❞r✐❧át❡r♦s q✉❡ ❞❡✜♥✐r❡♠♦s ❛ s❡❣✉✐r✳ ❉❡✜♥✐çã♦ ✸✳✹✳✶✳ ❯♠❛ r❡❣✐ã♦ ♥♦ ♣❧❛♥♦ ❝❤❛♠❛✲s❡ ❝♦♥✈❡①♦ s❡✱ q✉❛✐sq✉❡r q✉❡ s❡❥❛♠ ❞♦✐s ♣♦♥t♦s ❞✐st✐♥t♦s ❞❡ss❡ ❝♦♥❥✉♥t♦✱ ♦ s❡❣♠❡♥t♦ q✉❡ t❡♠ ❡ss❡s ♣♦♥t♦s ♣♦r ❡①tr❡♠✐❞❛❞❡s ❡stá ❝♦♥t✐❞♦ ♥❡ss❡ ❝♦♥❥✉♥t♦✳

  ❖❜s❡r✈❡ q✉❡ ❛ ✜❣✉r❛ ✶ ♠♦str❛ ✉♠ ❝♦♥❥✉♥t♦ ❝♦♥✈❡①♦ ❡ ❛ ✜❣✉r❛ ✷ ♠♦str❛ ✉♠ ❝♦♥❥✉♥t♦

  ✸✳✹ ◗✉❛❞r✐❧át❡r♦s ◆♦tá✈❡✐s ✺✷

  B B A A F igura2

  F igura1 1 , A 2 , n ❉❡✜♥✐çã♦ ✸✳✹✳✷✳ ❈♦♥s✐❞❡r❡♠♦s ✉♠ ♥ú♠❡r♦ n ♣n ➙ 3q ❞❡ ♣♦♥t♦s ♦r❞❡♥❛❞♦s A ☎ ☎ ☎ , A ❞❡ ♠♦❞♦ q✉❡ três ♣♦♥t♦s ❝♦♥s❡❝✉t✐✈♦s s❡❥❛♠ ♥ã♦ ❝♦❧✐♥❡❛r❡s ❡ ❝♦♥s✐❞❡r❡♠♦s ♦s s❡❣♠❡♥t♦s 1 A 2 , A 2 A 3 , A n 1

  ❝♦♥s❡❝✉t✐✈♦s A ☎ ☎ ☎ , A ✳ ❉❡♥♦♠✐♥❛✲s❡ ♣♦❧í❣♦♥♦ ❛ ✜❣✉r❛ ❝♦♥st✐t✉í❞❛ ♣❡❧♦s ♣♦♥t♦s ❞♦s n s❡❣♠❡♥t♦s ❝♦♥s❡❝✉t✐✈♦s✳

  ❉❡✜♥✐r❡♠♦s✱ ❛❣♦r❛✱ ♣♦❧í❣♦♥♦ ❝♦♥✈❡①♦✳ ❉❡ss❛ ❢♦r♠❛✱ ✐♥✐❝✐❛r❡♠♦s ❛ ♣ró①✐♠❛ s❡çã♦ ❝♦♠ ♦ ❝♦♥❤❡❝✐♠❡♥t♦ ♥❡❝❡ssár✐♦ ♣❛r❛ ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞♦s ❝♦♥❝❡✐t♦s s♦❜r❡ q✉❛❞r✐❧át❡r♦s✳ ❉❡✜♥✐çã♦ ✸✳✹✳✸✳ ❉❡♥♦♠✐♥❛✲s❡ ♣♦❧í❣♦♥♦ ❝♦♥✈❡①♦ ♣❛r❛ ❛q✉❡❧❡ ♣♦❧í❣♦♥♦ ❝✉❥♦ ✐♥t❡r✐♦r é ✉♠ ❝♦♥❥✉♥t♦ ❝♦♥✈❡①♦✳

  ◆❛ ✜❣✉r❛ ✶ t❡♠♦s ✉♠ ♣♦❧í❣♦♥♦ ❝♦♥✈❡①♦ ❡ ♥❛ ✜❣✉r❛ ✷ t❡♠♦s ✉♠ ♣♦❧í❣♦♥♦ ♥ã♦ ❝♦♥✈❡①♦✳ F igura 1 F igura 2

  ✸✳✹✳✶ P❛r❛❧❡❧♦❣r❛♠♦

  ❉❡✜♥✐çã♦ ✸✳✹✳✹✳ ❯♠ ♣❛r❛❧❡❧♦❣r❛♠♦ é ✉♠ q✉❛❞r✐❧át❡r♦ ❝♦♥✈❡①♦ q✉❡ ♣♦ss✉✐ ❧❛❞♦s ♦♣♦st♦s ♣❛r❛❧❡❧♦s✳

  C D A B ❋✐❣✉r❛ ✸✳✺✿ P❛r❛❧❡❧♦❣r❛♠♦ ABCD

  Pr♦♣♦s✐çã♦ ✸✳✹✳✶✳ ❯♠ q✉❛❞r✐❧át❡r♦ ❝♦♥✈❡①♦ é ✉♠ ♣❛r❛❧❡❧♦❣r❛♠♦ s❡✱ ❡ só s❡✱ s❡✉s â♥❣✉❧♦s

  ✸✳✹ ◗✉❛❞r✐❧át❡r♦s ◆♦tá✈❡✐s ✺✸

  ❉❡♠♦♥str❛çã♦✿ ❙✉♣♦♥❤❛✱ ♣r✐♠❡✐r♦✱ q✉❡ ♦ q✉❛❞r✐❧át❡r♦ ❝♦♥✈❡①♦ ABCD ♥❛ ✜❣✉r❛ ✸✳✻ é ✉♠ ♣❛r❛❧❡❧♦❣r❛♠♦✳

  D C A B AD BC A B k ÐÑ

  ❊♥tã♦ ÐÑ ❡✱ ❝♦♠♦ ♦s â♥❣✉❧♦s ♣ ❡ ♣ ❞♦ ♣❛r❛❧❡❧♦❣r❛♠♦ sã♦ ❝♦❧❛t❡r❛✐s ✐♥t❡r♥♦s

  ✆

  ♣ AB A B

  ❡♠ r❡❧❛çã♦ à tr❛♥s✈❡rs❛❧ ÐÑ ✱ t❡♠♦s ♣ ✏ 180 ✳ P♦r ♦✉tr♦ ❧❛❞♦✱ ❛s r❡t❛s AB ❡ DC

  ✆ ✆

  ♣ ♣ A D D C sã♦ ♣❛r❛❧❡❧❛s ❡✱ ♣♦rt❛♥t♦✱ t❡♠♦s q✉❡ ♦s â♥❣✉❧♦s ♣ ✏ 180 ❡ ♣ ✏ 180 ♣♦✐s sã♦✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ❝♦❧❛t❡r❛✐s ✐♥t❡r♥♦s ❝♦♠ r❡s♣❡✐t♦ ❛s r❡t❛s tr❛♥s✈❡rs❛✐s AD ❡ BC✳ ❉❡ss❛

  ✆

  A D C ❢♦r♠❛✱ ♦❜t❡♠♦s q✉❡ ♣ ✏ 180 ✁ ♣ ✏ ♣ ✳ Pr♦✈❛♥❞♦✱ ❞❡ss❛ ♠❛♥❡✐r❛ ♦ r❡s✉❧t❛❞♦✳

  A C B D ❉❛ ♠❡s♠❛ ❢♦r♠❛✱ s❡❥❛ ABCD ✉♠ q✉❛❞r✐❧át❡r♦ ❝♦♥✈❡①♦ t❛❧ q✉❡ ♣ ✏ ♣ ❡ ♣ ✏ ♣ ✳

  D C A B A C B D.

  ❋✐❣✉r❛ ✸✳✻✿ ♣ ✏ ♣ ❡ ♣ ✏ ♣

  ✆ ✆

  ♣ ♣ ♣ ♣ ♣ ♣ ♣ A B C D A B C D A B C D

  ❊♥tã♦ ♣ ✏ ♣ ❡✱ ❝♦♠♦ ♣ ✏ 360 ✱ t❡♠♦s ♣ ✏ ♣ ✏ 180 ✳

  ✆ ✆

  ♣ ♣ ♣ A D B C A B AD k ÐÑ BC

  ❆♥❛❧♦❣❛♠❡♥t❡ ♣ ✏ ♣ ✏ 180 ✳ ❈♦♠♦ ♣ ✏ 180 ✱ ❡♥tã♦ ÐÑ ✳ ❉❛ ♠❡s♠❛ ÐÑ

  ✆

  B C ♣ AB CD k ❢♦r♠❛✱ ♣ ✏ 180 ✱ ♥♦s ❞á ÐÑ ✱ ❞❡ ♠❛♥❡✐r❛ q✉❡ ABCD t❡♠ ❧❛❞♦s ♦♣♦st♦s ♣❛r❛✲ ❧❡❧♦s✱ ♣♦rt❛♥t♦ ✉♠ ♣❛r❛❧❡❧♦❣r❛♠♦✳ Pr♦♣♦s✐çã♦ ✸✳✹✳✷✳ ❯♠ q✉❛❞r✐❧át❡r♦ ❝♦♥✈❡①♦ é ✉♠ ♣❛r❛❧❡❧♦❣r❛♠♦ s❡✱ ❡ só s❡✱ s❡✉s ♣❛r❡s ❞❡ ❧❛❞♦s ♦♣♦st♦s ❢♦r❡♠ ✐❣✉❛✐s✳ ❉❡♠♦♥str❛çã♦✿ ❉❛❞♦ ♦ ♣❛r❛❧❡❧♦❣r❛♠♦ ABCD✱ ❝♦♥s✐❞❡r❡ s✉❛ ❞✐❛❣♦♥❛❧ BD✳ ❈♦♠♦ AD

  ADB CBD ABD CDB k BC ❡ AB k DC t❡♠♦s✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ q✉❡ ④ ✏ ④ ❡ ④ ✏ ④

  ✸✳✹ ◗✉❛❞r✐❧át❡r♦s ◆♦tá✈❡✐s ✺✹

  D C A B ✭❛❧t❡r♥♦s ✐♥t❡r♥♦s✮✳ ❆ss✐♠ s❡♥❞♦✱ ♣❡❧♦ t❡♦r❡♠❛ ✸✳✶✳✶✱ ❣❛r❛♥t✐♠♦s q✉❡ ♦s tr✐â♥❣✉❧♦s ABD ❡ CDB sã♦ ❝♦♥❣r✉❡♥t❡s✳ ❊✱ ♣♦rt❛♥t♦✱ AB ✏ CD ❡ AD ✏ BC✳

  ❘❡❝✐♣r♦❝❛♠❡♥t❡✱ s✉♣♦♥❤❛ q✉❡ ABCD s❡❥❛ ✉♠ q✉❛❞r✐❧át❡r♦ ❝♦♥✈❡①♦ ❞❡ ♠♦❞♦ q✉❡ AB

  ✏ CD ❡ AD ✏ BC✳ P♦rt❛♥t♦✱ ♣❡❧♦ t❡♦r❡♠❛ ✸✳✶✳✷✱ ♦s tr✐â♥❣✉❧♦s ABD ❡ CDB sã♦ ADB CBD ABD CDB

  ❝♦♥❣r✉❡♥t❡s✳ ❉❡ss❛ ❢♦r♠❛✱ s❡❣✉❡✲s❡ ♦s â♥❣✉❧♦s ④ ✏ ④ ❡ ④ ✏ ④ ✱ s❡♥❞♦ AD BC AB CD k ÐÑ k ÐÑ

  ❛❧t❡r♥♦s ✐♥t❡r♥♦s ❡✱ ♣♦rt❛♥t♦✱ ♥❡ss❛s ❝♦♥❞✐çõ❡s✱ ÐÑ ❡ ÐÑ ✳ ❈♦♥❝❧✉✐✲s❡✱ ❞❡ss❛ ❢♦r♠❛✱ q✉❡ ♦ q✉❛❞r✐❧át❡r♦ ❝♦♥✈❡①♦ ABCD é ✉♠ ♣❛r❛❧❡❧♦❣r❛♠♦✳ Pr♦♣♦s✐çã♦ ✸✳✹✳✸✳ ❯♠ q✉❛❞r✐❧át❡r♦ ❝♦♥✈❡①♦ é ✉♠ ♣❛r❛❧❡❧♦❣r❛♠♦ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ s✉❛s ❞✐❛❣♦♥❛✐s s❡ ✐♥t❡rs❡❝t❛♠ ♥♦s r❡s♣❡❝t✐✈♦s ♣♦♥t♦s ♠é❞✐♦s✳ ❉❡♠♦♥str❛çã♦✿ ❈♦♥s✐❞❡r❛♥❞♦ ♦ ♣❛r❛❧❡❧♦❣r❛♠♦ ABCD ❛❜❛✐①♦✱ tr❛❝❡ ❛s ❞✐❛❣♦♥❛✐s AC ❡

  ✳ D C A B ABD

  ❈♦♠♦ ABCD é ♣❛r❛❧❡❧♦❣r❛♠♦✱ t❡♠♦s q✉❡ AB é ♣❛r❛❧❡❧♦ ❛ DC ❡✱ ♣♦rt❛♥t♦✱ ④ ✏ CDB ④ BAC DCA

  ❡ ③ ✏ ④ ♣♦✐s sã♦✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ❛❧t❡r♥♦s ✐♥t❡r♥♦s✳ P♦r ♦✉tr♦ ❧❛❞♦✱ ♣❡❧♦ t❡♦r❡♠❛ ✸✳✶✳✶✱ ♦s tr✐â♥❣✉❧♦s ABM ❡ CDM sã♦✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ❝♦♥❣r✉❡♥t❡s✳ ❉❡ss❛ ❢♦r♠❛✱ AM ✏ CM ❡ BM ✏ DM ❡✱ ♣♦rt❛♥t♦✱ ♦ ♣♦♥t♦ M é ♦ ♣♦♥t♦ ♠é❞✐♦ ❞❛s ❞✐❛❣♦♥❛✐s✳

  ✸✳✹ ◗✉❛❞r✐❧át❡r♦s ◆♦tá✈❡✐s ✺✺

  D C M A B P♦r ♦✉tr♦ ❧❛❞♦✱ s❡❥❛ M ♦ ♣♦♥t♦ ♠é❞✐♦ ❞❛s ❞✐❛❣♦♥❛✐s ❞❡ss❡ q✉❛❞r✐❧át❡r♦✱ ♦✉ s❡❥❛✱

  AM AM B CM D ✏ CM ❡ BM ✏ DM✳ ❈♦♠♦ ④ ✏ ④ ✱ ♣♦✐s sã♦ â♥❣✉❧♦s ♦♣♦st♦s ♣❡❧♦ ✈ért✐❝❡✱ t❡♠♦s ♣❡❧♦ t❡♦r❡♠❛ ✸✳✶✳✶ q✉❡ ♦s tr✐â♥❣✉❧♦s ABM ❡ CDM sã♦ ❝♦♥❣r✉❡♥t❡s✳ ❆♥❛❧♦❣❛♠❡♥t❡✱

  ❝♦♥❝❧✉í♠♦s q✉❡ ♦s tr✐â♥❣✉❧♦s BCM ❡ DAM sã♦ ❝♦♥❣r✉❡♥t❡s✳ ❉❡ss❛ ❢♦r♠❛✱ AB ✏ CD ❡ BC

  ✏ AD ❡✱ ♣♦rt❛♥t♦✱ ABCD é ✉♠ ♣❛r❛❧❡❧♦❣r❛♠♦✳ ❆♦ ❡st✉❞❛r♠♦s tr✐â♥❣✉❧♦s ❡ s✉❛s ♣r♦♣r✐❡❞❛❞❡s✱ ❝♦♥✈é♠ ✈❡r✐✜❝❛r♠♦s ❛❧❣✉♥s ❛s♣❡❝t♦s

  ✐♠♣♦rt❛♥t❡s ♥❛ ❝♦♥str✉çã♦ ❞♦ tr✐â♥❣✉❧♦ ♠❡❞✐❛❧✱ ♦✉ s❡❥❛✱ ❞♦ tr✐â♥❣✉❧♦ ❢♦r♠❛❞♦ ♣❡❧♦ s❡✉s ♣♦♥t♦s ♠é❞✐♦s✳ ➱ ✐♥t❡r❡ss❛♥t❡ ❛♥❛❧✐s❛r♠♦s q✉❡ ♥❡ss❡s tr✐â♥❣✉❧♦s s❡ ❛♣❧✐❝❛♠ t♦❞❛s ❛s ♣r♦♣r✐❡❞❛❞❡s q✉❡ ❝❛r❛❝t❡r✐③❛♠ ♦ ♣❛r❛❧❡❧♦❣r❛♠♦✳

  ❉❛❞♦ ✉♠ tr✐â♥❣✉❧♦ ABC q✉❛❧q✉❡r✱ ♦ tr✐â♥❣✉❧♦ ❢♦r♠❛❞♦ ♣❡❧♦s ♣♦♥t♦s ♠é❞✐♦s ❞❛ ❧❛❞♦s ❞❡ss❡ tr✐â♥❣✉❧♦ é ❝❤❛♠❛❞♦ ❞❡ tr✐â♥❣✉❧♦ ♠❡❞✐❛❧✳

  A M N B P C ❋✐❣✉r❛ ✸✳✼✿ tr✐â♥❣✉❧♦ MNP é ♦ tr✐â♥❣✉❧♦ ♠❡❞✐❛❧ ❞❡ ABC

  ✳ ◆❛ ✜❣✉r❛ ✸✳✼ t❡♠♦s ✈ár✐♦s q✉❛❞r✐❧át❡r♦s✳ ❖❜s❡r✈❛♥❞♦ ♦ q✉❡ ❥á ❢♦✐ ❢❡✐t♦ ❛té ❛❣♦r❛ é

  ❢á❝✐❧ ✈❡r q✉❡ ♦s q✉❛❞r✐❧át❡r♦s BP NM✱ P CNM ❡ AMP N sã♦ ♣❛r❛❧❡❧♦❣r❛♠♦s✳

  ✸✳✹✳✷ ❚r❛♣é③✐♦

  ❉❡✜♥✐çã♦ ✸✳✹✳✺✳ ❯♠ tr❛♣é③✐♦ é ✉♠ q✉❛❞r✐❧át❡r♦ ❝♦♥✈❡①♦ q✉❡ ♣♦ss✉✐ ❞♦✐s ❧❛❞♦s ♣❛r❛❧❡❧♦s✳

  ✸✳✹ ◗✉❛❞r✐❧át❡r♦s ◆♦tá✈❡✐s ✺✻

  D C M N P Q A B ❋✐❣✉r❛ ✸✳✽✿ ❇❛s❡ ♠é❞✐❛ MN ❡ ♠❡❞✐❛♥❛ ❞❡ ❊✉❧❡r P Q✳

  ➱ ✐♠♣♦rt❛♥t❡ s❛❜❡r q✉❡ ♦ s❡❣♠❡♥t♦ q✉❡ ✉♥❡ ♦s ♣♦♥t♦s ♠é❞✐♦s ❞♦s ❧❛❞♦s ♥ã♦ ♣❛r❛❧❡❧♦s ❞❡ ✉♠ tr❛♣é③✐♦ é ❛ ❜❛s❡ ♠é❞✐❛ ❡ ♦ s❡❣♠❡♥t♦ q✉❡ ✉♥❡ ♦s ♣♦♥t♦s ♠é❞✐♦s ❞❛s ❞✐❛❣♦♥❛✐s ❞❡

  ✶

  ✉♠ tr❛♣é③✐♦ é ❛ ♠❡❞✐❛♥❛ ❞❡ ❊✉❧❡r ❬✼❪✳ Pr♦♣♦s✐çã♦ ✸✳✹✳✹✳ ❙❡❥❛ ABCD ✉♠ tr❛♣é③✐♦ ❞❡ ❜❛s❡s AB ❡ CD ❡ ❧❛❞♦s ♥ã♦ ♣❛r❛❧❡❧♦s AD ❡ BC✳ ❙❡❥❛♠✱ ❛✐♥❞❛✱ M ❡ N✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ♦s ♣♦♥t♦s ♠é❞✐♦s ❞♦s ❧❛❞♦s ♥ã♦ ♣❛r❛❧❡❧♦s AD

  ❡ BC✱ ❡ P ❡ Q ♦s ♣♦♥t♦s ♠é❞✐♦s ❞❛s ❞✐❛❣♦♥❛✐s AC ❡ BD✱ t❛♠❜é♠ r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ❊♥tã♦✿

  M N k ÐÑ AB M N k ÐÑ CD ✭✐✮ M, N, P ❡ Q sã♦ ❝♦❧✐♥❡❛r❡s ❡ ÐÝÑ ❡ ÐÝÑ ✳

  1

  1 ✭✐✐✮ MN ✏ ☎ ♣AB CDq ❡ P Q ✏ ☎ ⑤AB ✁ CD⑤✳

  2

  2 ❉❡♠♦♥str❛çã♦✿ P❛r❛ ❞❡♠♦♥str❛r ✭✐✮✱ ♦❜s❡r✈❡ ❛ ✜❣✉r❛ ✸✳✾✱ ♥♦t❡ q✉❡ MP é ❜❛s❡ ♠é❞✐❛ ❞♦ tr✐â♥❣✉❧♦ DCA✳

  D C M N P Q A B ❋✐❣✉r❛ ✸✳✾✿ ❇❛s❡ ♠é❞✐❛ MN ❡ ♠❡❞✐❛♥❛ ❞❡ ❊✉❧❡r P Q ❞❡ ✉♠ tr❛♣é③✐♦✳

  ❖❜s❡r✈❡ ♥❛ ✜❣✉r❛ ✸✳✾ q✉❡ M é ♣♦♥t♦ ♠é❞✐♦ ❞❡ AD ❡ P é ♣♦♥t♦ ♠é❞✐♦ ❞❡ AC✱ ❧♦❣♦ M P

  M P k ÐÑ CD é ❜❛s❡ ♠é❞✐❛ ❞♦ tr✐â♥❣✉❧♦ ACD✳ P♦rt❛♥t♦ ÐÝÑ ✳ P♦r ♦✉tr♦ ❧❛❞♦✱ MQ é ❜❛s❡

  ÐÑ M Q k AB

  ♠é❞✐❛ ❞♦ tr✐â♥❣✉❧♦ ABD✳ ▲♦❣♦✱ ÐÝÑ ✳ AB k ÐÑ CD M P k ÐÝÑ M Q

  ❈♦♠♦ ÐÑ ✱ s❡❣✉❡ q✉❡ ÐÝÑ ✳ P♦r ♦✉tr♦ ❧❛❞♦✱ ❡ss❛s r❡t❛s tê♠ ✉♠ ♣♦♥t♦ ❡♠ M P M Q

  ❝♦♠✉♠ ❡✱ ♣♦rt❛♥t♦✱ ❛s r❡t❛s ÐÝÑ ❡ ÐÝÑ sã♦ ❝♦✐♥❝✐❞❡♥t❡s✱ ♣r♦✈❛♥❞♦ ❞❡ss❛ ❢♦r♠❛ q✉❡ ♦s ♣♦♥t♦s M✱ P ❡ Q sã♦ ❝♦❧✐♥❡❛r❡s✳

  ▲❡♦♥❤❛r❞ ❊✉❧❡r✱ ♠❛t❡♠át✐❝♦ s✉íç♦ ❞♦ sé❝✉❧♦ ❳❱■■■✳

  ✸✳✹ ◗✉❛❞r✐❧át❡r♦s ◆♦tá✈❡✐s ✺✼

  N Q k ÐÑ CD N P k ÐÑ AB N Q N P ❙❡♥❞♦ ÐÑ ❡ ÐÑ ✱ ❝♦♥❝❧✉✐✲s❡ q✉❡ ❛s r❡t❛s ÐÑ ❡ ÐÑ sã♦ ❝♦✐♥❝✐❞❡♥t❡s ❡ ♦s

  M N k ÐÑ AB M N k ÐÑ CD ♣♦♥t♦s M✱ N✱ P ❡ Q sã♦ ❝♦❧✐♥❡❛r❡s ❡ ÐÝÑ ❡ ÐÝÑ ✳

  P❛r❛ ❞❡♠♦♥str❛r ♦ ✐t❡♠ ✭✐✐✮✱ ♦❜s❡r✈❛♥❞♦ ❛ ✜❣✉r❛ ✸✳✾ ❝♦♥❝❧✉í♠♦s q✉❡ M N

  ✏ MQ NQ. ✭✸✳✹✮ ❖❜s❡r✈❡ q✉❡ MQ ❡ NQ sã♦ ❜❛s❡s ♠é❞✐❛s✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ❞♦s tr✐â♥❣✉❧♦s ABD ❡

  AB CD BCD ✳ P♦rt❛♥t♦✱ MQ ✏ ❡ NQ ✏ ✳

  2

  2 ❙✉❜st✐t✉✐♥❞♦ ❛s ✐❣✉❛❧❞❛❞❡s ♥❛ ❡q✉❛çã♦ ✸✳✹✱ ❝♦♥❝❧✉✐✲s❡ q✉❡

  AB CD

  1 M N ✏ ✏ ☎ ♣AB CDq.

  2

  2

  2 ❉❡ ♠❛♥❡✐r❛ ❛♥á❧♦❣❛✱

  P Q ✏ MQ ✁ MP. ✭✸✳✺✮

  ❈♦♠♦ MQ ❡ MP sã♦✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ❜❛s❡ ♠é❞✐❛ ❞♦s tr✐â♥❣✉❧♦s ABD ❡ ACD✱ t❡♠♦s AB CD q✉❡ MQ ✏ ❡ MP ✏ ✳ ❙✉❜st✐t✉✐♥❞♦ ❛s ✐❣✉❛❧❞❛❞❡s ♥❛ ❡q✉❛çã♦ ✸✳✺✱ t❡♠♦s q✉❡

  2

  2 ✞✞ AB CD ✞✞

  1 P Q ✏ ✁ ☎ ⑤AB ✁ CD⑤. ✞✞ ✞✞ ✏

  2

  2

  2

  ✸✳✹✳✸ ❘❡tâ♥❣✉❧♦✱ ❧♦s❛♥❣♦ ❡ q✉❛❞r❛❞♦

  ❉❡✜♥✐çã♦ ✸✳✹✳✻✳ ❯♠ r❡tâ♥❣✉❧♦ é ✉♠ q✉❛❞r✐❧át❡r♦ ❝♦♥✈❡①♦ ♥♦ q✉❛❧ t♦❞♦s ♦s â♥❣✉❧♦s ✐♥t❡r♥♦s sã♦ ✐❣✉❛✐s✳

  D C A B ❉❡✜♥✐çã♦ ✸✳✹✳✼✳ ❯♠ ❧♦s❛♥❣♦ é ✉♠ q✉❛❞r✐❧át❡r♦ ❝♦♥✈❡①♦ q✉❡ ❛♣r❡s❡♥t❛ ♦s ❧❛❞♦s ❝♦♠ ♠❡s♠❛ ♠❡❞✐❞❛✳

  ✸✳✹ ◗✉❛❞r✐❧át❡r♦s ◆♦tá✈❡✐s ✺✽

  D A C B ❈♦♠♦ ♦s ❧❛❞♦s ♦♣♦st♦s ❞❡ ✉♠ r❡tâ♥❣✉❧♦ sã♦ s❡♠♣r❡ ♣❛r❛❧❡❧♦s✱ ❡♥tã♦ ♣♦r ❞❡✜♥✐çã♦ ❞❡

  ♣❛r❛❧❡❧♦❣r❛♠♦✱ t♦❞♦ r❡tâ♥❣✉❧♦ é ✉♠ ♣❛r❛❧❡❧♦❣r❛♠♦✳ ❉❡ ♠❛♥❡✐r❛ ❛♥á❧♦❣❛✱ ❝♦♥❝❧✉í♠♦s q✉❡ t♦❞♦ ❧♦s❛♥❣♦ é ✉♠ ♣❛r❛❧❡❧♦❣r❛♠♦✳ Pr♦♣♦s✐çã♦ ✸✳✹✳✺✳ ❯♠ ♣❛r❛❧❡❧♦❣r❛♠♦ é ✉♠ r❡tâ♥❣✉❧♦ s❡✱ ❡ só s❡✱ s✉❛s ❞✐❛❣♦♥❛✐s t✐✈❡r❡♠ ❝♦♠♣r✐♠❡♥t♦s ✐❣✉❛✐s✳ ❉❡♠♦♥str❛çã♦✿ ❙❡❥❛ ABCD ✉♠ r❡tâ♥❣✉❧♦ ❝✉❥❛s ❞✐❛❣♦♥❛✐s s❡❥❛♠ AC ❡ BD✳ P❡❧❛ ❞❡✜♥✐çã♦ ❞❛❞❛ ❞❡ r❡tâ♥❣✉❧♦✱ t❡♠♦s q✉❡ t♦❞♦ r❡tâ♥❣✉❧♦ é ✉♠ ♣❛r❛❧❡❧♦❣r❛♠♦ ❡✱ ♣♦rt❛♥t♦✱

  ✆

  AD DAB CBA ✏ BC✳ ❈♦♠♦ ④ ✏ ③ ✏ 90 ❡ AD é ✉♠ ❧❛❞♦ ❝♦♠✉♠ ❛♦s tr✐â♥❣✉❧♦s ABD ❡

  ABC ✱ ❡♥tã♦✱ ♣❡❧♦ t❡♦r❡♠❛ ✸✳✶✳✶✱ ❡❧❡s sã♦ ❝♦♥❣r✉❡♥t❡s ❡✱ ♣♦rt❛♥t♦✱ AC ✏ BD✳

  C D A B ❋✐❣✉r❛ ✸✳✶✵✿ ❘❡tâ♥❣✉❧♦ ABCD ❡ s✉❛s ❞✐❛❣♦♥❛✐s AC ❡ BD

  ❉❡ ♠❛♥❡✐r❛ ❛♥á❧♦❣❛✱ s❡♥❞♦ ♦ q✉❛❞r✐❧át❡r♦ ABCD ✉♠ ♣❛r❛❧❡❧♦❣r❛♠♦ ❝♦♠ AC ✏ BD ❡ ❝♦♠♦ AD ✏ BC ❡ AB é ✉♠ ❧❛❞♦ ❝♦♠✉♠ ❛♦s tr✐â♥❣✉❧♦s ABD ❡ ABC✱ t❡♠♦s ❛ss✐♠✱

  DAB CBA ♣❡❧♦ t❡♦r❡♠❛ ✸✳✶✳✷✱ q✉❡ ❡ss❡s tr✐â♥❣✉❧♦s sã♦ ❝♦♥❣r✉❡♥t❡s✳ P♦rt❛♥t♦✱ ④ ✏ ③ ✳ P♦r

  ✆

  ③ DAB CBA

  ♦✉tr♦ ❧❛❞♦✱ ❝♦♠♦ ABCD é ✉♠ ♣❛r❛❧❡❧♦❣r❛♠♦ t❡♠♦s q✉❡ ④ ✏ 180 ✱ ❞♦♥❞❡

  ✆

  DAB CBA ❝♦♥❝❧✉í♠♦s q✉❡ ④ ✏ 90 ✏ ③ ✳ ❆♥❛❧♦❣❛♠❡♥t❡✱ r❡♣❡t✐♥❞♦ ❛r❣✉♠❡♥t♦s s❡♠❡❧❤❛♥t❡s

  ADC BCD ❛♦s ✉t✐❧✐③❛❞♦s ❛♥t❡r✐♦r♠❡♥t❡✱ ❝♦♠ ♦s â♥❣✉❧♦s ④ ❡ ④ ✱ ♥♦s tr✐â♥❣✉❧♦s ACD ❡ BDC ACD

  ❡ BDC✱ ❝♦♥❝❧✉í✲s❡ q✉❡ ♦ ♣❛r❛❧❡❧♦❣r❛♠♦ ❆❇❈❉ é ✉♠ r❡tâ♥❣✉❧♦✳ ❈♦r♦❧ár✐♦ ✸✳✹✳✶✳ ❆ ♠❡❞✐❛♥❛ r❡❧❛t✐✈❛ à ❤✐♣♦t❡♥✉s❛ ❞❡ ✉♠ tr✐â♥❣✉❧♦ r❡tâ♥❣✉❧♦ é ✐❣✉❛❧ à ♠❡t❛❞❡ ❞❛ ♠❡❞✐❞❛ ❞❛ ❤✐♣♦t❡♥✉s❛✳ ❉❡♠♦♥str❛çã♦✿ ❙❡❥❛ ABC ✉♠ tr✐â♥❣✉❧♦ r❡tâ♥❣✉❧♦ ❡♠ A✳ ❚r❛❝❡✱ ♣♦r B✱ ✉♠❛ r❡t❛ ♣❛r❛❧❡❧❛ ❛ AC ❡✱ ♣♦r C✱ ✉♠❛ r❡t❛ ♣❛r❛❧❡❧❛ ❛ AB✳ ❙❡♥❞♦ D ♦ ♣♦♥t♦ ❞❡ ✐♥t❡rs❡❝çã♦ ❞❡ss❛s

  ✸✳✹ ◗✉❛❞r✐❧át❡r♦s ◆♦tá✈❡✐s ✺✾

  C D M A B BD ❈♦♠♦✱ ♣♦r ❝♦♥str✉çã♦✱ ♦ ❧❛❞♦ AC ❞♦ tr✐â♥❣✉❧♦ é ♣❛r❛❧❡❧♦ à r❡t❛ ÐÑ ✱ ❡ AB é ✉♠❛

  BD BAC ABD tr❛♥s✈❡rs❛❧ q✉❡ ✐♥t❡rs❡❝t❛ AC ❡ ÐÑ ✱ ❡♥tã♦ ③ ❡ ④ sã♦ ❝♦rr❡s♣♦♥❞❡♥t❡s✳ ❈♦♠♦

  ✆ ✆ ✆ ✆

  ③ BAC BAC ④ ABD ABD ACD

  ✏ 90 ✱ s❡❣✉❡ q✉❡✱ ③ ✏ 180 ❡ ④ ✏ 90 ✳ ❉❛ ♠❡s♠❛ ❢♦r♠❛✱ ④ ✏ 90

  ✆ ✆

  BDC ❡✱ ❝♦♠♦ ❛ s♦♠❛ ❞♦s â♥❣✉❧♦s ✐♥t❡r♥♦s ❞♦ q✉❛❞r✐❧át❡r♦ ABDC é 360 ✱ s❡❣✉❡ q✉❡ ④ ✏ 90 ✳ P♦rt❛♥t♦ ♦ q✉❛❞r✐❧át❡r♦ ABDC é r❡tâ♥❣✉❧♦✱ ❞♦♥❞❡ AD ✏ BC ❡ ♦ ♣♦♥t♦ M ❞❡ ✐♥t❡rs❡❝çã♦ ❞❛s ❞✐❛❣♦♥❛✐s AD ❡ BC é ♦ ♣♦♥t♦ ♠é❞✐♦ ❞❡ss❡s s❡❣♠❡♥t♦s✳ ▲♦❣♦✱

  BC ✏ AD ✏ 2 ☎ AM.

  Pr♦♣♦s✐çã♦ ✸✳✹✳✻✳ ❯♠ ♣❛r❛❧❡❧♦❣r❛♠♦ é ✉♠ ❧♦s❛♥❣♦ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ t✐✈❡r ❞✐❛❣♦♥❛✐s ♣❡r♣❡♥❞✐❝✉❧❛r❡s✳ ❉❡♠♦♥str❛çã♦✿ ❈♦♥s✐❞❡r❡ ♦ q✉❛❞r✐❧át❡r♦ ABCD ❞❛ ✜❣✉r❛ ✸✳✶✶ ✉♠ ❧♦s❛♥❣♦ ❞❡ ❞✐❛❣♦♥❛✐s AC

  ❡ BD✳ D A C M B

  ❋✐❣✉r❛ ✸✳✶✶✿ ▲♦s❛♥❣♦ ABCD✳

  ✸✳✺ ▲✉❣❛r❡s ●❡♦♠étr✐❝♦s ❇ás✐❝♦s ✻✵

  ❈♦♠♦✱ ♣♦r ❞❡✜♥✐çã♦✱ AB ✏ AD✱ CB ✏ CD ❡ AC é ✉♠ ❧❛❞♦ ❝♦♠✉♠ ❛♦s tr✐â♥❣✉❧♦s ABC

  ❡ ADC✱ t❡♠♦s✱ ♣❡❧♦ t❡♦r❡♠❛ ✸✳✶✳✷✱ q✉❡ ♦s tr✐â♥❣✉❧♦s ABC ❡ ADC sã♦ ❝♦♥❣r✉❡♥t❡s✳ DAM BAM DCM BCM

  P♦rt❛♥t♦✱ s❡❣✉❡✲s❡ ❞❡ss❛ ❝♦♥❣r✉ê♥❝✐❛✱ q✉❡ ④ ✏ ④ ❡ ④ ✏ ④ ✳ BAD

  ❉❡ss❛ ❢♦r♠❛✱ AM é ❜✐ss❡tr✐③ ❞♦ â♥❣✉❧♦ ④ ❞♦ tr✐â♥❣✉❧♦ ABD✳ ❈♦♠♦ ❡ss❡ tr✐â♥❣✉❧♦ é ✐s♦s❝❡❧❡s ❝♦♠ ❜❛s❡ BD✱ s❡❣✉❡✲s❡ q✉❡ AM é ❛❧t✉r❛ r❡❧❛t✐✈❛ à ❜❛s❡ BD ❞♦ r❡❢❡r✐❞♦ tr✐â♥❣✉❧♦✳ ❈♦♥s❡q✉❡♥t❡♠❡♥t❡ M é ♦ ♣♦♥t♦ ♠é❞✐♦ ❞❡ BD ❡✱ ❛ss✐♠ ♦s tr✐â♥❣✉❧♦s AMD ❡ AMB sã♦

  ♦

  AM D ❝♦♥❣r✉❡♥t❡s✳ P♦rt❛♥t♦✱ ❛ ♠❡❞✐❞❛ ❞♦ â♥❣✉❧♦ ④ é 90 ✳ ❉♦♥❞❡ s❡❣✉❡ q✉❡ ❛ r❡t❛ BD é ♣❡r♣❡♥❞✐❝✉❧❛r à r❡t❛ AM ✏ AC✱ ♦✉ s❡❥❛✱ ♦ q✉❡ q✉❡rí❛♠♦s ♣r♦✈❛r✳

  ❘❡❝✐♣r♦❝❛♠❡♥t❡✱ s❡❥❛ ABCD ✉♠ ♣❛r❛❧❡❧♦❣r❛♠♦ ❝✉❥❛s ❛s ❞✐❛❣♦♥❛✐s AC ❡ BD sã♦ ♣❡r✲ ♣❡♥❞✐❝✉❧❛r❡s✳ ❈♦♠♦ ABCD é ✉♠ ♣❛r❛❧❡❧♦❣r❛♠♦✱ t❡♠♦s q✉❡ s✉❛s ❞✐❛❣♦♥❛✐s s❡ ❝♦rt❛♠ ❛♦ ♠❡✐♦✳ ❉❡ss❛ ❢♦r♠❛✱ ♥♦ tr✐â♥❣✉❧♦ ADC✱ ♦ s❡❣♠❡♥t♦ DM é ♠❡❞✐❛♥❛ ❡ ❛❧t✉r❛ r❡❧❛t✐✈❛ ❛♦ ❧❛❞♦ AC✳ ❉❡ss❛ ❢♦r♠❛✱ t❡♠♦s q✉❡ AD ✏ CD✳ P♦r ♦✉tr♦ ❧❛❞♦✱ AD ✏ BC ❡ AB ✏ CD ❡✱ ♣♦rt❛♥t♦ ♦ ♣❛r❛❧❡❧♦❣r❛♠♦ ABCD é ✉♠ ❧♦s❛♥❣♦✳ ❉❡✜♥✐çã♦ ✸✳✹✳✽✳ ❯♠ q✉❛❞r✐❧át❡r♦ é ✉♠ q✉❛❞r❛❞♦ q✉❛♥❞♦ ❢♦r ✉♠ r❡tâ♥❣✉❧♦ ❡ ✉♠ ❧♦s❛♥❣♦✱ s✐♠✉❧t❛♥❡❛♠❡♥t❡✳

  ❆ss✐♠✱ q✉❛❞r❛❞♦s sã♦ q✉❛❞r✐❧át❡r♦s ❞❡ â♥❣✉❧♦s ✐❣✉❛✐s ❡ ❧❛❞♦s ✐❣✉❛✐s✳ ❙✉❛s ❞✐❛❣♦♥❛✐s

  ✆

  sã♦ t❛♠❜é♠ ✐❣✉❛✐s ❡ ♣❡r♣❡♥❞✐❝✉❧❛r❡s✱ s❡ ✐♥t❡rs❡❝t❛♠ ❛♦ ♠❡✐♦ ❡ ❢♦r♠❛♠ â♥❣✉❧♦s ❞❡ 45 ❝♦♠ ♦s ❧❛❞♦s ❞♦ q✉❛❞r✐❧át❡r♦✳

  ✸✳✺ ▲✉❣❛r❡s ●❡♦♠étr✐❝♦s ❇ás✐❝♦s

  ❉❡✜♥✐çã♦ ✸✳✺✳✶✳ ❉❛❞❛ ✉♠❛ ♣r♦♣r✐❡❞❛❞❡ r❡❧❛t✐✈❛ ❛ ♣♦♥t♦s ❞♦ ♣❧❛♥♦✱ ♦ ❧✉❣❛r ❣❡♦♠étr✐❝♦ ✭▲●✮ ❞♦s ♣♦♥t♦s q✉❡ ♣♦ss✉❡♠ ❛ ♣r♦♣r✐❡❞❛❞❡ é ♦ s✉❜❝♦♥❥✉♥t♦ ❞♦ ♣❧❛♥♦ q✉❡ s❛t✐s❢❛③ ❛s ❞✉❛s ❝♦♥❞✐çõ❡s ❛ s❡❣✉✐r✿

  ✭✐✮ ❚♦❞♦ ♣♦♥t♦ ❞♦ s✉❜❝♦♥❥✉♥t♦ ♣♦ss✉✐ ❛ ❞❛❞❛ ♣r♦♣r✐❡❞❛❞❡❀ ✭✐✐✮ ❚♦❞♦ ♣♦♥t♦ ❞♦ ♣❧❛♥♦ q✉❡ ♣♦ss✉✐ ❛ ♣r♦♣r✐❡❞❛❞❡ ♣❡rt❡♥❝❡ ❛♦ s✉❜❝♦♥❥✉♥t♦✳

  ♣A Bq ❉❡✜♥✐çã♦ ✸✳✺✳✷✳ ❉❛❞♦ ✉♠ s❡❣♠❡♥t♦ ❞❡ r❡t❛ AB✱ s❡❥❛ M ✏ ♦ ♣♦♥t♦ ♠é❞✐♦

  2 ❞❡ss❡ s❡❣♠❡♥t♦✳ ❈❤❛♠❛♠♦s ❞❡ ♠❡❞✐❛tr✐③ ❞♦ s❡❣♠❡♥t♦ AB ❛ r❡t❛ r ♣❛ss❛♥❞♦ ♣♦r M ❡ q✉❡ é ♣❡r♣❡♥❞✐❝✉❧❛r à r❡t❛ AB✳ Pr♦♣♦s✐çã♦ ✸✳✺✳✶✳ ❉❛❞♦s ♦s ♣♦♥t♦s A ❡ B ♥♦ ♣❧❛♥♦✱ ❛ ♠❡❞✐❛tr✐③ ❞♦ s❡❣♠❡♥t♦ AB é ♦ LG

  ❞♦s ♣♦♥t♦s ❞♦ ♣❧❛♥♦ q✉❡ ❡q✉✐❞✐st❛♠ ❞❡ A ❡ ❞❡ B ❡ ♦s ♣♦♥t♦s q✉❡ ❡q✉✐❞✐st❛♠ ❞❡ ❞♦✐s

  ✸✳✺ ▲✉❣❛r❡s ●❡♦♠étr✐❝♦s ❇ás✐❝♦s ✻✶

  ❉❡♠♦♥str❛çã♦✿ ❙❡❥❛ M ♦ ♣♦♥t♦ ♠é❞✐♦ ❡ r ❛ ♠❡❞✐❛tr✐③ ❞❡ AB✳ ❉❛❞♦ P ♣❡rt❡♥❝❡♥t❡ ❛ r❡t❛ r✱ t❡♠♦s q✉❡ ♦ tr✐â♥❣✉❧♦ ABP é ✐sós❝❡❧❡s ✉♠❛ ✈❡③ q✉❡ P M é ♠❡❞✐❛♥❛ ❡ ❛❧t✉r❛ ❞❡ss❡ tr✐â♥❣✉❧♦✳ r

  P B M A ❘❡❝✐♣r♦❝❛♠❡♥t❡✱ s❡❥❛ P ✉♠ ♣♦♥t♦ ♥♦ ♣❧❛♥♦ t❛❧ q✉❡ AP ✏ BP ✳ ❊♥tã♦ ♦ tr✐â♥❣✉❧♦

  ABP é ✐sós❝❡❧❡s ❞❡ ❜❛s❡ AB✳ ❙❡❣✉❡ q✉❡ ❛ ♠❡❞✐❛♥❛ ❡ ❛ ❛❧t✉r❛ r❡❧❛t✐✈❛s ❛ AB ❝♦✐♥❝✐❞❡♠✳

  ❈♦♠♦ ❛ ♠❡❞✐❛♥❛ ❞♦ tr✐â♥❣✉❧♦ AP B r❡❧❛t✐✈❛ ❛ ❜❛s❡ AB é ♦ s❡❣♠❡♥t♦ P M✱ s❡❣✉❡ q✉❡ P M P M

  é ♣❡r♣❡♥❞✐❝✉❧❛r ❛♦ s❡❣♠❡♥t♦ AB✳ ▲♦❣♦ ÐÝÑ é ❛ ♠❡❞✐❛tr✐③ ❞❡ AB✳ AOB

  Pr♦♣♦s✐çã♦ ✸✳✺✳✷✳ ❙❡❥❛ P ✉♠ ♣♦♥t♦ ❞❡ ✉♠ ❞❡t❡r♠✐♥❛❞♦ â♥❣✉❧♦ ③ ✳ ❙❡ P ♣❡rt❡♥❝❡ ❛ ✁ ✠ ✁ ✠

  ÝÑ ÝÑ P, Ý OA P, Ý OB

  ❡ss❡ â♥❣✉❧♦✱ ❡♥tã♦ d ✏ d s❡✱ ❡ s♦♠❡♥t❡ s❡✱ P ♣❡rt❡♥❝❡ ❛ ❜✐ss❡tr✐③ ❞♦ AOB

  â♥❣✉❧♦ ③ ✳ AOB

  ❉❡♠♦♥str❛çã♦✿ ❙✉♣♦♥❤❛ q✉❡ P s❡❥❛ ✉♠ ♣♦♥t♦ ♥♦ ✐♥t❡r✐♦r ❞♦ â♥❣✉❧♦ ③ ✱ t❛❧ q✉❡ P M

  ✏ P N✱ ♦♥❞❡ M ❡ N sã♦ ♦s ♣és ❞❛s ♣❡r♣❡♥❞✐❝✉❧❛r❡s ❜❛✐①❛❞❛s ❞❡ P r❡s♣❡❝t✐✈❛♠❡♥t❡ OA OB

  às s❡♠✐rr❡t❛s ÝÝÑ ❡ ÝÝÑ ✳ A M P O N B

  ❊♥tã♦✱ ♣❡❧♦ t❡♦r❡♠❛ ✸✳✶✳✷✱ ♦s tr✐â♥❣✉❧♦s r❡tâ♥❣✉❧♦s MOP ❡ NOP sã♦ ❝♦♥❣r✉❡♥t❡s✱ M OP N OP

  ✉♠❛ ✈❡③ q✉❡ OP é ❧❛❞♦ ❝♦♠✉♠ ❡ P M ✏ P N✳ ❉❡ss❛ ❢♦r♠❛✱ ④ ✏ ④ ❡✱ ♣♦rt❛♥t♦✱ AOB

  ♦ ♣♦♥t♦ P ♣❡rt❡♥❝❡ ❛ ❜✐ss❡tr✐③ ❞♦ â♥❣✉❧♦ ③ ✳ AOB

  ❘❡❝✐♣r♦❝❛♠❡♥t❡✱ s✉♣♦♥❤❛ q✉❡ P ♣❡rt❡♥❝❡ à ❜✐ss❡tr✐③ ❞♦ â♥❣✉❧♦ ③ ❡ s❡❥❛♠ M ❡ N✱ OA OB r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ♦s ♣és ❞❛s ♣❡r♣❡♥❞✐❝✉❧❛r❡s ❜❛✐①❛❞❛s ❞❡ P às s❡♠✐rr❡t❛s ÝÝÑ ❡ ÝÝÑ ✳ ❖s

  ✸✳✻ ❚❡♦r❡♠❛ ❞❡ ❚❤❛❧❡s ✻✷ tr✐â♥❣✉❧♦s r❡tâ♥❣✉❧♦s OMP ❡ ONP sã♦ ❝♦♥❣r✉❡♥t❡s ♣❡❧♦ ❝♦r♦❧ár✐♦ ✸✳✷✳✸✱ ✉♠❛ ✈❡③ q✉❡

  ✆

  ④ M OP N OP OM P ON P

  ✏ ④ ✱ ④ ✏ ④ ✏ 90 ❡ OP é ❧❛❞♦ ❝♦♠✉♠ ❛♦s ♠❡s♠♦s✳ ❉❛í✱ P M ✏ P N✱ ✁ ✠ ✁ ✠

  ÝÑ ÝÑ P, Ý OA P, Ý OB

  ♦✉ s❡❥❛✱ d ✏ d ✳

  ✸✳✻ ❚❡♦r❡♠❛ ❞❡ ❚❤❛❧❡s

  ◆❡st❛ s❡çã♦ ❞♦ tr❛❜❛❧❤♦ ✐r❡♠♦s ✉t✐❧✐③❛r ✉♠ ✐♠♣♦rt❛♥t❡ ❝♦♥❥✉♥t♦ ❞❡ ❢❡rr❛♠❡♥t❛s q✉❡ sã♦ ✐♠♣r❡s❝✐♥❞í✈❡✐s ❡ ♥♦s ♣❡r♠✐t✐rã♦ r❡❛❧✐③❛r ♦ ❡st✉❞♦ s✐st❡♠át✐❝♦ ❞❛s ❛s♣❡❝t♦s ♠étr✐❝♦s ❞❛ ●❡♦♠❡tr✐❛ ❊✉❝❧✐❞✐❛♥❛✱ ❝♦♠♣❛r❛♥❞♦ r❛③õ❡s ❞❡ ❝♦♠♣r✐♠❡♥t♦s ❞❡ s❡❣♠❡♥t♦s✳

  ❈♦♥s✐❞❡r❡ ❛ s❡❣✉✐♥t❡ s✐t✉❛çã♦✿ três r❡t❛s ♣❛r❛❧❡❧❛s r✱ s ❡ t sã♦ ✐♥t❡rs❡❝t❛❞❛s ♣❡❧❛s r❡t❛s

  ✶

  u ❡ u ✱ ❝♦♥❢♦r♠❡ ❛ ✜❣✉r❛ ✸✳✶✷✳

  ✶

  u u

  ✶

  r A A

  ✶

  B B s

  ✶

  C C t ❋✐❣✉r❛ ✸✳✶✷✿ r❡t❛s ♣❛r❛❧❡❧❛s ❝♦rt❛❞❛s ♣♦r tr❛♥s✈❡rs❛✐s ❝♦♠ AB ✏ BC

  ❉✐✈✐❞✐♥❞♦ ♦s s❡❣♠❡♥t♦s AB ❡ BC ❡♠ m ❡ n ♣❛rt❡s ✐❣✉❛✐s✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ♥❛ ✜❣✉r❛ AB m

  ✸✳✶✷✱ ❡♥tã♦✱ t❡rí❛♠♦s ✏ ✱ ❝♦♠ m ❡ n P N✳ ❊ ✉t✐❧✐③❛♥❞♦ ✈ár✐❛s ✈❡③❡s ❛s ❛♣❧✐❝❛çõ❡s BC n

  ❞♦ t❡♦r❡♠❛ ❞❛ ❜❛s❡ ♠é❞✐❛ ❞❡ ✉♠ tr❛♣é③✐♦✱ ❣❛r❛♥t✐♠♦s q✉❡

  ✶ ✶

  AB A B m .

  ✏ ✏

  ✶ ✶

  BC B C n ❉❛í✱ ❝♦♥❝❧✉í♠♦s q✉❡

  ✶ ✶

  AB A B ✏

  ✶ ✶

  BC B C é ✈á❧✐❞❛ s❡♠♣r❡ q✉❡ ♦ ♣r✐♠❡✐r♦ ♦✉ ♦ s❡❣✉♥❞♦ ♠❡♠❜r♦ ❢♦r ✉♠ ♥ú♠❡r♦ r❛❝✐♦♥❛❧✳

  ✶ ✶ ✶ ✶

  B C ❙✉♣♦♥❤❛✱ ❛❣♦r❛ q✉❡ AB ✏ BC✱ ❡♥tã♦ t❡r❡♠♦s q✉❡ A ✏ B ✳ ❖♥❞❡✱ ♣♦❞❡✲s❡

  ♦❜s❡r✈❛r q✉❡

  ✶ ✶

  AB A B ✏ ✏ 1.

  ✶ ✶

  BC B C AB ❈♦♥s✐❞❡r❛♥❞♦ AB ✘ BC ❡ ❛✐♥❞❛ q✉❡ s❡❥❛ ✉♠ ♥ú♠❡r♦ r❛❝✐♦♥❛❧✱ ♣❡❧♦ ❡①❡♠♣❧♦ ♥❛

  ✸✳✻ ❚❡♦r❡♠❛ ❞❡ ❚❤❛❧❡s ✻✸

  ✜❣✉r❛ ✸✳✶✸✱ t❡♠♦s q✉❡ AB

  3 . ✏

  BC

  2

  ✶

  u u

  ✶

  A A r

  ✶

  X X

  ✶

  Y Y

  ✶

  B B s

  ✶

  Z Z

  ✶

  C C t

  ❋✐❣✉r❛ ✸✳✶✸✿ r❡t❛s ♣❛r❛❧❡❧❛s ❝♦rt❛❞❛s ♣♦r tr❛♥s✈❡rs❛✐s ❝♦♠ AB ✘ BC ❉✐✈✐❞✐♥❞♦✱ ❛ss✐♠✱ ♦s s❡❣♠❡♥t♦s AB ❡ BC ❡♠ três ❡ ❞✉❛s ♣❛rt❡s ✐❣✉❛✐s r❡s♣❡❝t✐✈❛♠❡♥t❡✱

  ♦❜t❡♠♦s X✱ Y ❡ Z ❡♠ u✱ t❛✐s q✉❡ AX

  ✏ XY ✏ Y B ✏ BZ ✏ ZC

  ✶

  ❡ tr❛ç❛♥❞♦ ♣♦r X✱ Y ❡ Z ♣❛r❛❧❡❧❛s às r❡t❛s r✱ s ❡ t✱ ❛s q✉❛✐s ✐♥t❡rs❡❝t❛♠ u r❡s♣❡❝t✐✈❛♠❡♥t❡

  ✶ ✶ ✶

  ❡♠ X ✱ Y ❡ Z ✱ ❡♥tã♦ ❛♣❧✐❝❛♥❞♦ ❛❧❣✉♠❛s ✈❡③❡s ♦ t❡♦r❡♠❛ ❞❛ ❜❛s❡ ♠é❞✐❛ ❞❡ ✉♠ tr❛♣é③✐♦✱ t❡♠♦s

  

✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶

  A

  X Y B Z C ✏ X ✏ Y ✏ B ✏ Z

  ❡✱ ♣♦rt❛♥t♦✱

  ✶ ✶

  AB A B

  3 . ✏ ✏

  ✶ ✶

  BC B C

  2 ❉✐❛♥t❡ ❞♦ ❡①♣♦st♦✱ ✉♠❛ ♣❡r❣✉♥t❛ ♥❛t✉r❛❧ ❛ ❢❛③❡r ❛❣♦r❛ é ❛ s❡❣✉✐♥t❡✿ s❡rá q✉❡ ❛ r❡❧❛çã♦ ♣❛r❛ ❛ r❛③ã♦ ❡♥tr❡ ♦s s❡❣♠❡♥t♦s t❛♠❜é♠ é ✈á❧✐❞❛ ♣❛r❛ q✉❛♥❞♦ ✉♠ ❞♦s ♠❡♠❜r♦s

  ❞❛ ✐❣✉❛❧❞❛❞❡ ❢♦r ✉♠ ♥ú♠❡r♦ ✐rr❛❝✐♦♥❛❧❄ ❆ r❡s♣♦st❛ é s✐♠✳ ❊ ♣❛r❛ ♦ ❧❡✐t♦r ❡♥t❡♥❞❡r ♦ s✐♠ ❞❛❞♦ ❛ ♣❡r❣✉♥t❛ ✐♥✐❝✐❛❧✱ ❞❡✐①❛r❡♠♦s ❝♦♠♦ r❡❢❡rê♥❝✐❛ ❬✻❪✱ ♣✳ ✶✸✾✱ ♣❛r❛ q✉❡ ♦ ♠❡s♠♦ ♣♦ss❛ ♣❡sq✉✐s❛r s♦❜r❡ t❛❧ ❛✜r♠❛çã♦✳

  ❊ss❛ ❜r❡✈❡ ❞✐s❝✉ssã♦ q✉❡ ❛❝❛❜❛♠♦s ❞❡ t❡r r❡✢❡t❡ ✉♠ ❞♦s r❡s✉❧t❛❞♦s ❢✉♥❞❛♠❡♥t❛✐s ❞❛

  ✷

  ●❡♦♠❡tr✐❛ ❊✉❝❧✐❞✐❛♥❛ ♣❧❛♥❛✱ ❞❡♥♦♠✐♥❛❞♦ ❚❡♦r❡♠❛ ❞❡ ❚❤❛❧❡s ✱ ♦ q✉❛❧ ✐r❡♠♦s ❡♥✉♥❝✐❛r ❛ s❡❣✉✐r✳

  ✶ ✶

  Pr♦♣♦s✐çã♦ ✸✳✻✳✶✳ ❙❡❥❛♠ r✱ s ❡ t r❡t❛s ♣❛r❛❧❡❧❛s✳ ❊s❝♦❧❤❡♠♦s ♣♦♥t♦s A✱ A P r✱ B✱ B P s

  ✶ ✶ ✶ ✶

  ❡ C✱ C P t✱ ❞❡ ♠♦❞♦ q✉❡ A✱ B✱ C ❡ A ✱ B ✱ C s❡❥❛♠ ❞♦✐s t❡r♥♦s ❞❡ ♣♦♥t♦s ❝♦❧✐♥❡❛r❡s✳

  ✸✳✻ ❚❡♦r❡♠❛ ❞❡ ❚❤❛❧❡s ✻✹

  ❊♥tã♦

  ✶ ✶

  AB A B .

  ✏

  ✶ ✶

  BC B C ❚ã♦ ✐♠♣♦rt❛♥t❡ q✉❛♥t♦ ♦ t❡♦r❡♠❛ ❞❡ ❚❤❛❧❡s é ❛ s✉❛ r❡❝í♣r♦❝❛ ♣❛r❝✐❛❧✱ ❛ q✉❛❧ s❡rá

  ❡♥✉♥❝✐❛❞❛ ❛ s❡❣✉✐r✱ t❛♠❜é♠ ❞❡✈✐❞❛ ❛ ❚❤❛❧❡s ❞❡ ▼✐❧❡t♦✳

  ✶ ✶

  ❈♦r♦❧ár✐♦ ✸✳✻✳✶✳ ❙❡❥❛♠ ❞❛❞♦s✱ ♥♦ ♣❧❛♥♦✱ r❡t❛s r✱ s ❡ ♣♦♥t♦s A✱ A P r✱ B✱ B P s✱ ❝♦♠

  ✶ ✶

  AB A B ÐÝÑ

  ÐÑ ✶ ✶ AB A B

  ❳ ✏ tC✉✳ ❙❡ ✏ ✱ ❡♥tã♦ ❛ r❡t❛ r é ♣❛r❛❧❡❧❛ à r❡t❛ s✳

  ✶

  BC B C t C

  ✶

  B B s

  ✶ ✷

  s B

  ✶

  A A r

  ❉❡♠♦♥str❛çã♦✿ ❉❛❞♦ ♦ ♣♦♥t♦ B ❡ ❛ r❡t❛ r✱ s❛❜❡♠♦s ❞♦s ❛①✐♦♠❛s ❞❡ ❊✉❝❧✐❞❡s q✉❡ ❡①✐st❡

  ✶ ✷

  ✉♠❛ ú♥✐❝❛ r❡t❛ s ♣❛ss❛♥❞♦ ♣♦r B ❡ q✉❡ é ♣❛r❛❧❡❧❛ ❛ r❡t❛ r✳ ❙❡❥❛ B ♦ ♣♦♥t♦ ❞❡ ✐♥t❡rs❡❝çã♦

  ✶ ✶ ✶

  B ❞❛ r❡t❛ s ❝♦♠ ❛ r❡t❛ A ✳ ❚r❛ç❛♥❞♦ ✉♠❛ r❡t❛ t q✉❡ ♣❛ss❛ ♣❡❧♦ ♣♦♥t♦ C ❡ é ♣❛r❛❧❡❧❛ ❛s

  ✶ ✷

  AB A B

  ✶

  r❡t❛s r ❡ s ✱ ♦❜t❡♠♦s✱ ♣❡❧♦ ❚❡♦r♠❛ ❞❡ ❚❛❧❡s✱ q✉❡ ✏ ✳ P♦r ♦✉tr♦ ❧❛❞♦✱ ❞❡❝♦rr❡

  ✷

  BC B C ❞❡ ♥♦ss❛s ❤✐♣♦t❡s❡s q✉❡✿

  

✶ ✶ ✶ ✷

  A B A B .

  ✏

  

✶ ✷

  B C B C

  ✶ ✷ ✶

  ❉♦♥❞❡ s❡❣✉❡ q✉❡ B ✏ B ✱ ♦✉ s❡❥❛✱ s ✏ s ❡✱ ♣♦rt❛♥t♦✱ s k r✳ ❆ s❡❣✉✐r ❡♥✉♥❝✐❛r❡♠♦s ❡ ❞❡♠♦♥str❛r❡♠♦s ✉♠❛ ❛♣❧✐❝❛çã♦ ✐♥t❡r❡ss❛♥t❡ ❞♦ ❚❡♦r❡♠❛ ❞❡

  ❚❛❧❡s✱ ❝♦♥❤❡❝✐❞♦ ♥❛ ❧✐t❡r❛t✉r❛ ❝♦♠♦ ❚❡♦r❡♠❛ ❞❛ ❇✐ss❡tr✐③✳ Pr♦♣♦s✐çã♦ ✸✳✻✳✷✳ ❙❡❥❛ ABC ✉♠ tr✐â♥❣✉❧♦ t❛❧ q✉❡ ♦s ❝♦♠♣r✐♠❡♥t♦s ❞❡ AB ❡ AC sã♦ ❞✐❢❡r❡♥t❡s✳ ▲♦❣♦✱

  ✸✳✻ ❚❡♦r❡♠❛ ❞❡ ❚❤❛❧❡s ✻✺

  ✭✐✮ ❙❡ P é ♦ ♣é ❞❛ ❜✐ss❡tr✐③ ✐♥t❡r♥❛ ❡ Q é ♦ ♣é ❞❛ ❜✐sst❡r✐③ ❡①t❡r♥❛ r❡❧❛t✐✈❛s ❛♦ ❧❛❞♦ BC

  ✱ ❡♥tã♦ BP BQ BA .

  ✏ ✏ P C QC AC

  ✭✐✐✮ ❙❡ AB ✏ c✱ AC ✏ b ❡ BC ✏ a✱ ❡♥tã♦ t❡♠♦s ✩

  ✩ ac ac

  ✬ BQ

  ✬ BP ✬ ✏ ✏

  ✬ ✬ ✬ ✬ b c ⑤b ✁ c⑤ ✫ ✫

  ❡ ✬ ✬ ab ab ✬ ✬ ✬ ✬ ✪ P C . ✬ QC .

  ✏ ✏ ✪ b c

  ⑤b ✁ c⑤ A C Q B P BQ BA

  ❉❡♠♦♥str❛çã♦✿ P❛r❛ ♦ ✐t❡♠ ✭✐✮✱ ♠♦str❡♠♦s q✉❡ ✏ ✳ ❙✉♣♦♥❤❛ q✉❡ AB ➔ AC✳ QC AC

  ✶

  AQ ❚r❛❝❡ ♣❡❧♦ ♣♦♥t♦ B✱ ❛ ♣❛r❛❧❡❧❛ à r❡t❛ ÐÑ ❡ ♠❛rq✉❡ s❡✉ ♣♦♥t♦ ❞❡ ✐♥t❡rs❡çã♦ B ❝♦♠ ♦ ❧❛❞♦ AC

  ❞♦ tr✐â♥❣✉❧♦✳

  X A

  ✶

  B Q B C ÐÝÑ

  ✶

  QA BB AQ BAX k ❈♦♠♦ ÐÑ ❡ ÐÑ é ❜✐ss❡tr✐③ ❞❡ ④ ✱ ♦❜t❡♠♦s

  ✶ ✶

  ④ ABB BAQ QAX BB A.

  ✏ ③ ✏ ④ ✏ ④

  ✶ ✶ ✶

  P♦rt❛♥t♦✱ ♦ tr✐â♥❣✉❧♦ ABB é ✐sós❝❡❧❡s ❞❡ ❜❛s❡ BB ✱ ❞❡ ♠❛♥❡✐r❛ q✉❡ AB ✏ AB✳ ❆❣♦r❛✱ ÐÝÑ

  ✶

  AQ BB CQ ❛♣❧✐❝❛♥❞♦ ♦ t❡♦r❡♠❛ ❞❡ ❚❤❛❧❡s às r❡t❛s ♣❛r❛❧❡❧❛s ÐÑ ❡ ✱ ✐♥t❡rs❡❝t❛❞❛s ♣❡❧❛s r❡t❛s ÐÑ

  AC ❡ ÐÑ ✱ ♦❜t❡♠♦s

  ✶

  BQ AB BA .

  ✏ ✏ QC AC AC

  ✸✳✼ ❙❡♠❡❧❤❛♥ç❛ ❞❡ ❚r✐â♥❣✉❧♦s ✻✻

  BP BA ❆ ♣r♦✈❛ ❞❡ q✉❡ ❛ ✐❣✉❛❧❞❛❞❡ ✏ é s❛t✐s❢❡✐t❛✱ ❞❡✐①❛r❡♠♦s ♣❛r❛ ♦ ❧❡✐t♦r ❝♦♠♦ ❡①❡r❝í❝✐♦

  P C AC ❡ ♦ ❝❛s♦ ❡♠ q✉❡ AB → AC ♣♦❞❡ s❡r tr❛t❛❞♦ ❞❡ ♠♦❞♦ ❛♥á❧♦❣♦✳

  ◗✉❛♥t♦ ❛♦ ✐t❡♠ ✭✐✐✮✱ s❡❣✉❡ ✐♠❡❞✐❛t❛♠❡♥t❡ ❞♦ ✐t❡♠ ✭✐✮✳ ❙❡♥❞♦ BP ✏ x ❡ CP ✏ y✱ x c t❡♠♦s x y ✏ a ❡✱ ♣❡❧♦ ✐t❡♠ ✭✐✮✱ ✏ ✳ y b

  A c b x y

  B C P ❘❡s♦❧✈❡♥❞♦ ♦ s✐st❡♠❛ ❞❡ ❞✉❛s ❡q✉❛çõ❡s

  ✩ x y ✬

  ✏ a ✬ ✫ x c ✬ ✬ ✪

  ✏ y b ac ab

  ♦❜t❡♠♦s x ✏ ❡ y ✏ ✳ b c b c ❆s ❞❡♠❛✐s r❡❧❛çõ❡s ❞♦ ✐t❡♠ ✭✐✐✮ sã♦ ♣r♦✈❛❞❛s ❞❡ ♠♦❞♦ ❛♥á❧♦❣♦✳

  ✸✳✼ ❙❡♠❡❧❤❛♥ç❛ ❞❡ ❚r✐â♥❣✉❧♦s

  ◆❡st❛ s❡çã♦ ✐r❡♠♦s ❡st✉❞❛r ♦ ❝♦♥❝❡✐t♦ ❞❡ s❡♠❡❧❤❛♥ç❛ ❞❡ tr✐â♥❣✉❧♦s✱ ♦ q✉❛❧ ❣❡♥❡r❛❧✐③❛ ❛ ♥♦çã♦ ❞❡ ❝♦♥❣r✉ê♥❝✐❛ ❡ ❛❧❣✉♠❛s ❛♣❧✐❝❛çõ❡s ❞❡ss❡ ❝♦♥❝❡✐t♦✳ ❉❡✜♥✐çã♦ ✸✳✼✳✶✳ ❉♦✐s tr✐â♥❣✉❧♦s sã♦ s❡♠❡❧❤❛♥t❡s q✉❛♥❞♦ ❡①✐st✐r ✉♠❛ ❝♦rr❡s♣♦❞ê♥❝✐❛ ❜✐✉♥í✈♦❝❛ ❡♥tr❡ ♦s ✈ért✐❝❡s ❞❡ ✉♠ ❡ ♦✉tr♦ tr✐â♥❣✉❧♦✱ ❞❡ t❛❧ ❢♦r♠❛ q✉❡ ♦s â♥❣✉❧♦s ❡♠ ✈ért✐❝❡s ❝♦rr❡s♣♦♥❞❡♥t❡s s❡❥❛♠ ✐❣✉❛✐s ❡ ❛ r❛③ã♦ ❡♥tr❡ ♦s ❝♦♠♣r✐♠❡♥t♦s ❞♦s ❧❛❞♦s ❝♦rr❡s♣♦♥❞❡♥t❡s s❡❥❛ s❡♠♣r❡ ❛ ♠❡s♠❛✳

  ◗✉❛♥❞♦ ♣✉❞❡r♠♦s ❞✐❧❛t❛r ❡✴♦✉ ❣✐r❛r ❡✴♦✉ r❡✢❡t✐r ❡✴♦✉ tr❛♥s❧❛❞❛r ✉♠ ❞❡❧❡s✱ ♦❜t❡♥❞♦ ♦ ♦✉tr♦ ❛♦ ✜♥❛❧ ❞❡ t❛✐s ♦♣❡r❛çõ❡s✱ ❡♥tã♦ ♣♦❞❡♠♦s ❢❛❧❛r q✉❡ ♦s tr✐â♥❣✉❧♦s sã♦ s❡♠❡❧❤❛♥t❡s✳

  ❆ s❡❣✉✐r✱ ✐♥tr♦❞✉③✐r❡♠♦s ♦ ❝♦♥❝❡✐t♦ ❞❡ r❛③ã♦ ❞❡ s❡♠❡❧❤❛♥ç❛ ❡ ❞❡♣♦✐s ❡♥✉♥❝✐❛r❡♠♦s ♦s ❝❛s♦s ❞❡ s❡♠❡❧❤❛♥ç❛ ❞❡ tr✐â♥❣✉❧♦s ✉s✉❛✐s✳

  ✶ ✶ ✶ ✶

  B C ❙❡❥❛♠ ABC ❡ A ❞♦✐s tr✐â♥❣✉❧♦s ♦♥❞❡ ❤á ❛ ❝♦rr❡s♣♦♥❞ê♥❝✐❛ ❞♦s ✈ért✐❝❡s A Ø A ✱

  ✶ ✶ ✶ ✶ ✶

  B A A B B C C .

  Ø B ✱ C Ø C ✱ ♦♥❞❡ ♣ ✏ ♣ ✱ ♣ ✏ ① ❡ ♣ ✏ ①

  ✸✳✼ ❙❡♠❡❧❤❛♥ç❛ ❞❡ ❚r✐â♥❣✉❧♦s ✻✼

  A

  ✶

  A

  ✶

  kb kc

  ✶ ✶

  c b

  ✶ ✶ ✶

  B ✶ C C B a ka

  ❊♥tã♦ ❡①✐st❡ ✉♠ ♥ú♠❡r♦ k r❡❛❧ ❡ ♣♦s✐t✐✈♦ t❛❧ q✉❡ AB BC AC ✏ ✏ ✏ k.

  ✶ ✶ ✶ ✶ ✶ ✶

  A B B C A C ❚❛❧ r❡❛❧ ♣♦s✐t✐✈♦ k é ❞❡♥♦♠✐♥❛❞♦ r❛③ã♦ ❞❡ s❡♠❡❧❤❛♥ç❛ ❡♥tr❡ ♦s tr✐â♥❣✉❧♦s ABC ❡

  ✶ ✶ ✶ ✶ ✶ ✶

  A B C B C ✱ ♥❡ss❛ ♦r❞❡♠✳ ❊s❝r❡✈❡♠♦s ABC ✒ A ♣❛r❛ ✐♥❞✐❝❛r q✉❡ ♦s tr✐â♥❣✉❧♦s ABC ❡

  ✶ ✶ ✶ ✶ ✶ ✶

  A B C sã♦ s❡♠❡❧❤❛♥t❡s ❝♦♠ ❛ ❝♦rr❡s♣♦♥❞ê♥❝✐❛ A Ø A ✱ B Ø B ✱ C Ø C ✳ ❆ ♣r♦♣♦s✐çã♦ ❛ s❡❣✉✐r ❡♥✉♥❝✐❛ ♦ ♣r✐♠❡✐r♦ ❝❛s♦ ❞❡ s❡♠❡❧❤❛♥ç❛ ❞❡ tr✐â♥❣✉❧♦s ❡ s❡rá ❝♦✲

  ♥❤❡❝✐❞♦ ❝♦♠♦ ♦ ❝❛s♦ ▲▲▲✳

  ✶ ✶ ✶

  B C Pr♦♣♦s✐çã♦ ✸✳✼✳✶✳ ❙❡❥❛♠ ABC ❡ A tr✐â♥❣✉❧♦s ♥♦ ♣❧❛♥♦ t❛✐s q✉❡

  AB BC AC .

  ✏ ✏

  

✶ ✶ ✶ ✶ ✶ ✶

  A B B C A C

  ✶ ✶ ✶ ✶ ✶

  ❊♥tã♦ ∆ABC ✒ ∆A ✱ ❝♦♠ ❛ ❝♦rr❡s♣♦♥❞ê♥❝✐❛ ❞❡ ✈ért✐❝❡s A Ø A ✱ B Ø B ✱

  ✶ ✶ ✶ ✶

  C A A B B C C .

  Ø C ✳ ❊♠ ♣❛rt✐❝✉❧❛r✱ ♣ ✏ ♣ ✱ ♣ ✏ ① ❡ ♣ ✏ ①

  ✶ ✶ ✶

  B C ❉❡♠♦♥str❛çã♦✿ ❙❡❥❛♠ ABC ❡ A ♦s tr✐â♥❣✉❧♦s ❡♠ q✉❡stã♦✳

  A

  ✶

  A

  ✶ ✶

  B C C B

  ✶ ✶ ✶ ✶

  B C ❙❡♥❞♦ k ♦ ✈❛❧♦r ❝♦♠✉♠ ❞❛s r❛③õ❡s ❞♦ ❡♥✉♥❝✐❛❞♦✱ t❡♠♦s AB ✏ k ☎ A ✱ BC ✏ k ☎ B

  ✶ ✶ ✷

  C ❡ AC ✏ k ☎ A ✳ ❙✉♣♦♥❤❛✱ s❡♠ ♣❡r❞❛ ❞❡ ❣❡♥❡r❛❧✐❞❛❞❡✱ k → 1 ❡ ♠❛rq✉❡ ♦ ♣♦♥t♦ B P AB

  ✷ ✶ ✶ B .

  t❛❧ q✉❡ AB ✏ A

  ✸✳✼ ❙❡♠❡❧❤❛♥ç❛ ❞❡ ❚r✐â♥❣✉❧♦s ✻✽

  A

  

✷ ✷

  C B B D C ❋✐❣✉r❛ ✸✳✶✹✿ ♦ ❝❛s♦ ❞❡ s❡♠❡❧❤❛♥ç❛ ▲▲▲

  ✷ ✷

  ❙❡♥❞♦ C ❛ ✐♥t❡rs❡çã♦✱ ❝♦♠ ♦ ❧❛❞♦ AC✱ ❞❛ r❡t❛ q✉❡ ♣❛ss❛ ♣♦r B ❡ é ♣❛r❛❧❡❧❛ ❛♦ ❧❛❞♦ BC

  ✱ s❡❣✉❡ ❞♦ t❡♦r❡♠❛ ❞❡ ❚❤❛❧❡s q✉❡

  ✷ ✷

  AC AB

  1 ,

  ✏ ✏ AC AB k

  1

  ✷ ✶ ✶ C .

  ❞❡ ♠❛♥❡✐r❛ q✉❡ AC ✏ ☎ AC ✏ A k

  ✷

  ❚r❛❝❡✱ ❛❣♦r❛✱ ✉♠❛ r❡t❛ ♣❛r❛❧❡❧❛ ❛♦ ❧❛❞♦ AB ♣❛ss❛♥❞♦ ♣♦r C ✱ ❛ q✉❛❧ ✐♥t❡rs❡❝t❛ ♦

  ✷ ✷

  C DB ❧❛❞♦ BC ♥♦ ♣♦♥t♦ D✱ ❝♦♥❢♦r♠❡ ✜❣✉r❛ ✸✳✶✹✳ ❊♥tã♦✱ B é ✉♠ ♣❛r❛❧❡❧♦❣r❛♠♦✱ ❡ ♣❡❧♦ t❡♦r❡♠❛ ❞❡ ❚❤❛❧❡s✱ t❡♠♦s

  ✷ ✷

  AC AB

  1 . ✏ ✏

  AC AB k

  1

  ✷ ✷ ✶ ✶

  C C ▲♦❣♦✱ B ✏ ☎ BC ✏ B ✳ k

  ❉❡ss❛ ❢♦r♠❛ ♠♦str❛♠♦s q✉❡

  ✷ ✶ ✶ ✷ ✶ ✶ ✷ ✷ ✶ ✶ AB B , AC C C C .

  ✏ A ✏ A ❡ B ✏ B

  ✷ ✷ ✶ ✶ ✶

  C B C P♦r ♦✉tr♦ ❧❛❞♦✱ ♦❜s❡r✈❡ q✉❡ ♦s tr✐â♥❣✉❧♦s AB ❡ A sã♦ ❝♦♥❣r✉❡♥t❡s✱ ♣❡❧♦ ❝❛s♦

  ✶ ✶ ✶

  A A B B C C ▲▲▲ ❞❡ ❝♦♥❣r✉ê♥❝✐❛✱ t❡♠♦s q✉❡ ♣ ✏ ♣ ✱ ♣ ✏ ① ❡ ♣ ✏ ① ❡✱ ♣♦rt❛♥t♦✱ ♦s tr✐â♥❣✉❧♦s

  ✶ ✶ ✶

  ABC B C ❡ A sã♦ s❡♠❡❧❤❛♥t❡s✳

  P❛r❛ ♦s ❞♦✐s ❝❛s♦s s❡❣✉✐♥t❡s ❞❡ s❡♠❡❧❤❛♥ç❛ ❞❡ tr✐â♥❣✉❧♦s✱ ❞❡✐①❛r❡♠♦s ❛ ❝❛r❣♦ ❞♦ ❧❡✐t♦r ❛ r❡❛❧✐③❛r ❛s s✉❛s ❞❡♠♦♥str❛çõ❡s✳ ❖ ❝r✐tér✐♦ ♣❛r❛ ❛ s❡♠❡❧❤❛♥ç❛ ❞❡ ❞♦✐s tr✐â♥❣✉❧♦s ❛ s❡❣✉✐r é ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ ♦ ❝❛s♦ ▲❆▲ ❞❡ s❡♠❡❧❤❛♥ç❛✳

  ✶ ✶ ✶

  B C Pr♦♣♦s✐çã♦ ✸✳✼✳✷✳ ❙❡❥❛♠ ABC ❡ A tr✐â♥❣✉❧♦s ♥♦ ♣❧❛♥♦ t❛✐s q✉❡

  AB BC

  ✶ B B .

  ✏ ✏ k e ♣ ✏ ①

  ✶ ✶ ✶ ✶

  A B B C

  ✶ ✶ ✶ ✶ ✶ ✶

  B C ❊♥tã♦✱ ∆ABC ✒ ∆A ✱ ❝♦♠ ❛ ❝♦rr❡s♣♦♥❞ê♥❝✐❛ ❞❡ ✈ért✐❝❡s A Ø A ✱ B Ø B ✱ C Ø C ✳

  ✸✳✼ ❙❡♠❡❧❤❛♥ç❛ ❞❡ ❚r✐â♥❣✉❧♦s ✻✾

  AC

  ✶ ✶

  A A , ♣ C C ❊♠ ♣❛rt✐❝✉❧❛r✱ ♣ ✏ ♣ ✏ ① ❡ ✏ k✳

  ✶ ✶

  A C A

  ✶

  A

  ✶

  kc

  ✶

  c

  ✶ ✶ ✶ ✶

  B C ka B C a

  ❊ss❡ ♣ró①✐♠♦ ❝r✐tér✐♦ é ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ ❝❛s♦ ❆❆ ❞❡ s❡♠❡❧❤❛♥ç❛ ❞❡ tr✐â♥❣✉❧♦s✳

  ✶ ✶ ✶

  B C Pr♦♣♦s✐çã♦ ✸✳✼✳✸✳ ❙❡❥❛♠ ABC ❡ A tr✐â♥❣✉❧♦s ♥♦ ♣❧❛♥♦✱ t❛✐s q✉❡

  ✶ ✶

  ♣ ♣ A A e B B .

  ✏ ♣ ✏ ①

  ✶ ✶ ✶

  B C ❊♥tã♦ ♦s tr✐â♥❣✉❧♦s ABC ❡ A sã♦ s❡♠❡❧❤❛♥t❡s✱ ❝♦♠ ❛ ❝♦rr❡s♣♦♥❞ê♥❝✐❛ ❞❡ ✈ért✐❝❡s

  ✶ ✶ ✶

  A , B , C Ø A Ø B Ø C ✳ ❊♠ ♣❛rt✐❝✉❧❛r✱

  AB BC AC .

  ✏ ✏

  

✶ ✶ ✶ ✶ ✶ ✶

  A B B C A C A

  ✶

  A

  ✶ ✶

  B C B C ❯♠ ❞♦s ❛ss✉♥t♦s q✉❡ ❞❡✈❡♠♦s ❡①♣❧♦r❛r ✐♥t❡♥s❛♠❡♥t❡ ♥♦s ❝❛s♦s ❞❡ s❡♠❡❧❤❛♥ç❛ ❞❡ tr✐â♥✲

  ❣✉❧♦s é ♦ ❝❤❛♠❛❞♦ r❡❧❛çõ❡s ♠étr✐❝❛s ♥♦ tr✐â♥❣✉❧♦ r❡tâ♥❣✉❧♦✳ ❆ s❡❣✉✐r ❡♥✉♥❝✐❛r❡♠♦s ✉♠❛ ♣r♦♣♦s✐çã♦ ❝♦♠ ❛s r❡❧❛çõ❡s ♠étr✐❝❛s q✉❡ ❡♥✈♦❧✈❡♠ ♦s ❡❧❡♠❡♥t♦s ❞❡ ✉♠ tr✐â♥❣✉❧♦ r❡tâ♥❣✉❧♦✳ Pr♦♣♦s✐çã♦ ✸✳✼✳✹✳ ❙❡❥❛ ABC ✉♠ tr✐â♥❣✉❧♦ r❡tâ♥❣✉❧♦✱ ❝♦♠ ❝❛t❡t♦s AB ✏ c, AC ✏ b ❡ ❤✐♣♦t❡♥✉s❛ BC ✏ a✳ ❙❡♥❞♦ H ♦ ♣é ❞❛ ❛❧t✉r❛ r❡❧❛t✐✈❛ à ❤✐♣♦t❡♥✉ss❛✱ CH ✏ m✱ BH ✏ n ❡ AH ✏ h✱ t❡♠♦s✿ ✭✐✮ a ☎ h ✏ b ☎ c. 2 2

  ✸✳✼ ❙❡♠❡❧❤❛♥ç❛ ❞❡ ❚r✐â♥❣✉❧♦s 2 2 2 ✼✵ c .

  ✭✐✐✐✮ a ✏ b 2 ✭✐✈✮ h ✏ m ☎ n.

  P❛r❛ t♦❞❛s ❛s ❞❡♠♦♥str❛çõ❡s ✐r❡♠♦s ♥♦s ♦r✐❡♥t❛r ♣❡❧❛ ✜❣✉r❛ ✸✳✶✺✳ A c b h

  B C n m H a

  ❋✐❣✉r❛ ✸✳✶✺✿ r❡❧❛çõ❡s ♠étr✐❝❛s ♥✉♠ tr✐â♥❣✉❧♦ r❡tâ♥❣✉❧♦✳ AHB CAB

  ❉❡♠♦♥str❛çã♦✿ P❛r❛ ♣r♦✈❛r ♦s ✐t❡♥s ♣iq ❡ ♣iiq✱ ♦❜s❡r✈❡♠♦s q✉❡ ④ ✏ ③ ❡ t❛♠❜é♠ ABH CBA q✉❡ ④ ✏ ③ ✳ ❉❡ss❛ ❢♦r♠❛✱ ♣❡❧❛ ♣r♦♣♦s✐çã♦ ✸✳✼✳✸✱ ♦s tr✐â♥❣✉❧♦s ❇❆❍ ❡ ❇❈❆ sã♦ s❡♠❡❧❤❛♥t❡s✱ ❝♦♠ ❛ ❝♦rr❡s♣♦♥❞ê♥❝✐❛ ❞♦s ✈ért✐❝❡s A Ø C, H Ø A, B Ø B✳ ❆ss✐♠✱

  AH AB BH AH ✏ ❡ ✏

  AC BC AB BC ♦✉✱ ❛✐♥❞❛✱ h c n c

  . ✏ ❡ ✏ b a c a

  ❉♦♥❞❡✱ s❡❣✉❡ q✉❡ 2 a 2 ☎ h ✏ b ☎ c ❡ c ✏ a ☎ n.

  ❆ r❡❧❛çã♦ b ✏ a ☎ m é ♣r♦✈❛❞❛ ❞❡ ♠❛♥❡✐r❛ ❛♥á❧♦❣❛✳ ❙♦♠❛♥❞♦ ♠❡♠❜r♦ ❛ ♠❡♠❜r♦ ❛s ❞✉❛s r❡❧❛çõ❡s ❞♦ ✐t❡♠ ♣iiq✱ t❡r❡♠♦s ❛ ✐❣✉❛❧❞❛❞❡ a♣m 2 2 2 2 2 n c c q ✏ b ✳ ▼❛s✱ ✉♠❛ ✈❡③ q✉❡ m n ✏ a✱ t❡r❡♠♦s a ✏ b ✳ Pr♦✈❛♥❞♦✱ ❞❡ss❛ ❢♦r♠❛✱

  ♦ ✐t❡♠ ♣iiiq✳ 2 ◆♦✈❛♠❡♥t❡✱ ♠✉❧t✐♣❧✐❝❛♥❞♦ ♠❡♠❜r♦ ❛ ♠❡♠❜r♦ ❛s ❞✉❛s r❡❧❛çõ❡s ❞♦ ✐t❡♠ ♣iiq✱ ♦❜t❡♠♦s 2 a ☎ m ☎ n ✏ ♣b ☎ cq ♦✉ ❛✐♥❞❛✱ 2

  ✂ ✡ b 2 ☎ c m ,

  ☎ n ✏ ✏ h a ♦♥❞❡ ✉t✐❧✐③❛♠♦s ♦ ✐t❡♠ ♣iq ♥❛ ú❧t✐♠❛ ✐❣✉❛❧❞❛❞❡ ❛❝✐♠❛✳

  ➱ ó❜✈✐♦ q✉❡ ♦ ♣r♦❢❡ss♦r ❛♦ ❛♣r❡s❡♥t❛r ♦ ❛ss✉♥t♦✱ ♣♦❞❡r✐❛ ❡①♣❧♦r❛r ❛ s❡♠❡❧❤❛♥ç❛ ❞❡ tr✐â♥❣✉❧♦s ♣❛r❛ ❛s ❞❡♠♦♥str❛çõ❡s ❞♦s ✐t❡♥s ♣iiiq ❡ ♣ivq✳ ❊st❛ ♣❛rt❡ é ❝r✐tér✐♦ ❞♦ ♣r♦❢❡ss♦r✱ ✜❝❛♥❞♦✱ ✐ss♦✱ ❛♣❡♥❛s ❝♦♠♦ s✉❣❡stã♦✳

  ✸✳✼ ❙❡♠❡❧❤❛♥ç❛ ❞❡ ❚r✐â♥❣✉❧♦s ✼✶

  ✸

  ❖ ✐t❡♠ ♣iiiq ❞❛ ♣r♦♣♦s✐çã♦ ❛❝✐♠❛ é ♦ ❝♦♥t❡ú❞♦ ❞♦ ❢❛♠♦s♦ t❡♦r❡♠❛ ❞❡ P✐tá❣♦r❛s ✳ ❆❧✲ ❣✉♥s r❡s✉❧t❛❞♦s ✐♥t❡r❡ss❛♥t❡s ❞❡ss❡ t❡♦r❡♠❛ ♣♦❞❡♠ s❡r ♦❜s❡r✈❛❞♦s q✉❛♥❞♦ ❞❡t❡r♠✐♥❛♠♦s ❛ ❞✐❛❣♦♥❛❧ ❞❡ ✉♠ q✉❛❞r❛❞♦ ♦✉ ❛ ❛❧t✉r❛ ❞❡ ✉♠ tr✐â♥❣✉❧♦ ❡q✉✐❧át❡r♦✱ ♦s q✉❛✐s ❡♥✉♥❝✐❛r❡♠♦s ❡ ❞❡♠♦♥str❛r❡♠♦s ❛ s❡❣✉✐r✳

  ❄ 2. ❈♦r♦❧ár✐♦ ✸✳✼✳✶✳ ❆s ❞✐❛❣♦♥❛✐s ❞❡ ✉♠ q✉❛❞r❛❞♦ ❞❡ ❧❛❞♦ a ♠❡❞❡♠ a ❉❡♠♦♥str❛çã♦✿ ❙❡ ABCD é ✉♠ q✉❛❞r❛❞♦ ❞❡ ❧❛❞♦ a ❡ ❞✐❛❣♦♥❛✐s AC ❡ BD✱ ❡♥tã♦ ♦ tr✐â♥❣✉❧♦ ❢♦r♠❛❞♦ ♣♦r ❞♦✐s ❞❡ s❡✉s ❧❛❞♦s ❝♦♥s❡❝✉t✐✈♦s ❡ ❛ ❞✐❛❣♦♥❛❧ q✉❡ ✉♥❡ ♦s s❡✉s ✈ért✐❝❡s é r❡tâ♥❣✉❧♦ ❡ ✐sós❝❡❧❡s✳ ◆❛ ✜❣✉r❛ ✸✳✶✻✱ t❡♠♦s r❡♣r❡s❡♥t❛❞♦ ❛ s✐t✉❛çã♦ ❛❝✐♠❛✳

  C D a a

  A B ❋✐❣✉r❛ ✸✳✶✻✿ ❞✐❛❣♦♥❛❧ ❞❡ ✉♠ q✉❛❞r❛❞♦ ❡♠ ❢✉♥çã♦ ❞❡ s❡✉ ❧❛❞♦✳

  ❉❛í✱ ♦ t❡♦r❡♠❛ ❞❡ P✐tá❣♦r❛s ❢♦r♥❡❝❡ ❄ 2 2 AC AB BC

  ✏ ❄ 2 2 a a

  ✏ ❄ 2. ✏ a

  ❄ a

  3 ❈♦r♦❧ár✐♦ ✸✳✼✳✷✳ ❆s ❛❧t✉r❛s ❞❡ ✉♠ tr✐â♥❣✉❧♦ ❡q✉✐❧át❡r♦ ❞❡ ❧❛❞♦ a ♠❡❞❡♠ ✳

  2 ❉❡♠♦♥str❛çã♦✿ ❙❡❥❛♠ ABC ✉♠ tr✐â♥❣✉❧♦ ❡q✉✐❧át❡r♦ ❞❡ ❧❛❞♦ a ❡ M ♦ ♣♦♥t♦ ♠é❞✐♦ ❞♦ ❧❛❞♦ BC✳

  ✸

  ✸✳✽ ❆❧❣✉♥s ❚❡♦r❡♠❛s ❞❡ ❙❡♠❡❧❤❛♥ç❛ ❞❡ ❚r✐â♥❣✉❧♦s ✼✷ A a a

  B C M

  2 ❋✐❣✉r❛ ✸✳✶✼✿ ❛❧t✉r❛ ❞❡ ✉♠ tr✐â♥❣✉❧♦ ❡q✉✐❧át❡r♦✳

  ❆♣❧✐❝❛♥❞♦ ♦ t❡♦r❡♠❛ ❞❡ P✐tá❣♦r❛s ❛♦ tr✐â♥❣✉❧♦ r❡tâ♥❣✉❧♦ ACM ❞❛ ✜❣✉r❛ ✸✳✶✼✱ ♦❜t❡♠♦s 2 2 2 AM ✏ AC ✁ CM 2

  ✁ ✠ 2 a ✏ a ✁ 2

  2 3a . ✏

  4 ❉♦♥❞❡ s❡❣✉❡ ♦ r❡s✉❧t❛❞♦✳

  ✸✳✽ ❆❧❣✉♥s ❚❡♦r❡♠❛s ❞❡ ❙❡♠❡❧❤❛♥ç❛ ❞❡ ❚r✐â♥❣✉❧♦s

  ◆❡st❛ s❡çã♦ ❛❜♦r❞❛r❡♠♦s ❛❧❣✉♥s r❡s✉❧t❛❞♦s r❡❧❡✈❛♥t❡s ❞♦ ♣♦♥t♦ ❞❡ ✈✐st❛ ❤✐stór✐❝♦ s♦❜r❡ s❡♠❡❧❤❛ç❛s ❞❡ tr✐â♥❣✉❧♦s✳ ❖ ♣r✐♠❡✐r♦ r❡s✉❧t❛❞♦ ❡st❛❜❡❧❡❝❡ ✉♠❛ r❡❝í♣r♦❝❛ ❞♦ t❡♦r❡♠❛ ❞❛ ❜✐ss❡tr✐③✳ ❱❡r✐✜q✉❡ q✉❡✱

  ❡♠ t♦❞♦ tr✐â♥❣✉❧♦✱ ❛s ❜✐ss❡tr✐③❡s ✐♥t❡r♥❛ ❡ ❡①t❡r♥❛ r❡❧❛t✐✈❛s ❛ ✉♠ ♠❡s♠♦ ✈ért✐❝❡ sã♦ ♣❡r✲ ♣❡♥❞✐❝✉❧❛r❡s✳ ❱❡❥❛♠♦s ❡st❡ r❡s✉❧t❛❞♦✳

  BC Pr♦♣♦s✐çã♦ ✸✳✽✳✶✳ ❙❡❥❛ ABC ✉♠ tr✐â♥❣✉❧♦ ❡ P ❡ Q ♣♦♥t♦s s♦❜r❡ ❛ r❡t❛ ÐÑ ✱ ❝♦♠ P P BC ❡ Q ❘ BC✳ ❙❡

  BP BQ

  ✆

  ③ P AQ ,

  ✏ 90 ❡ ✏ P C QC BAC

  ❡♥tã♦ AP é ❛ ❜✐ss❡tr✐③ ✐♥t❡r♥❛ ❡ AQ é ❛ ❜✐ss❡tr✐③ ❡①t❡r♥❛ ❞❡ ③ ✳ AQ

  ❉❡♠♦♥str❛çã♦✿ ❚r❛❝❡✱ ♣❡❧♦ ♣♦♥t♦ P ✱ ❛ r❡t❛ ♣❛r❛❧❡❧❛ ❛ ÐÑ ✱ ❡ s❡❥❛♠ U ❡ V s❡✉s ♣♦♥t♦s AB AC

  ❞❡ ✐♥t❡rs❡çã♦ ❝♦♠ ❛s r❡t❛s ÐÑ ❡ ÐÑ ✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳

  ✸✳✽ ❆❧❣✉♥s ❚❡♦r❡♠❛s ❞❡ ❙❡♠❡❧❤❛♥ç❛ ❞❡ ❚r✐â♥❣✉❧♦s ✼✸ A

  V B Q P C U ❋✐❣✉r❛ ✸✳✶✽✿ r❡❝í♣r♦❝❛ ❞♦ t❡♦r❡♠❛ ❞❛ ❜✐ss❡tr✐③✳

  BU P BAQ P V C QAC U BP ABQ ❈♦♠♦ ③ ✏ ③ ❡ ③ ✏ ③ ❡ ③ ✏ ③ ✱ ♣♦✐s sã♦✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ â♥✲

  ❣✉❧♦s ❛❧t❡r♥♦s ✐♥t❡r♥♦s✱ ❝♦rr❡s♣♦♥❞❡♥t❡s ❡ ♦♣♦st♦s ♣❡❧♦ ✈ért✐❝❡✳ ❙❡❣✉❡✲s❡✱ ♣❡❧❛ ♣r♦♣♦s✐çã♦ ✸✳✼✳✸✱ q✉❡

  BP P U P C P V .

  ✏ ❡ ✏ BQ QA CQ AQ BP P C

  ▼❛s✱ ❝♦♠♦ ✏ ♣♦r ❤✐♣ót❡s❡✱ ❛s r❡❧❛çõ❡s ❞❡ s❡♠❡❧❤❛♥ç❛ ❛❝✐♠❛ ♥♦s ❞✐③ q✉❡ BQ QC P U P V

  , ✏

  QA AQ AP AQ U V AQ U V AP k ÐÑ

  ♦✉ s❡❥❛✱ P U ✏ P V ✳ ❆❣♦r❛✱ ❞❡ ÐÑ ❑ ÐÑ ❡ ÐÑ t❡♠♦s ÐÑ ❑ ÐÑ ✳ ❉❡ss❛ ❢♦r♠❛✱ U AV

  ❝♦♠♦ AP é ♠❡❞✐❛♥❛ ❡ ❛❧t✉r❛ ❞♦ tr✐â♥❣✉❧♦ AUV ✱ t❡♠♦s q✉❡ AP é ❜✐ss❡tr✐③ ❞❡ ③ ❡✱ BAC

  ❞❛í✱ ❞❡ ③ ✳ P♦r ✜♠✱ ❝♦♠♦ ❛ ❜✐ss❡tr✐③ ❡①t❡r♥❛ r❡❧❛t✐✈❛ ❛♦ ✈ért✐❝❡ A é ♣❡r♣❡♥❞✐❝✉❧❛r à ❜✐ss❡tr✐③ ✐♥✲

  BAC t❡r♥❛✱ t❡♠♦s q✉❡ AQ é ❜✐ss❡tr✐③ ❡①t❡r♥❛ ❞❡ ③ ✳ P♦❞❡♠♦s✱ ❛❣♦r❛✱ ❡♥✉♥❝✐❛r ❡ ♣r♦✈❛r ♦ ♣r✐♠❡✐r♦ t❡♦r❡♠❛ ❞❡st❛ s❡çã♦✱ ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ ♦

  ✹

  t❡♦r❡♠❛ ❞❡ ❆♣♦❧ô♥✐♦✱ ❞❡✈✐❞♦ ❛ ❆♣♦❧ô♥✐♦ ❞❡ P❡r❣❛ ✳ ❚❡♦r❡♠❛ ✸✳✽✳✶✳ ❉❛❞♦s ✉♠ ♥ú♠❡r♦ r❡❛❧ ♣♦s✐t✐✈♦ k ✘ 1 ❡ ♣♦♥t♦s B ❡ C ♥♦ ♣❧❛♥♦✱ ♦ ❧✉❣❛r ❣❡♦♠étr✐❝♦ ❞♦s ♣♦♥t♦s A ❞♦ ♣❧❛♥♦ t❛✐s q✉❡ AB ✏ k ☎ AC é ♦ ❝ír❝✉❧♦ ❝♦♠ ❞✐â♠❡tr♦ P Q✱

  BC ♦♥❞❡ P P BC ❡ Q P ÐÑ sã♦ ♦s ♣♦♥t♦s t❛✐s q✉❡

  BP BQ ✏ ✏ k. P C QC

  ❆♣♦❧ô♥✐♦ ❞❡ P❡r❣❛✱ ♠❛t❡♠át✐❝♦ ❣r❡❣♦ ❞♦ sé❝✉❧♦ ■■■ ❛✳❈✳ ❆♣♦❧ô♥✐♦ ❞❡✉ ❣r❛♥❞❡s ❝♦♥tr✐❜✉✐çõ❡s à ●❡♦✲ ♠❡tr✐❛ ❊✉❝❧✐❞✐❛♥❛✱ ♥♦t❛❞❛♠❡♥t❡ ❛♦ ❡st✉❞♦ ❞❛s ❝ô♥✐❝❛s✳

  ✸✳✽ ❆❧❣✉♥s ❚❡♦r❡♠❛s ❞❡ ❙❡♠❡❧❤❛♥ç❛ ❞❡ ❚r✐â♥❣✉❧♦s ✼✹ A

  ✶

  B Q Q P C P ❋✐❣✉r❛ ✸✳✶✾✿ ❝ír❝✉❧♦ ❞❡ ❆♣♦❧ô♥✐♦ s♦❜r❡ BC ♥❛ r❛③ã♦ k✳

  BC ❉❡♠♦♥str❛çã♦✿ ❙❡ A ✘ P ❡ A ✘ Q é ✉♠ ♣♦♥t♦ t❛❧ q✉❡ AB ✏ k ☎ AC✱ ❡♥tã♦ A ❘ ÐÑ ✳

  ✶ ✶

  , Q BC ❙❡❥❛♠✱ ♣♦✐s P P ÐÑ r❡s♣❡❝t✐✈❛♠❡♥t❡ ♦s ♣és ❞❛ ❜✐ss❡tr✐③ ✐♥t❡r♥❛ ❡ ❡①t❡r♥❛ ❞♦ tr✐â♥❣✉❧♦ ABC

  ✱ tr❛ç❛❞❛s ❛ ♣❛rt✐r ❞❡ A✳ ❙❡❣✉❡ ❞❛ ♣r♦♣♦s✐çã♦ ✸✳✻✳✷ q✉❡

  ✶ ✶

  BP BQ BA ✏ ✏ ✏ k

  ✶ ✶

  P C Q C AC

  ✶ ✶ ✆ ✶ ✶

  P AQ ❡✱ ♦❜t❡♠♦s P ✏ P ❡ Q ✏ Q✳ ▼❛s✱ ❝♦♠♦ ④ ✏ 90 ✱ s❡❣✉❡ q✉❡ A ♣❡rt❡♥❝❡ ❛♦ ❝ír❝✉❧♦ ❞❡ ❞✐â♠❡tr♦ P Q✳

  ❘❡❝✐♣r♦❝❛♠❡♥t❡✱ s❡ A ✘ P, Q é ✉♠ ♣♦♥t♦ s♦❜r❡ ♦ ❝ír❝✉❧♦ ❞❡ ❞✐â♠❡tr♦ P Q✱ ❡♥tã♦ BP BQ

  ✆

  A BC P AQ ❘ ÐÑ ❡ ③ ✏ 90 ✳ ❈♦♠♦ ✏ ✏ k✱ s❡❣✉❡ ❞❛ ♣r♦♣♦s✐çã♦ ✸✳✽✳✶ q✉❡✱ ♥♦

  P C QC tr✐â♥❣✉❧♦ ABC✱ AP é ❜✐ss❡tr✐③ ✐♥t❡r♥❛ ❡ AQ é ❜✐ss❡tr✐③ ❡①t❡r♥❛✳ P♦rt❛♥t♦✱ ♣❡❧♦ t❡♦r❡♠❛ ❞❛ ❜✐ss❡tr✐③✱ ❡♥✉♥❝✐❛❞♦ ♥❛ ♣r♦♣♦s✐çã♦ ✸✳✻✳✷✱ t❡♠♦s

  AB BP ✏ ✏ k,

  AC P C ♦♥❞❡ A ♣❡rt❡♥❝❡ ❛♦ ❧✉❣❛r ❣❡♦♠étr✐❝♦ ❞❡s❡❥❛❞♦✳

  ❖ ❧✉❣❛r ❣❡♦♠étr✐❝♦ ❞❡s❝r✐t♦ ♥♦ ❡♥✉♥❝✐❛❞♦ ❞♦ t❡♦r❡♠❛ ✸✳✽✳✶ é ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ ♦ ❝ír❝✉❧♦ ❞❡ ❆♣♦❧ô♥✐♦ r❡❧❛t✐✈♦ ❛ ♣B, Cq✱ ♥❛ r❛③ã♦ k ✘ 1✳ ❊♠ ♣❛rt✐❝✉❧❛r✱ ♦❜s❡r✈❡ q✉❡ ♦s ❝ír❝✉❧♦s ❞❡ ❆♣♦❧ô♥✐♦ r❡❧❛t✐✈♦s ❛ ♣B, Cq ❡ ♣C, Bq✱ ❛♠❜♦s ♥❛ r❛③ã♦ k ✘ 1✱ sã♦ ❞✐st✐♥t♦s✳

  ❆ ♣ró①✐♠❛ ❛♣❧✐❝❛çã♦ ❞❡ s❡♠❡❧❤❛♥ç❛ é ✉♠ ❢❛♠♦s♦ t❡♦r❡♠❛ s♦❜r❡ q✉❛r✐❧át❡r♦s ✐♥s❝r✐tí✲

  ✺

  ✈❡✐s✱ ❞❡✈✐❞♦ ❛ ❈❧á✉❞✐♦ Pt♦❧♦♠❡✉ ✱ ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ t❡♦r❡♠❛ ❞❡ Pt♦❧♦♠❡✉✳

  ❈❧á✉❞✐♦ Pt♦❧♦♠❡✉✱ ♠❛t❡♠át✐❝♦ ❡ ❛strô♥♦♠♦ ❣r❡❣♦ ❞♦ sé❝✉❧♦ ■■ ❞✳❈✳✱ ❞❡✉ ❣r❛♥❞❡s ❝♦♥tr✐❜✉✐çõ❡s à

●❡♦♠❡tr✐❛ ❊✉❝❧✐❞✐❛♥❛✳ Pt♦❧♦♠❡✉ é ♠❛✐s ❝♦♥❤❡❝✐❞♦ ♣♦r s❡✉s tr❛❜❛❧❤♦s ❝♦♠♦ ❛strô♥♦♠♦✱ ♣r✐♥❝✐♣❛❧♠❡♥t❡

♣♦r ❤❛✈❡r ♣r♦♣♦st♦ ❛ ✭❡q✉✐✈♦❝❛❞❛✮ ❚❡♦r✐❛ ●❡♦❝ê♥tr✐❝❛✱ s❡❣✉♥❞♦ ❛ q✉❛❧ ❛ ❚❡rr❛ ♦❝✉♣❛✈❛ ♦ ❝❡♥tr♦ ❞♦

❯♥✐✈❡rs♦✳

  ✸✳✽ ❆❧❣✉♥s ❚❡♦r❡♠❛s ❞❡ ❙❡♠❡❧❤❛♥ç❛ ❞❡ ❚r✐â♥❣✉❧♦s ✼✺ Pr♦♣♦s✐çã♦ ✸✳✽✳✷✳ ❙❡ ABCD é ✉♠ q✉❛❞r✐❧át❡r♦ ✐♥s❝r✐tí✈❡❧ ❞❡ ❞✐❛❣♦♥❛✐s AC ❡ BD✱ ❡♥tã♦

  BP ✱ t❡♠♦s ❞❛s ❡q✉❛çõ❡s ✸✳✻ ❡ ✸✳✼ q✉❡

  AB ☎ CD AD ☎ BC ✏ BD.

  AC . ❈♦♥❝❧✉í✲s❡ q✉❡

  AC AD ☎ BC

  AB ☎ CD

  BD ✏

  AC . ❊♥tã♦✱

  AD ☎ BC

  AC ❡ BP ✏

  AB ☎ CD

  DP ✏

  ✭✸✳✼✮ ❈♦♠♦ BD ✏ DP

  AB ☎ CD AD ☎ BC ✏ AC ☎ BD. ❉❡♠♦♥str❛çã♦✿ ❈♦♥s✐❞❡r❡ ✉♠ q✉❛❞r✐❧át❡r♦ ABCD ✐♥s❝r✐tí✈❡❧ ❡♠ ✉♠❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ❞❡ ❝❡♥tr♦ O✳ ▼❛rq✉❡ ♦ ♣♦♥t♦ P s♦❜r❡ ❛ ❞✐❛❣♦♥❛❧ BD ❞♦ q✉❛❞r✐❧át❡r♦ ABCD✱ t❛❧ q✉❡

  ✏ AD AC .

  ✸✳✼✳✸✱ ❞❡ t❛❧ ❢♦r♠❛ q✉❡ BP BC

  ✭✸✳✻✮ ❉❛ ♠❡s♠❛ ❢♦r♠❛✱ ♦s tr✐â♥❣✉❧♦s ADC ❡ BP C t❛♠❜é♠ sã♦ s❡♠❡❧❤❛♥t❡s ♣❡❧❛ ♣r♦♣♦s✐çã♦

  ✏ AC CD .

  ✱ ♦s tr✐â♥❣✉❧♦s ABC ❡ DP C sã♦ s❡♠❡❧❤❛♥t❡s ♣❡❧♦ ♣r♦♣♦s✐çã♦ ✸✳✼✳✸ ❡✱ ❞❛í✱ AB DP

  ✱ ♣♦✐s ♣♦ss✉❡♠ ♦ ♠❡s♠♦ ❛r❝♦ ② BC

  ✏ ④ BDC

  ❈♦♠♦ ③ BAC

  ✳ A B C D O P

  ✏ ③ ACB

  ④ P CD

  ❆ s❡❣✉✐r✱ ✉♠❛ ❝♦♥s❡q✉ê♥❝✐❛ ✐♥t❡r❡ss❛♥t❡ ❞♦ t❡♦r❡♠❛ ❞❡ Pt♦❧♦♠❡✉✳

  ✸✳✽ ❆❧❣✉♥s ❚❡♦r❡♠❛s ❞❡ ❙❡♠❡❧❤❛♥ç❛ ❞❡ ❚r✐â♥❣✉❧♦s ✼✻ ❈♦r♦❧ár✐♦ ✸✳✽✳✶✳ ❙❡ ABC é ✉♠ tr✐â♥❣✉❧♦ ❡q✉✐❧át❡r♦ ❡ P é ✉♠ ♣♦♥t♦ q✉❛❧q✉❡r s♦❜r❡ ♦

  BC ❛r❝♦ ♠❡♥♦r ② ❞❡ s❡✉ ❝ír❝✉❧♦ ❝✐r❝✉♥s❝r✐t♦✱ ❡♥tã♦ P B P C ✏ P A.

  A C B P ❋✐❣✉r❛ ✸✳✷✵✿ ❝♦r♦❧ár✐♦ ❞♦ t❡♦r❡♠❛ ❞❡ Pt♦❧♦♠❡✉

  ❉❡♠♦♥str❛çã♦✿ ❆♣❧✐❝❛♥❞♦ ♦ t❡♦r❡♠❛ ❞❡ Pt♦❧♦♠❡✉ ❛♦ q✉❛❞r✐❧át❡r♦ ABP C ❞❛ ✜❣✉r❛ ✸✳✷✵✱ ♦❜t❡♠♦s

  AB ☎ P C AC ☎ P B ✏ AP ☎ BC. ❊♠ s❡❣✉✐❞❛✱ ❝❛♥❝❡❧❛♥❞♦ ♦s ❝♦♠♣r✐♠❡♥t♦s ✐❣✉❛✐s AB ✏ AC ✏ BC✱ ❝❤❡❣❛♠♦s ❛♦ r❡s✉❧✲ t❛❞♦ ❡s♣❡r❛❞♦✳

  ✻

  ▼♦str❛r❡♠♦s✱ ❛❣♦r❛✱ ❞♦✐s t❡♦r❡♠❛s ✐♠♣♦rt❛♥t❡s ❞❡✈✐❞♦ ❛ ▲❡♦♥❤❛r❞ ❊✉❧❡r ✳ ❖ ♣r✐♠❡✐r♦ ❞❡❧❡s é ❝♦♥❤❡❝✐❞♦ ♥❛ ❧✐t❡r❛t✉r❛ ❝♦♠♦ ♦ t❡♦r❡♠❛ ❞❡ ❊✉❧❡r✳ ❚❡♦r❡♠❛ ✸✳✽✳✷✳ ❙❡ O, G ❡ H sã♦ r❡s♣❡❝t✐✈❛♠❡♥t❡ ♦ ❝✐r❝✉♥❝❡♥tr♦✱ ♦ ❜❛r✐❝❡♥tr♦ ❡ ♦ ♦rt♦❝❡♥tr♦ ❞❡ ✉♠ tr✐â♥❣✉❧♦ ABC✱ ❡♥tã♦✿

  ✭✐✮ AH ✏ 2 ☎ OM✱ ♦♥❞❡ M é ♦ ♣♦♥t♦ ♠é❞✐♦ ❞❡ BC✳ ✭✐✐✮ H, G ❡ O sã♦ ❝♦❧✐♥❡❛r❡s✱ ❝♦♠ G P HO ❡ HG ✏ 2 ☎ GO.

  ❉❡♠♦♥str❛çã♦✿ ❈♦♠♦ O é ❝✐r❝✉♥❝❡♥tr♦ ❞♦ tr✐â♥❣✉❧♦ ABC ❞❛ ✜❣✉r❛ ✸✳✷✶✱ t❡♠♦s q✉❡ ♦ tr✐â♥❣✉❧♦ OBC é ✐sós❝❡❧❡s ❞❡ ❜❛s❡ BC ❡ M é ♣♦♥t♦ ♠é❞✐♦ ❞❛ ❜❛s❡ BC✱ ❧♦❣♦ OM é ❛❧t✉r❛ ❞♦ tr✐â♥❣✉❧♦ OBC ❡♠ r❡❧❛çã♦ à ❜❛s❡ BC✳ ❉❡ss❛ ❢♦r♠❛✱ t❡♠♦s q✉❡ OM é ♣❛r❛❧❡❧♦

  OM N HAB ❛♦ s❡❣♠❡♥t♦ AH ❡ MN é ♣❛r❛❧❡❧♦ ❛♦ ❧❛❞♦ AB✱ ❧♦❣♦ ♦s â♥❣✉❧♦s ④ ❡ ④ sã♦ ✐❣✉❛✐s✳

  ▲❡♦♥❤❛r❞ ❊✉❧❡r é s✉íç♦ ❡ ✈✐✈❡✉ ♥♦ sé❝✉❧♦ ❳❱■■■✱ é ❛té ❤♦❥❡ ♦ ♠❛t❡♠át✐❝♦ q✉❡ ♠❛✐s ♣✉❜❧✐❝♦✉ tr❛❜❛❧❤♦s r❡❧❡✈❛♥t❡s✳

  ✸✳✽ ❆❧❣✉♥s ❚❡♦r❡♠❛s ❞❡ ❙❡♠❡❧❤❛♥ç❛ ❞❡ ❚r✐â♥❣✉❧♦s ✼✼ A N H O G B M C

  ❋✐❣✉r❛ ✸✳✷✶✿ ❛ r❡t❛ ❞❡ ❊✉❧❡r ❞❛ r❡t❛ HO ❞❡ ABC✳ ❆♥❛❧♦❣❛♠❡♥t❡ ♦ s❡❣♠❡♥t♦ ON é ❛❧t✉r❛ ❞♦ tr✐â♥❣✉❧♦ OAC✱ ♦♥❞❡ ON k BH ❡ ♦s

  ON M HBA â♥❣✉❧♦s ④ ❡ ④ sã♦ ✐❣✉❛✐s✳

  P❡❧❛ ♣r♦♣♦s✐çã♦ ✸✳✼✳✸✱ ♦s tr✐â♥❣✉❧♦s OMN ❡ HAB sã♦ s❡♠❡❧❤❛♥t❡s✳ ❉❛í✱ ❝♦♥❝❧✉✐✲s❡ q✉❡ OM M N

  1 . ✏ ✏

  AH AB

  2 ❉♦♥❞❡✱ s❡❣✉❡ q✉❡

  AH ✏ 2 ☎ OM.

  

  P❛r❛ ❞❡♠♦♥str❛r ♦ ✐t❡♠ ♣iq✱ ❝♦♥s✐❞❡r❡ G ♦ ♣♦♥t♦ ❞❡ ✐♥t❡rs❡çã♦ ❞♦s s❡❣♠❡♥t♦s AM

  ✶ ✶

  OG M HG A OM k ÐÑ AH

  ❡ HO✱ ❡♥tã♦ ④ ✏ ④ ✭♦♣♦st♦ ♣❡❧♦ ✈ért✐❝❡✮✳ ❚❛♠❜é♠✱ ❝♦♠♦ ÐÝÑ ✱ s❡❣✉❡

  ✶ ✶ ✶ ✶

  OM G HAG q✉❡ ④ ✏ ④ ✭❛❧t❡r♥♦s ✐♥t❡r♥♦s✮✳ P♦rt❛♥t♦✱ ♦s tr✐â♥❣✉❧♦s MOG ❡ AHG sã♦ ❝♦♥❣r✉❡♥t❡s ♣❡❧❛ ♣r♦♣♦s✐çã♦ ✸✳✼✳✸ ❡✱ ❞❛í✱

  ✶ ✶

  OG M G OM

  1 ,

  ✏ ✏ ✏

  ✶ ✶

  HG AG AH

  2

  ✶

  ♦♥❞❡ ✉t✐❧✐③❛♠♦s ♦ ✐t❡♠ ♣iq ♥❛ ú❧t✐♠❛ ✐❣✉❛❧❞❛❞❡ ❛❝✐♠❛✳ ❙❡❣✉❡ ❞❛í q✉❡ G ❡ G sã♦ ♣♦♥t♦s ❞♦ s❡❣♠❡♥t♦ AM✱ t❛✐s q✉❡

  ✶

  AG AG .

  ✏

  ✶

  M G M G

  ✶

  P♦rt❛♥t♦✱ ❞❡✈❡♠♦s t❡r G ✏ G ✳ HO

  ◆❛s ♥♦t❛çõ❡s ❞♦ r❡s✉❧t❛❞♦ ❛❝✐♠❛✱ ❞✐③❡♠♦s q✉❡ ÐÑ é ❛ r❡t❛ ❞❡ ❊✉❧❡r ❡ HO é ❛ ♠❡❞✐❛♥❛ ❞❡ ❊✉❧❡r ❞♦ tr✐â♥❣✉❧♦ ABC✳

  ✼

  ❚❡♦r❡♠❛ ✸✳✽✳✸✳ ❊♠ t♦❞♦ tr✐â♥❣✉❧♦✱ ♦ ❝✐r❝✉♥❝❡♥tr♦ ❞♦ tr✐â♥❣✉❧♦ órt✐❝♦ ❝♦✐♥❝✐❞❡ ❝♦♠ ♦ ♣♦♥t♦ ♠é❞✐♦ ❞❛ ♠❡❞✐❛♥❛ ❞❡ ❊✉❧❡r✳ ❆❞❡♠❛✐s✱ ♦ ❝ír❝✉❧♦ ❝✐r❝✉♥s❝r✐t♦ ❛♦ tr✐â♥❣✉❧♦ órt✐❝♦ é

  ❚r✐â♥❣✉❧♦ órt✐❝♦ é ♦ tr✐â♥❣✉❧♦ ❝✉❥♦s ✈ért✐❝❡s sã♦ ♦s ♣és ❞❛s ❛❧t✉r❛s ❞❡ ✉♠ tr✐â♥❣✉❧♦ ♥ã♦ r❡tâ♥❣✉❧♦✳

  ✸✳✽ ❆❧❣✉♥s ❚❡♦r❡♠❛s ❞❡ ❙❡♠❡❧❤❛♥ç❛ ❞❡ ❚r✐â♥❣✉❧♦s ✼✽ t❛♠❜é♠ ❝✐r❝✉♥s❝r✐t♦ ❛♦ tr✐â♥❣✉❧♦ ♠❡❞✐❛❧✱ ♣❛ss❛ ♣❡❧♦s ♣♦♥t♦s ♠é❞✐♦s ❞♦s s❡❣♠❡♥t♦s q✉❡ ✉♥❡♠ ♦s ✈ért✐❝❡s ❞♦ tr✐â♥❣✉❧♦ ❛♦ ♦rt♦❝❡♥tr♦ ❡ t❡♠ r❛✐♦ ✐❣✉❛❧ à ♠❡t❛❞❡ ❞♦ r❛✐♦ ❞♦ ❝ír❝✉❧♦ ❝✐r❝✉♥s❝r✐t♦ ❛♦ tr✐â♥❣✉❧♦✳ ❉❡♠♦♥str❛çã♦✿ ❙❡❥❛♠ ABC ✉♠ tr✐â♥❣✉❧♦ ❝♦♠ ❝✐r❝✉♥❝❡♥tr♦ O ❡ ♦rt♦❝❡♥tr♦ H✱ t❛❧ q✉❡

  ✶

  M é ♦ ♣♦♥t♦ ♠é❞✐♦ ❞❡ BC✱ A é ♦ ♣♦♥t♦ ♠é❞✐♦ ❞❡ AH ❡ R é ♦ r❛✐♦ ❞♦ ❝ír❝✉❧♦ ❝✐r❝✉♥s❝r✐t♦

  ❛ ABC✱ ❝♦♥❢♦r♠❡ ✜❣✉r❛ ✸✳✷✷✳ A

  ✶

  A H O

  ✶

  O B H M C a ❋✐❣✉r❛ ✸✳✷✷✿ ♦ ❝ír❝✉❧♦ ❞❡ ❊✉❧❡r ❞♦ tr✐â♥❣✉❧♦ ABC✳

  ✶ ✶

  k OM ❈♦♠♦ AA ❡✱ ♣❡❧♦ t❡♦r❡♠❛ ✸✳✽✳✷✱ AA ✏ OM✱ s❡❣✉❡✲s❡ q✉❡ ♦ q✉❛❞r✐❧át❡r♦

  ✶

  AA M O M

  ✶ ✶

  H H k OM

  P♦r ♦✉tr♦ ❧❛❞♦✱ ❝♦♠♦ A ❡ t❛♠❜é♠ ♣❡❧♦ t❡♦r❡♠❛ ✸✳✽✳✷ A ✏ OM✱ s❡❣✉❡✲s❡

  ✶

  HM O q✉❡ ♦ q✉❛❞r✐❧át❡r♦ A ❞❛ ✜❣✉r❛ ✸✳✷✸ é ✉♠ ♣❛r❛❧❡❧♦❣r❛♠♦✳ A

  ✶

  A H

  ✶

  O O B H M C a ❋✐❣✉r❛ ✸✳✷✸✿ ♦ ❝ír❝✉❧♦ ❞❡ ❊✉❧❡r ❞♦ tr✐â♥❣✉❧♦ ABC✳

  ✶

  P♦rt❛♥t♦ s✉❛s ❞✐❛❣♦♥❛✐s ❝♦rt❛♠✲s❡ ❛♦ ♠❡✐♦✱ ❞❡ ❢♦r♠❛ q✉❡✱ s❡♥❞♦ O ♦ ♣♦♥t♦ ♠é❞✐♦ ❞❡

  ✸✳✽ ❆❧❣✉♥s ❚❡♦r❡♠❛s ❞❡ ❙❡♠❡❧❤❛♥ç❛ ❞❡ ❚r✐â♥❣✉❧♦s ✼✾

  ✶ ✶

  HO M

  ✱ ❝♦♥❝❧✉í♠♦s q✉❡ O t❛♠❜é♠ é ♠é❞✐♦ ❞❡ A ✳ ❆ss✐♠✱

  1 R

  ✶ ✶ ✶ ✶ O A M M .

  ✏ O ✏ ☎ A ✏ a

  2

  2 ❙❡❥❛✱ ❛❣♦r❛✱ H ♦ ♣é ❞❛ ❛❧t✉r❛ r❡❧❛t✐✈❛ ❛ BC✳ ❈♦♠♦ ❡♠ t♦❞♦ tr✐â♥❣✉❧♦ r❡tâ♥❣✉❧♦ ❛

  ♠❡❞✐❛♥❛ r❡❧❛t✐✈❛ à ❤✐♣♦t❡♥✉s❛ é ✐❣✉❛❧ à ♠❡t❛❞❡ ❞❛ ♠❡s♠❛✱ ✈✐st♦ ♥♦ ❝♦r♦❧ár✐♦ ✸✳✹✳✶✱ s❡❣✉❡

  ✶

  H a M q✉❡ ♥♦ tr✐â♥❣✉❧♦ A ✱ ❞❛ ✜❣✉r❛ ✸✳✷✷✱ t❡♠♦s

  1 R

  ✶ ✶ O H M . a

  ✏ ☎ A ✏

  2

  2 R

  ✶ ✶ a

  ❊♠ ♣❛rt✐❝✉❧❛r✱ ♦ ❝ír❝✉❧♦ ❞❡ ❝❡♥tr♦ O ❡ r❛✐♦ ✐❣✉❛❧ ❛ ♣❛ss❛ ♣♦r A ✱ M ❡ H ✳ P♦r ✜♠✱

  2 ❝♦♠ ♦ ♠❡s♠♦ ❛r❣✉♠❡♥t♦ é ✈á❧✐❞♦ ❡♠ r❡❧❛çã♦ ❛s ♦✉tr❛ ❞✉❛s ❛❧t✉r❛s✳

  ❖ ❝ír❝✉❧♦ ❞❡s❝r✐t♦ ♥❛ ❞❡♠♦♥str❛çã♦ ❞❡ss❡ t❡♦r❡♠❛ é ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ ♦ ❝ír❝✉❧♦ ❞❡ ❊✉❧❡r ❞♦ tr✐â♥❣✉❧♦ ABC✳

  ✽

  ❖ ❚❡♦r❡♠❛ ❛ s❡❣✉✐r é ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ ❚❡♦r❡♠❛ ❞❡ ❙t❡✇❛rt ✱ ❡❧❡ ♠♦str❛ ❛ r❡❧❛çã♦ ❡♥tr❡ ❛s ♠❡❞✐❞❛s ❞♦s ❧❛❞♦s ❞❡ ✉♠ tr✐â♥❣✉❧♦ ❡ ✉♠❛ ❝❡✈✐❛♥❛ q✉❛❧q✉❡r✳ ❚❡♦r❡♠❛ ✸✳✽✳✹✳ ❉❛❞♦s ✉♠ tr✐â♥❣✉❧♦ ABC ❡ ✉♠ ♣♦♥t♦ D ❞♦ ❧❛❞♦ BC✱ ✈❛❧❡ ❛ r❡❧❛çã♦ c

  ☎ n b ☎ m ✁ d ☎ a ✏ a ☎ m ☎ n, s❡♥❞♦ a, b ❡ c ❛s ♠❡❞✐❞❛s ❞♦s ❧❛❞♦s BC, AC ❡ AB✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ d ❛ ♠❡❞✐❞❛ ❞❛ ❝❡✈✐❛♥❛ ❡ m ❡ n ❛s ♠❡❞✐❞❛s ❞♦s s❡❣♠❡♥t♦s ❞❡t❡r♠✐♥❛❞♦s ♣❡❧❛ ❝❡✈✐❛♥❛ AD ♥♦ ❧❛❞♦ BC✳

  A c b h d k

  B C H D a m n a

  ❋✐❣✉r❛ ✸✳✷✹✿ ❊❧❡♠❡♥t♦s ❞♦ tr✐â♥❣✉❧♦ ♣❛r❛ ♦ ❚❡♦r❡♠❛ ❞❡ ❙t❡✇❛rt✳ B

  ❉❡♠♦♥str❛çã♦✿ ❙✉♣♦♥❞♦ ♦ â♥❣✉❧♦ ♣ ❛❣✉❞♦✱ ❝♦♠♦ ♥❛ ✜❣✉r❛ ✸✳✷✹✱ s❡❥❛ k ❛ ♠❡❞✐❞❛

  ▼❛tt❤❡✇ ❙t❡✇❛rt ✭✶✼✶✼✲✶✼✽✺✮ ❢♦✐ ✉♠ ♠❛t❡♠át✐❝♦ ❡s❝♦❝ês✳ ❙✉❛ ♦❜r❛ ♠❛✐s ❝♦♥❤❡❝✐❞❛ é ❙♦♠❡ ❣❡♥❡r❛❧

  ✸✳✽ ❆❧❣✉♥s ❚❡♦r❡♠❛s ❞❡ ❙❡♠❡❧❤❛♥ç❛ ❞❡ ❚r✐â♥❣✉❧♦s ✽✵ ❞❛ ♣r♦❥❡çã♦ ❞❛ ❝❡✈✐❛♥❛ AD s♦❜r❡ ♦ ❧❛❞♦ BC✳ ◆♦ tr✐â♥❣✉❧♦ ABH a ✱ ✈❛❧❡♠ ❛s r❡❧❛çõ❡s✱ ♣r♦✈❡♥✐❡♥t❡s ❞♦ t❡♦r❡♠❛ ❞❡ P✐tá❣♦r❛s✿ c 2

  ✁ k 2 ,

  ☎ n b 2 ☎ m ✁ d 2 ☎ a ✏ a ☎ m ☎ n.

  ❙♦♠❛♥❞♦✲s❡ ♦s ❞♦✐s ♠❡♠❜r♦s ❞❛s ✐❣✉❛❧❞❛❞❡s✱ ✈❡♠✿ c 2 ☎ n b 2 ☎ m ✏ m ☎ n ☎ ♣m nq d 2 ☎ ♣m nq. ❈♦♠♦ m n ✏ a✱ t❡♠♦s✿ c 2

  ❡ b 2 ☎ m ✏ n 2 ☎ m d 2 ☎ m 2 ☎ m ☎ n ☎ k.

  ▼✉❧t✐♣❧✐❝❛♥❞♦ ✸✳✽ ♣♦r n ❡ ✸✳✾ ♣♦r m✱ ♦❜t❡♠♦s✿ c 2 ☎ n ✏ m 2 ☎ n d 2 ☎ n ✁ 2 ☎ m ☎ n ☎ k

  2 ☎ n ☎ k ✭✸✳✾✮

  ❞❛í b 2 ✏ n 2 d 2

  2 ☎ n ☎ k k 2 d 2

  ✏ ♣m ✁ kq 2 h 2 ❡ h 2 ✏ d 2 ✁ k 2 .

  ▲♦❣♦✱ b 2 ✏ n 2

  ✏ ♣n kq 2 h 2 ❡ h 2 ✏ d 2 ✁ k 2 .

  ✁ 2 ☎ m ☎ k ✭✸✳✽✮ Pr♦❝❡❞❡♥❞♦ ❞❛ ♠❡s♠❛ ❢♦r♠❛ ♥♦ tr✐â♥❣✉❧♦ ACH a ✱ t❡♠♦s b 2

  ❞♦♥❞❡ c 2 ✏ m 2 d 2

  ✁ k 2 ,

  ▲♦❣♦✱ c 2 ✏ m 2 ✁ 2 ☎ m ☎ k k 2 d 2

  ❯♠❛ ❞❡♠♦♥str❛çã♦ ❛♥á❧♦❣❛ ♣♦❞❡ s❡r ❢❡✐t❛ ♥♦ ❝❛s♦ ❞♦ â♥❣✉❧♦ B s❡r ♦❜t✉s♦ ♦✉ r❡t♦✳

  ✸✳✽ ❆❧❣✉♥s ❚❡♦r❡♠❛s ❞❡ ❙❡♠❡❧❤❛♥ç❛ ❞❡ ❚r✐â♥❣✉❧♦s ✽✶ ❱❡❥❛♠♦s✱ ❛❣♦r❛✱ ❞♦✐s ❡①❡♠♣❧♦s q✉❡ ✐❧✉str❛♠ ❡ss❡ t❡♦r❡♠❛✳

  ❊①❡♠♣❧♦ ✸✳✽✳✶✳ ❊♠ t♦❞♦ tr✐â♥❣✉❧♦✱ ❛ s♦♠❛ ❞♦s q✉❛❞r❛❞♦s ❞❛s ♠❡❞✐❞❛s ❞❛s ♠❡❞✐❛♥❛s é

  3 ✐❣✉❛❧ ❛ ❞❛ s♦♠❛ ❞♦s q✉❛❞r❛❞♦s ❞❛s ♠❡❞✐❞❛s ❞♦s ❧❛❞♦s✳

  4 ❈♦♥s✐❞❡r❡ ♦ tr✐â♥❣✉❧♦ ABC✱ ❞❡ ❧❛❞♦s BC ✏ a, AC ✏ b ❡ AB ✏ c✳ ❙✉❛s ♠❡❞✐❛♥❛s sã♦

  AM , BN a b c ✏ m ✏ m ❡ CO ✏ m ✳

  A N O B M C a ❆♣❧✐❝❛♥❞♦ ♦ ❚❡♦r❡♠❛ ❞❡ ❙t❡✇❛rt ♣❛r❛ ❛ ♠❡❞✐❛♥❛ m ✱ ♦❜té♠✲s❡✿ 2 a a a a 2 2 b c

  ☎ ☎ ✁ m a ☎ a ✏ a ☎ ☎

  2

  2

  2

  2 ♦✉ 2 2 2 b c a 2 a . ✁ m ✏

  2

  2 b c

  4 ❆♥❛❧♦❣❛♠❡♥t❡✱ ♦❜tê♠✲s❡ ♣❛r❛ ❛s ♠❡❞✐❛♠❛s m ❡ m ✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✿ 2 2 2 2 2 2 a c b a b c 2 2 e . ✁ m b ✏ ✁ m c ✏

  2

  2

  4

  2

  2

  4 ❙♦♠❛♥❞♦✲s❡ ❛s três ú❧t✐♠❛s ❡q✉❛çõ❡s✱ ♦❜t❡♠♦s 2 2 2

  3 2 2 2 m m m b c a b c ✏ ☎ ♣a q,

  4 q✉❡ é ♦ q✉❡ q✉❡rí❛♠♦s ♣r♦✈❛r✳ ❊①❡♠♣❧♦ ✸✳✽✳✷✳ ❊♠ ✉♠ tr✐â♥❣✉❧♦ r❡tâ♥❣✉❧♦✱ ❛ s♦♠❛ ❞♦s q✉❛❞r❛❞♦s ❞❛s ♠❡❞✐❞❛s ❞❛s

  5 ❞✐stâ♥❝✐❛s ❞♦ ✈ért✐❝❡ ❞♦ â♥❣✉❧♦ r❡t♦ ❛♦s ♣♦♥t♦s ❞❡ tr✐s❡❝çã♦ ❞❛ ❤✐♣♦t❡♥✉s❛ é ✐❣✉❛❧ ❛ ❞♦

  9 q✉❛❞r❛❞♦ ❞❛ ♠❡❞✐❞❛ ❞❛ ❤✐♣♦t❡♥✉s❛✳ 1 ❙❡❥❛ ABC ✉♠ tr✐â♥❣✉❧♦ r❡tâ♥❣✉❧♦ ❡♠ A✱ ❞❡ ❧❛❞♦s BC ✏ a, AC ✏ b ❡ AB ✏ c✳ ❙❡❥❛♠ 2 T ❡ T ♦s ♣♦♥t♦s q✉❡ ❞✐✈✐❞❡♠ ❛ ❤✐♣♦t❡♥✉s❛ ❡♠ três ♣❛rt❡s ✐❣✉❛✐s✳

  ▲♦❣♦✱ a

  BT 1 1 T 2 2 C . ✏ T ✏ T ✏

  ✸✳✾ ➪r❡❛s ❞❡ ❋✐❣✉r❛s P❧❛♥❛s ✽✷

  B T 1 T 2 A 1 1 C ❙❡ AT ✏ d ✱ ❛♣❧✐❝❛♥❞♦✱ t❛♠❜é♠✱ ♦ ❚❡♦r❡♠❛ ❞❡ ❙t❡✇❛rt ♥♦ tr✐â♥❣✉❧♦ ABC ♣❛r❛ ❛ 1

  ❝❡✈✐❛♥❛ AT ✱ ♦❜t❡♠♦s 2 2 2 2 a 2a a 2a b 2c 2a 2 2 2 b c 1 1 . ☎ ☎ ✁ d ☎ a ✏ a ☎ ☎ ô ✁ d ✏

  3 2

  3 2

  3

  3

  3

  3

  9 ❆♥❛❧♦❣❛♠❡♥t❡✱ s❡ AT ✏ d ✱ ❛♣❧✐❝❛♥❞♦✲s❡ ♦ ❚❡♦r❡♠❛ ❞❡ ❙t❡✇❛rt ♥♦ tr✐â♥❣✉❧♦ ABC 2

  ♣❛r❛ ❛ ❝❡✈✐❛♥❛ AT ✱ ♦❜t❡♠♦s✿ 2 2 2 2b c 2a 2 2 . ✁ d ✏

  3

  3

  9 ❙♦♠❛♥❞♦✲s❡ ❛s ❞✉❛s ú❧t✐♠❛s ❡q✉❛çõ❡s✱ ♦❜t❡♠♦s 2 2 2 2 2 4a b c . 1 2

  ✁ d ✁ d ✏ 2

  9 2 2 c P♦rt❛♥t♦✱ ♣❡❧♦ ❚❡♦r❡♠❛ ❞❡ P✐tá❣♦r❛s✱ ♦❜t❡♠♦s b ✏ a ✳ ▲♦❣♦✱ 2 2 2 2 4a a , 1 2

  ✁ d ✁ d ✏

  9 ♦✉ s❡❥❛✱ 2 2 2 5a d d . 1 2

  ✏

  9 Pr♦✈❛♥❞♦✱ ❞❡ss❛ ❢♦r♠❛✱ ♦ ❡①❡♠♣❧♦✳

  ✸✳✾ ➪r❡❛s ❞❡ ❋✐❣✉r❛s P❧❛♥❛s

  ◗✉❛♥❞♦ ♠❡❞✐♠♦s ✉♠❛ ❣r❛♥❞❡③❛ s✐❣♥✐✜❝❛ ❞✐③❡r q✉❡ ❡st❛♠♦s ❝♦♠♣❛r❛♥❞♦ ❝♦♠ ♦✉tr❛ ❞❡ ♠❡s♠❛ ❡s♣é❝✐❡ t♦♠❛❞❛ ❝♦♠♦ ✉♥✐❞❛❞❡✳ ◆❡st❛ s❡çã♦✱ ❡st❛❜❡❧❡❝❡r ❡ss❛s ♠❡❞✐❞❛s ♣❛r❛ ❛ r❡❣✐ã♦ ❞♦ ♣❧❛♥♦ ❞❡❧✐♠✐t❛❞♦ ♣♦r ✉♠❛ ✜❣✉r❛ ♣❧❛♥❛✳ ❊st❛ ♠❡❞✐❞❛ s❡rá ❞❡♥♦♠✐♥❛❞❛ ár❡❛ ❞❡st❛ r❡❣✐ã♦✳

  P❛r❛ ❡♥❝♦♥tr❛r ❛ ár❡❛ ❞❡ ✉♠❛ ❞❡t❡r♠✐♥❛❞❛ r❡❣✐ã♦ ❞♦ ♣❧❛♥♦✱ ❞❡✈❡r❡♠♦s ❝♦♠♣❛r❛r

  ✸✳✾ ➪r❡❛s ❞❡ ❋✐❣✉r❛s P❧❛♥❛s ✽✸ s✉❛ s✉♣❡r❢í❝✐❡ ❝♦♠ ❛ ❞❡ ✉♠❛ ♦✉tr❛ r❡❣✐ã♦ t♦♠❛❞❛ ❝♦♠♦ ✉♥✐❞❛❞❡✳ ❖ r❡s✉❧t❛❞♦ ❞❡ss❛

  ❝♦♠♣❛r❛çã♦ s❡rá ✉♠ ♥ú♠❡r♦ q✉❡ ❞❡✈❡rá ❡①♣r✐♠✐r q✉❛♥t❛s ✈❡③❡s ❡ss❛ r❡❣✐ã♦ ❝♦♥té♠ ❛ r❡❣✐ã♦ t♦♠❛❞❛ ❝♦♠♦ ✉♥✐❞❛❞❡ ❞❡ ár❡❛✳ ❆❞♦t❛♠♦s ❝♦♠♦ ✉♥✐❞❛❞❡ ❞❡ ár❡❛ ♦ q✉❛❞r❛❞♦ ❝✉❥♦ ❧❛❞♦ ♠❡❞❡ ✉♠❛ ✉♥✐❞❛❞❡ ❞❡ ❝♦♠♣r✐✲

  ♠❡♥t♦✳ ❈❤❛♠❛r❡♠♦s ❞❡ q✉❛❞r❛❞♦ ✉♥✐tár✐♦✳ ❂ ✶ ✉♥✐❞❛❞❡ ❞❡ ár❡❛

  1 ❋✐❣✉r❛ ✸✳✷✺✿ ❛ ✉♥✐❞❛❞❡ ❞❡ ár❡❛✳

  ❈❛s♦ ♦ ❧❛❞♦ ❞❡ ✉♠ q✉❛❞r❛❞♦ t❡♥❤❛ 1 unidade ♣uq de comprimento✱ ❛ ✉♥✐❞❛❞❡ ❞❡ ár❡❛ 2 s❡rá ❝❤❛♠❛❞❛ ❞❡ ✉♥✐❞❛❞❡s q✉❛❞r❛❞❛s ❡ r❡♣r❡s❡♥t❛❞♦ ♣♦r u ✳ ❆ ár❡❛ ❞❡❧✐♠✐t❛❞❛ ♣❡❧♦s ❧❛❞♦s ❞❡ ✉♠ ♣♦❧í❣♦♥♦ ❡①♣r✐♠❡ q✉❛♥t❛s ✈❡③❡s ❡ss❡ ♣♦❧í❣♦♥♦ ❝♦♥té♠ ❛ ✉♥✐❞❛❞❡ ❞❡ ár❡❛✱ ❝♦♠♦ ❥á ❢♦✐ ❞✐t♦✳ ■st♦ é ❢á❝✐❧ ♣❡r❝❡❜❡r✱ ♣♦r ❡①❡♠♣❧♦✱ q✉❛♥❞♦ ❞❡s❡❥❛♠♦s ❝♦♥❤❡❝❡r ❛ ár❡❛ ❞❡ ✉♠ r❡tâ♥❣✉❧♦ ❝✉❥♦s ❧❛❞♦s ♠❡❞❡♠ 5 u ❡ 3 u✳

  3 u 5 u ❆ ✉♥✐❞❛❞❡ ❞❡ ár❡❛ ❝❛❜❡ ✶✺ ✈❡③❡s ♥♦ r❡tâ♥❣✉❧♦ ❡✱ ♣♦r ✐ss♦✱ s✉❛ ár❡❛ é ❞❡ 15 ✉♥✐❞❛❞❡s 2 q✉❛❞r❛❞❛s ♣15 u q✳ ➱ ❢á❝✐❧ ✈❡r q✉❡ s❡ ❛s ♠❡❞✐❞❛s ❞♦s ❧❛❞♦s ❞❡ ✉♠ r❡tâ♥❣✉❧♦ sã♦ ♥ú♠❡r♦s

  ✐♥t❡✐r♦s a ❡ b✱ ❡♥tã♦ ❛ s✉❛ ár❡❛ é ♦ ♣r♦❞✉t♦ ❞❡ss❡s ♥ú♠❡r♦s✿ S ✏ a ☎ b. ❊♠ ♣❛rt✐❝✉❧❛r✱ s❡ ❛ ♠❡❞✐❞❛ ❞♦ ❧❛❞♦ ❞❡ ✉♠ q✉❛❞r❛❞♦ é ✉♠ ♥ú♠❡r♦ ✐♥t❡✐r♦ n✱ s✉❛ 2

  ár❡❛ é ✐❣✉❛❧ ❛ n ✳ ◆ã♦ ♣♦❞❡♠♦s r❡str✐♥❣✐r ❡ss❡ ❝♦♥❝❡✐t♦ ❛♣❡♥❛s ♣❛r❛ ♦s ♥ú♠❡r♦s ✐♥t❡✐r♦s✳ ❉❡✈❡♠♦s ♣❡r❝❡❜❡r q✉❡ é ♣♦ssí✈❡❧ ❞❡t❡r♠✐♥❛r ❛ ár❡❛ ✉s❛♥❞♦ ♠❡❞✐❞❛s ♣❛r❛ ♦s ❧❛❞♦s ❝♦♠ ♥ú♠❡r♦s r❛❝✐♦♥❛✐s✳ ❈♦♠♦ ❡①❡♠♣❧♦✱ ✈❡r✐✜q✉❡ q✉❛❧ ❛ ár❡❛ ❞❡ ✉♠ r❡tâ♥❣✉❧♦ ❝✉❥♦s ❧❛❞♦s

  5 cm ♠❡❞❡♠ 9, 6 cm ❡ ✳

  3

  ✸✳✾ ➪r❡❛s ❞❡ ❋✐❣✉r❛s P❧❛♥❛s ✽✹

  ✸✳✾✳✶ ➪r❡❛ ❞♦ P❛r❛❧❡❧♦❣r❛♠♦

  ❈♦♥s✐❞❡r❡♠♦s ♦ ♣❛r❛❧❡❧♦❣r❛♠♦ ABCD✱ ♥❛ ✜❣✉r❛ ❛❜❛✐①♦✱ ❝♦♠ ❜❛s❡ AB ✏ a ❡ ❛❧t✉r❛ h ✳

  D C h a

  A B ◆❛ ♣r♦♣♦s✐çã♦ ❛❜❛✐①♦✱ ♠♦str❛r❡♠♦s q✉❡ q✉❛♥❞♦ ❝♦♥❤❡❝❡♠♦s ❛ ár❡❛ ❞❡ ✉♠ r❡tâ♥❣✉❧♦

  ♣♦❞❡♠♦s ❡①♣❧✐❝✐t❛r ❛ ár❡❛ ❞❡ ✉♠ ♣❛r❛❧❡❧♦❣r❛♠♦✳ Pr♦♣♦s✐çã♦ ✸✳✾✳✶✳ ❆ ár❡❛ S ❞❡ ✉♠ ♣❛r❛❧❡❧♦❣r❛♠♦ ❞❡ ❜❛s❡ a ❡ ❛❧t✉r❛ h é ✐❣✉❛❧ ❛ S ✏ a☎h✳ ❉❡♠♦♥str❛çã♦✿ ❈♦♥s✐❞❡r❡ ✉♠ ♣❛r❛❧❡❧♦❣r❛♠♦ ABCD ❞❡ ❜❛s❡ a ❡ ❛❧t✉r❛ h✱ ❝♦♠♣❧❡t❡♠♦s ♦ ♣❛r❛❧❡❧♦❣r❛♠♦ ❝♦♠ ♦s tr✐â♥❣✉❧♦s r❡tâ♥❣✉❧♦s ❝♦♥❣r✉❡♥t❡s ✭✈❡r✐✜q✉❡✮ BEC ❡ DF A ♣❛r❛ ♦❜t❡r♠♦s ♦ r❡tâ♥❣✉❧♦ AECF ✳

  F D C h a x

  A B E ❆❣♦r❛ ✈❡r✐✜❝❛♠♦s q✉❡ ❛ ár❡❛ ❞♦ ♣❛r❛❧❡❧♦❣r❛♠♦ é ❞❛❞♦ ♣❡❧❛ ❞✐❢❡r❡♥ç❛ ❡♥tr❡ ❛ ár❡❛ ❞♦ r❡tâ♥❣✉❧♦ ❡ ❛ ár❡❛ ❞♦s ❞♦✐s tr✐â♥❣✉❧♦s✳

  S ABCD AECF BEC DF A ABCD ✏ S ✁ 2 ☎ S ♣ou S q ✏ S ✏ ♣a xq ☎ h ✁ x ☎ h ✏ a ☎ h

  ✸✳✾✳✷ ➪r❡❛ ❞♦ ❚r✐â♥❣✉❧♦

  Pr♦♣♦s✐çã♦ ✸✳✾✳✷✳ ❆ ár❡❛ S ❞♦ tr✐â♥❣✉❧♦ ❞❡ ❜❛s❡ a ❡ ❛❧t✉r❛ h é ❞❛❞❛ ♣❡❧♦ s❡♠✐♣r♦❞✉t♦ ❞❛ ❜❛s❡ a ❡ ❛❧t✉r❛ h✳ ❉❡♠♦♥str❛çã♦✿ P❛r❛ ♦❜t❡r ❛ ár❡❛ ❞❡ ✉♠ tr✐â♥❣✉❧♦ ABC✱ ❡s❝♦❧❤❛ ✉♠ ❧❛❞♦ ♣❛r❛ ❝❤❛♠❛r ❞❡ ❜❛s❡✳ ❙❡❥❛ ❡♥tã♦ BC ❛ ❜❛s❡ ❡ s✉♣♦♥❤❛ q✉❡ ❛ ❜❛s❡ t❡♥❤❛ ❝♦♠♣r✐♠❡♥t♦ a ❡ q✉❛ ❛ ❛❧t✉r❛ r❡❧❛t✐✈❛ ❛ ❡ss❛ ❜❛s❡ t❡♥❤❛ t❛♠❛♥❤♦ h✳

  ✸✳✾ ➪r❡❛s ❞❡ ❋✐❣✉r❛s P❧❛♥❛s ✽✺

  A D h a

  B C ❋✐❣✉r❛ ✸✳✷✻✿ tr✐â♥❣✉❧♦ ABC ❞❡ ❜❛s❡ a ❡ ❛❧t✉r❛ h✳

  P❡❧♦s ✈ért✐❝❡s C ❡ A✱ tr❛❝❡✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ r❡t❛s ♣❛r❛❧❡❧❛s ❛♦s ❧❛❞♦s AB ❡ BC✳ ❙❡❥❛ D

  ♦ ♣♦♥t♦ ❞❡ ✐♥t❡rs❡❝çã♦ ❡♥tr❡ ❡ss❛s ❞✉❛s r❡t❛s✳ ❈♦♥s✐❞❡r❛♥❞♦ ♦ ♣❛r❛❧❡❧♦❣r❛♠♦ ABCD t❡♠♦s q✉❡ s✉❛ ár❡❛ é a ☎ h✳ ❚❡♠♦s✱ ❛❣♦r❛✱ q✉❛ ❛ ár❡❛ ❞♦ tr✐â♥❣✉❧♦ ABC é ❛ ♠❡t❛❞❡ ❞❛ ár❡❛ ❞♦ ♣❛r❛❧❡❧♦❣r❛♠♦ ABCD✳ a

  ☎ h S . ✏

  2

  ✸✳✾✳✸ ➪r❡❛ ❞♦ ❚r❛♣é③✐♦

  Pr♦♣♦s✐çã♦ ✸✳✾✳✸✳ ❆ ár❡❛ S ❞♦ tr❛♣é③✐♦ ❞❡ ❜❛s❡ a ❡ ❛❧t✉r❛ h é ❞❛❞♦ ♣❡❧♦ ♣r♦❞✉t♦ ❞❛ ❜❛s❡

  ✾

  ♠é❞✐❛ ❞♦ tr❛♣é③✐♦ ♣❡❧❛ ❛❧t✉r❛✳ ❉❡♠♦♥str❛çã♦✿ ❈♦♥s✐❞❡r❡♠♦s ♦ tr❛♣é③✐♦ ABCD ❝♦♠ ❜❛s❡ ♠❛✐♦r BC ✏ a✱ ❜❛s❡ ♠❡♥♦r AD

  ✏ b ❡ ❛❧t✉r❛ h✱ ❝♦♠♦ ♥❛ ✜❣✉r❛✳ b A D h

  B E C b a ✁ b a

  ❚r❛ç❛♥❞♦ ♦ s❡❣♠❡♥t♦ DE ♣❛r❛❧❡❧♦ ❛ AB✱ ♦ tr❛♣é③✐♦ ABCD ✜❝❛ ❞✐✈✐❞✐❞♦ ♥♦ ♣❛r❛❧❡❧♦✲ ❣r❛♠♦ ABED ❞❡ ❜❛s❡ b ❡ ❛❧t✉r❛ h ❡ ♥♦ tr✐â♥❣✉❧♦ ECD ❞❡ ❜❛s❡ ♣a ✁ bq ❡ ❛❧t✉r❛ h✳ ❈♦♠♦ ❥á s❛❜❡♠♦s ❛s ár❡❛s ❞❡ss❛s ✜❣✉r❛s✱ ✜❝❛ ❝❧❛r♦ q✉❡ ❛ ár❡❛ S ❞♦ tr❛♣é③✐♦ é ❛ s♦♠❛ ❞❛s ár❡❛s ❞♦ ♣❛r❛❧❡❧♦❣r❛♠♦ ❡ ❞♦ tr✐â♥❣✉❧♦✱ ♦✉ s❡❥❛

  ❈❤❛♠❛♠♦s ❞❡ ❜❛s❡ ♠é❞✐❛ ❞❡ ✉♠ tr❛♣é③✐♦ ❛♦ s❡❣♠❡♥t♦ ❞❡ r❡t❛ q✉❡ ✉♥❡ ♦s ♣♦♥t♦s ♠é❞✐♦s ❞♦s ❧❛❞♦s

  ✸✳✾ ➪r❡❛s ❞❡ ❋✐❣✉r❛s P❧❛♥❛s ✽✻

  ✈ért✐❝❡ ♣❡r❝♦rr❡ ✉♠❛ r❡t❛ ♣❛r❛❧❡❧❛ à ❜❛s❡✳ B A C A

  ❈♦♠♦ M é ♣♦♥t♦ ♠é❞✐♦ ❞❡ BC✱ ❡♥tã♦ ♦s tr✐â♥❣✉❧♦s ABM ❡ ACM ♣♦ss✉❡♠ ❛ ♠❡s♠❛ ♠❡❞✐❞❛ ❞❛ ❜❛s❡ ❡ ♠❡s♠❛ ❛❧t✉r❛✱ ❡✱ ♣♦rt❛♥t♦ ♣♦ss✉❡♠ ❛ ♠❡s♠❛ ár❡❛✳ ❈♦♠♦ s❛❜❡♠♦s ♥❛ ❧✐t❡r❛t✉r❛✱ tr✐â♥❣✉❧♦s q✉❡ ❛♣r❡s❡♥t❛♠ ❛ ♠❡s♠❛ ár❡❛ sã♦ ❞❡♥♦♠✐♥❛❞♦s tr✐â♥❣✉❧♦s ❡q✉✐✈❛✲ ❧❡♥t❡s✳

  A B C M S 1 S 2 S 1 ✏ S 2

  Pr♦♣r✐❡❞❛❞❡ ✷ ❊♠ ✉♠ tr✐â♥❣✉❧♦✱ ✉♠❛ ♠❡❞✐❛♥❛ ❞✐✈✐❞❡ ❡ss❡ tr✐â♥❣✉❧♦ ❡♠ ❞✉❛s r❡❣✐õ❡s ❝♦♠ ár❡❛s ✐❣✉❛✐s✳

  BC tê♠ ♠❡s♠❛ ár❡❛ ♣♦✐s ♣♦ss✉❡♠ ♠❡s♠❛ ❜❛s❡ ❡ ♠❡s♠❛ ❛❧t✉r❛✳

  ✶

  r ❆ r❡t❛ r é ♣❛r❛❧❡❧❛ ❛ BC✳ ❖s tr✐â♥❣✉❧♦s ABC ❡ A

  ✶

  Pr♦♣r✐❡❞❛❞❡ ✶ ❆ ár❡❛ ❞❡ ✉♠ tr✐â♥❣✉❧♦ ♥ã♦ s❡ ❛❧t❡r❛ q✉❛♥❞♦ s✉❛ ❜❛s❡ ♣❡r♠❛♥❡❝❡ ✜①❛ ❡ ♦ t❡r❝❡✐r♦

  S ✏ b ☎ h

  ✸✳✾✳✹ Pr♦♣r✐❡❞❛❞❡s ●❡♦♠étr✐❝❛s ■♠♣♦rt❛♥t❡s

  2 .

  ♣a bq ☎ h

  2 ✏

  2b ☎ h a ☎ h ✁ b ☎ h

  2 ✏

  ♣a ✁ bq ☎ h

  ❱❡❥❛♠♦s✱ ❛❣♦r❛✱ ✉♠❛ ❛♣❧✐❝❛çã♦ ❞❡ss❛ ♣r♦♣r✐❡❞❛❞❡ ❛tr❛✈és ❞❡ ✉♠ ❡①❡♠♣❧♦ s✐♠♣❧❡s✳

  ✸✳✾ ➪r❡❛s ❞❡ ❋✐❣✉r❛s P❧❛♥❛s ✽✼

  ❊①❡♠♣❧♦ ✸✳✾✳✶✳ ❖ tr✐â♥❣✉❧♦ ABC t❡♠ ár❡❛ ✐❣✉❛❧ ❛ 30✳ ❖ ❧❛❞♦ BC ❡stá ❞✐✈✐❞✐❞♦ ❡♠ q✉❛tr♦ ♣❛rt❡s ✐❣✉❛✐s ♣❡❧♦s ♣♦♥t♦s D, E ❡ F ✱ ❡ ♦ ❧❛❞♦ AC ❡stá ❞✐✈✐❞✐❞♦ ❡♠ três ♣❛rt❡s ✐❣✉❛✐s ♣❡❧♦s ♣♦♥t♦s G ❡ H✳ ◗✉❛❧ é ❛ ár❡❛ ❞♦ tr✐â♥❣✉❧♦ GDE❄

  A G H B E D F C

  ✶✵

  BG ❖❜s❡r✈❡ ♦ tr✐â♥❣✉❧♦ ABC ♥❛ ✜❣✉r❛ ✸✳✷✼ ❝♦♠ ❛s ❝❡✈✐❛♥❛s ❡ BH✳

  A G

  10 H

  10

  10 B E D F C ❋✐❣✉r❛ ✸✳✷✼✿ tr✐â♥❣✉❧♦ ABC ❞❡ ❝❡✈✐❛♥❛s BG ❡ BH

  P❡❧❛ ♣r♦♣r✐❡❞❛❞❡ ✷ ❞❛ s✉❜s❡çã♦ ✸✳✾✳✹ ♦s tr✐â♥❣✉❧♦s BAG, BGH ❡ BHC ❞❛ ✜❣✉r❛ ✸✳✷✼ tê♠ ♠❡s♠❛ ár❡❛✳ ❈❛❞❛ ✉♠ t❡♠ ár❡❛ ✐❣✉❛❧ ❛ 10 ❡ ♦ tr✐â♥❣✉❧♦ BGC t❡♠ ár❡❛ ✐❣✉❛❧ ❛

  20 ✳ ❖❜s❡r✈❡ ❛❣♦r❛ ♦ tr✐â♥❣✉❧♦ BGC ❝♦♠ ❛s ❝❡✈✐❛♥❛s GD, GE ❡ GF ✳

  A G

  10 H

  5

  5

  5

  5 B D E F C ❋✐❣✉r❛ ✸✳✷✽✿ tr✐â♥❣✉❧♦ ABC ❝❡✈✐❛♥❛s GD✱ GE ❡ GF ✳

  P❡❧❛ ♠❡s♠❛ ♣r♦♣r✐❡❞❛❞❡ ✷ ❞❛ s✉❜s❡çã♦ ✸✳✾✳✹✱ ♦s tr✐â♥❣✉❧♦s GBD, GDE, GEF ❡ GF C ✶✵ ❞❛ ✜❣✉r❛ ✸✳✷✽ tê♠ ♠❡s♠❛ ár❡❛✳ ▲♦❣♦✱ ❝❛❞❛ ✉♠ ❞❡❧❡s t❡♠ ár❡❛ 5✳ ❆ ár❡❛ ❞♦

  ❈❡✈✐❛♥❛ é q✉❛❧q✉❡r s❡❣♠❡♥t♦ ❞❡ r❡t❛ q✉❡ ✉♥❡ ✉♠ ✈ért✐❝❡ ❞♦ tr✐â♥❣✉❧♦ ❛ ✉♠ ♣♦♥t♦ q✉❛❧q✉❡r ✐♥t❡r✐♦r

  ✸✳✾ ➪r❡❛s ❞❡ ❋✐❣✉r❛s P❧❛♥❛s ✽✽ tr✐â♥❣✉❧♦ GDE é ✐❣✉❛❧ ❛ 5✳ ◆♦t❡ q✉❡ ♣❛r❛ ❛ s♦❧✉çã♦ ❞♦ ♣r♦❜❧❡♠❛ ♥ã♦ ❢♦✐ ♥❡❝❡ssár✐♦ ♦ ✉s♦

  ☎ h

  ❆ss✐♠✱ t❡♠♦s ❛ ❝♦♥✜r♠❛çã♦ ❞❛ ♣r♦♣r✐❡❞❛❞❡ ✸✳ Pr♦♣r✐❡❞❛❞❡ ✹ ❆ r❛③ã♦ ❡♥tr❡ ❛s ár❡❛s ❞❡ tr✐â♥❣✉❧♦s s❡♠❡❧❤❛♥t❡s é ✐❣✉❛❧ ❛♦ q✉❛❞r❛❞♦ ❞❛ r❛③ã♦ ❞❡ s❡♠❡❧❤❛♥ç❛✳ ❖❜s❡r✈❡ ♥❛ ✜❣✉r❛ ❛❜❛✐①♦ ❞♦✐s tr✐â♥❣✉❧♦s s❡♠❡❧❤❛♥t❡s ABC ❡ DEF ❝♦♠ ❜❛s❡s a ❡ b ❡

  2 ✏ a 1 a 2 .

  2 a 2 ☎ h

  ☎ h

  S 1 S 2 ✏ a 1

  2 ✱ t❡♠♦s q✉❡ ❛ r❡❧❛çã♦ ❡♥tr❡ ❛s ár❡❛s S 1 ❡ S 2 é ❞❛❞❛ ♣♦r

  2 ❡ S 2 ✏ a 2

  ❡①♣❧í❝✐t♦ ❞❡ ❢ór♠✉❧❛s✱ ✉♠❛ ✈❡③ q✉❡ ❛♣❡♥❛s ✉♠❛ ♣r♦♣r✐❡❞❛❞❡ s✐♠♣❧❡s ❡ ❝♦♥✈❡♥✐❡♥t❡♠❡♥t❡ ❛♣❧✐❝❛❞❛ r❡s♦❧✈❡✉ ❛ q✉❡stã♦✳

  ☎ h

  ❋✐❣✉r❛ ✸✳✷✾✿ tr✐â♥❣✉❧♦s ABD ❡ ADC tê♠ ♠❡s♠❛ ❛❧t✉r❛ ❡ ❜❛s❡s ❞✐❢❡r❡♥t❡s✳ ❈♦♠♦ S 1 ✏ a 1

  A B C S 1 S 2 a 1 a 2 h D

  ❡♥tr❡ s✉❛s ❜❛s❡s✳ ❆ ❛✜r♠❛çã♦ t❡♠ ❝♦♠♣r♦✈❛çã♦ ✐♠❡❞✐❛t❛ ❛ ♣❛rt✐r ❞❛ ❢ór♠✉❧❛ q✉❡ ❞❡✜♥❡ ❛ ár❡❛ ❞♦ tr✐â♥❣✉❧♦✳

  Pr♦♣r✐❡❞❛❞❡ ✸ ❙❡ ❞♦✐s tr✐â♥❣✉❧♦s tê♠ ♠❡s♠❛ ❛❧t✉r❛✱ ❡♥tã♦ ❛ r❛③ã♦ ❡♥tr❡ s✉❛s ár❡❛s é ✐❣✉❛❧ ❛ r❛③ã♦

  ❱❡r✐✜❝❛♥❞♦ ❛ s♦❧✉çã♦ é ✐♥t❡r❡ss❛♥t❡ ♦❜s❡r✈❛r q✉❡✱ q✉❛♥❞♦ ✉s❛♠♦s ❡ss❛s ♣r♦♣r✐❡❞❛❞❡s✱ ♦s ❝á❧❝✉❧♦s s❡ t♦r♥❛♠ ♠❛✐s ❢á❝❡✐s ❡ ❛tr❛t✐✈♦s ♣❛r❛ ❛ r❡s♦❧✉çã♦ ❞♦ ♣r♦❜❧❡♠❛✳

  ❛❧t✉r❛s h 1 ❡ h 2 ✳ ❙❡♥❞♦ ♦s tr✐â♥❣✉❧♦s s❡♠❡❧❤❛♥t❡s✱ ❡♥tã♦ ❛ r❛③ã♦ ❡♥tr❡ ❛s ❜❛s❡s é ❛ ♠❡s♠❛ r❛③ã♦ ❡♥tr❡

  ✸✳✶✵ ➪r❡❛ ❞♦ ❈ír❝✉❧♦ ✽✾

  D A h 2 h 1 B C E F a b

  ❛s ❛❧t✉r❛s✳ ❊ss❛ r❛③ã♦ é ✉♠ ♥ú♠❡r♦ q✉❡ ❝❤❛♠❛♠♦s ❞❡ r❛③ã♦ ❞❡ s❡♠❡❧❤❛♥ç❛✳ a h 1 ✏ ✏ k. 1 b h 2 2 ❋❛③❡♥❞♦ ❛ r❛③ã♦ ❡♥tr❡s ❛s ár❡❛s S ❡ S ❞❡ss❡s tr✐â♥❣✉❧♦s✱ t❡♠♦s a 1

  ☎ h S 1 a h 1 2

  2 ✏ ✏ ☎ ✏ k ☎ k ✏ k b 2 S 2 b h 2

  ☎ h

  2 ❉❡ ♠♦❞♦ ❣❡r❛❧✱ ♣♦❞❡♠♦s ❛✜r♠❛r q✉❡ ❛ r❛③ã♦ ❡♥tr❡ ❛s ár❡❛s ❞❡ ✜❣✉r❛s s❡♠❡❧❤❛♥t❡s q✉❛✐sq✉❡r é ✐❣✉❛❧ ❛♦ q✉❛❞r❛❞♦ ❞❛ r❛③ã♦ ❞❡ s❡♠❡❧❤❛♥ç❛✱ ✈✐st♦ q✉❡ q✉❛❧q✉❡r ✜❣✉r❛ ♣❧❛♥❛

  ♣♦❞❡ s❡r ❞✐✈✐❞✐❞❛ ❡♠ tr✐â♥❣✉❧♦s ❡ ❛ ♣r♦♣r✐❡❞❛❞❡ ✹ ♥♦s ❛❥✉❞❛ ❛ ♣r♦✈❛r ✐ss♦✱ ♣♦ré♠ ♥ã♦ ❢❛r❡♠♦s ❡ss❛ ❞❡♠♦♥str❛çã♦✳

  ✸✳✶✵ ➪r❡❛ ❞♦ ❈ír❝✉❧♦

  ❆♥t❡s ❞❡ ❢❛❧❛r♠♦s s♦❜r❡ ❛ ár❡❛ ❞♦ ❝ír❝✉❧♦✱ é ✐♥t❡r❡ss❛♥t❡ ❢❛③❡r ✉♠ ❝♦♠❡♥tár✐♦ s♦❜r❡ ♦ ❝♦♠♣r✐♠❡♥t♦ ❞❡ ✉♠❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ❡ ♦ s❡✉ ❞✐â♠❡tr♦✳ ❆ r❛③ã♦ ❡♥tr❡ ❡ss❛s ❞✉❛s ❣r❛♥❞❡③❛s ❞á s❡♠♣r❡ ♦ ♠❡s♠♦ ✈❛❧♦r✱ ♦✉ s❡❥❛✱ ✐♥❞❡♣❡♥❞❡ ❞♦ t❛♠❛♥❤♦ ❞❛ ❝✐r❝✉♥❢❡rê♥❝✐❛✱ ♣♦✐s t♦❞❛s ❛s ❝✐r❝✉♥❢❡rê♥❝✐❛s sã♦ s❡♠❡❧❤❛♥t❡s ❡♥tr❡ s✐✳ ❙❡ C é ♦ ❝♦♠♣r✐♠❡♥t♦ ❞❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ❞❡ r❛✐♦ R✱ ❡♥tã♦

  C ✏ π

  2R ♣♦r ❞❡✜♥✐çã♦✳

  ❉❡✈❡♠♦s✱ ❡♥tã♦✱ ❞❡✜♥✐r ❛❣♦r❛ ♦ q✉❡ s❡r✐❛ ♦ ❝♦♠♣r✐♠❡♥t♦ ❞❡ ✉♠❛ ❝✐r❝✉♥❢❡rê♥❝✐❛✳ ❉❡✜♥✐çã♦ ✸✳✶✵✳✶✳ ❖ ❝♦♠♣r✐♠❡♥t♦ ❞❡ ✉♠❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ é ♦ ♥ú♠❡r♦ r❡❛❧ ❝✉❥❛s ❛♣r♦①✐✲ ♠❛çõ❡s ♣♦r ❢❛❧t❛ sã♦ ♦s ♣❡rí♠❡tr♦s ❞♦s ♣♦❧í❣♦♥♦s r❡❣✉❧❛r❡s ✐♥s❝r✐t♦s ❡ ❝✉❥❛s ❛♣r♦①✐♠❛çõ❡s ♣♦r ❡①❝❡ss♦ sã♦ ♦s ♣❡rí♠❡tr♦s ❞♦s ♣♦❧í❣♦♥♦s r❡❣✉❧❛r❡s ❝✐r❝✉♥s❝r✐t♦s✳

  ✸✳✶✵ ➪r❡❛ ❞♦ ❈ír❝✉❧♦ ✾✵

  ❯s❛♥❞♦ ❛ ❞❡✜♥✐çã♦ ❞❡ ❛♣r♦①✐♠❛çã♦ ♣♦r ❢❛❧t❛✱ ❝♦♥s✐❞❡r❡ ✉♠ ♣♦❧í❣♦♥♦ r❡❣✉❧❛r ❝♦♠ n ❧❛❞♦s ✭n s✉✜❝✐❡♥t❡♠❡♥t❡ ❣r❛♥❞❡✮ ✐♥s❝r✐t♦ ❡♠ ✉♠❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ❞❡ r❛✐♦ R✳ ■r❡♠♦s

  ❞✐✈✐❞✐r ♦ ♣♦❧í❣♦♥♦ r❡❣✉❧❛r ❡♠ tr✐â♥❣✉❧♦s ✐sós❝❡❧❡s ✐❣✉❛✐s✱ t♦❞♦s ❝♦♠ ✈ért✐❝❡ ♥♦ ❝❡♥tr♦ ❞❛ ❝✐r❝✉♥❢❡rê♥❝✐❛✳ ❈❛❞❛ tr✐â♥❣✉❧♦ t❡♠ ❞♦✐s ❧❛❞♦s ✐❣✉❛✐s ❛ R✱ ✉♠ ❧❛❞♦ ✐❣✉❛❧ ❛ a✱ q✉❡ é ❧❛❞♦ ❞♦ ♣♦❧í❣♦♥♦✱ ❡ ❛❧t✉r❛ h r❡❧❛t✐✈❛ ❛ ❡ss❛ ❜❛s❡✳

  R h a

  O R

  n

  ❆ ár❡❛ S ❞❡ss❡ ♣♦❧í❣♦♥♦ é ❞❛❞♦ ♣♦r a p n ☎ h ♣n ☎ aq ☎ h ☎ h

  S , n ✏ n ☎ ✏ ✏ n

  2

  2

  2 n ♦♥❞❡ p é ♦ ♣❡rí♠❡tr♦ ❞♦ ♣♦❧í❣♦♥♦ ❞❡ n ❧❛❞♦s✳ ◗✉❛♥❞♦ n ❝r❡s❝❡ ✐♥❞❡✜♥✐❞❛♠❡♥t❡✱ p ❛♣r♦①✐♠❛✲s❡ ❞♦ ❝♦♠♣r✐♠❡♥t♦ ❞❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ❡ h ❛♣r♦①✐♠❛✲s❡ ❞♦ r❛✐♦✳ P♦rt❛♥t♦✱ ❢❛③❡♥❞♦ n n

  ❛r❜✐tr❛r✐❛♠❡♥t❡ ❣r❛♥❞❡ t❡♠♦s q✉❡ ❛ ár❡❛ S ❛♣r♦①✐♠❛✲s❡ ❞❛ ár❡❛ S ❞♦ ❝ír❝✉❧♦ ❞❡ r❛✐♦ R

  ✳ ❖✉ s❡❥❛✱ 2π 2

  ☎ R ☎ R S . ✏ ✏ π ☎ R

  2

  ✸✳✶✵✳✶ ➪r❡❛s ❞❡ s❡t♦r❡s

  ❆❧❣✉♠❛s ✈❡③❡s s❡rá ♥❡❝❡ssár✐♦ ❝❛❧❝✉❧❛r ár❡❛s ❞❡ s❡t♦r❡s✳ ◆♦t❡ q✉❡ ❛ ár❡❛ ❞❡ ✉♠ s❡t♦r ❞❡ ✉♠ ❝ír❝✉❧♦ é ♣r♦♣♦r❝✐♦♥❛❧ ❛♦ â♥❣✉❧♦ ❝❡♥tr❛❧ ♦✉✱ ❛✐♥❞❛✱ ♣r♦♣♦r❝✐♦♥❛❧ ❛♦ ❝♦♠♣r✐♠❡♥t♦ ❞♦ s❡✉ ❛r❝♦✳ ❙❡ ❞♦❜r❛r♠♦s ♦ â♥❣✉❧♦ ❝❡♥tr❛❧✱ ❞♦❜r❛♠♦s ❛ ár❡❛ ❞♦ s❡t♦r✱ tr✐♣❧✐❝❛♥❞♦ ♦ â♥❣✉❧♦ ❝❡♥tr❛❧✱ tr✐♣❧✐❝❛♠♦s ❛ ár❡❛ ❞♦ s❡t♦r✱ ❡ ❛ss✐♠ ♣♦r ❞✐❛♥t❡✳

  ❋❛③❡♥❞♦ ✉♠❛ r❡❣r❛ ❞❡ três s✐♠♣❧❡s ❡♥tr❡ ❛s ár❡❛s ❡ ♦s â♥❣✉❧♦s ❞♦ ❝ír❝✉❧♦ ❡ ❞♦ s❡t♦r ❝✐r❝✉❧❛r✱ ♥ã♦ ♥❡❝❡ss❛r✐❛♠❡♥t❡ ♥❡ss❛ ♦r❞❡♠✱ ♦❜t❡♠♦s ❛s s❡❣✉✐♥t❡s r❡❧❛çõ❡s✿

  ✭✐✮ q✉❛♥❞♦ ♦s â♥❣✉❧♦s sã♦ ♠❡❞✐❞♦s ❡♠ ❣r❛✉s α 2 S ,

  ✏ ☎ π ☎ R

  ✸✳✶✵ ➪r❡❛ ❞♦ ❈ír❝✉❧♦ ✾✶

  B R L α

  A O ✭✐✐✮ q✉❛♥❞♦ ♦s â♥❣✉❧♦s sã♦ ♠❡❞✐❞♦s ❡♠ r❛❞✐❛♥♦s

  α 2 S .

  ✏ ☎ R

  2 P♦r ♦✉tr♦ ❧❛❞♦✱ ❝♦♠♦ ❛ ár❡❛ ❞♦ s❡t♦r ❝✐r❝✉❧❛r t❛♠❜é♠ é ♣r♦♣♦r❝✐♦♥❛❧ ❛♦ ❝♦♠♣r✐♠❡♥t♦ L

  ❞♦ s❡✉ ❛r❝♦✱ ♣♦❞❡♠♦s ❡①♣r✐♠✐r s✉❛ ár❡❛ ❞❡ss❛ ❢♦r♠❛ L L 2

  ☎ R S . ✏ ☎ π ☎ R ✏

  2

  2 ☎ π ☎ R

  L ☎ R

  ❊ss❛ ❢ór♠✉❧❛ é ❜❛st❛♥t❡ ✐♥t❡r❡ss❛♥t❡✱ ♣♦✐s ♦ ❢♦r♠❛t♦ ❞á ❛ ✐❞❡✐❛ ❞❡ ✉♠ ✏tr✐â♥❣✉❧♦✑

  2 ❞❡ ❜❛s❡ L ❡ ❛❧t✉r❛ R✳

  ✸✳✶✶ Pr♦❜❧❡♠❛s ✾✷

  ✸✳✶✶ Pr♦❜❧❡♠❛s

  ✶✳ ❙❡ ABC é ✉♠ tr✐â♥❣✉❧♦ ✐sós❝❡❧❡s ❞❡ ❜❛s❡ BC✱ ♣r♦✈❡ q✉❡ ❛ ❜✐ss❡tr✐③✱ ❛ ♠❡❞✐❛♥❛ ❡ ❛ ❛❧t✉r❛ r❡❧❛t✐✈❛s ❛ BC ❝♦✐♥❝✐❞❡♠✳

  ✷✳ ❙❡❥❛♠ ∆ABC ✉♠ tr✐â♥❣✉❧♦ ❡ P ✱ M ❡ H r❡s♣❡❝t✐✈❛♠❡♥t❡ ♦s ♣és ❞❛ ❜✐ss❡tr✐③ ✐♥t❡r♥❛✱ ♠❡❞✐❛♥❛ ❡ ❛❧t✉r❛ r❡❧❛t✐✈❛s ❛♦ ❧❛❞♦ BC✳ ❙❡ P ❡ H ♦✉ M ❡ H ❝♦✐♥❝✐❞✐r❡♠✱ ♣r♦✈❡ q✉❡ ∆ABC

  é ✐sós❝❡❧❡s ❞❡ ❜❛s❡ BC A

  ✸✳ ❊♠ ✉♠ tr✐â♥❣✉❧♦ ❆❇❈ t❡♠♦s ˆ ✏ 90 ✳ ❙❡♥❞♦ P P AC ♦ ♣é ❞❛ ❜✐ss❡tr✐③ ✐♥t❡r♥❛ r❡❧❛t✐✈❛ ❛ B ❡ s❛❜❡♥❞♦ q✉❡ ❛ ❞✐stâ♥❝✐❛ ❞❡ P ❛♦ ❧❛❞♦ BC é ✐❣✉❛❧ ❛ 2 cm✱ ❝❛❧❝✉❧❡ ♦ ❝♦♠♣r✐♠❡♥t♦ ❞♦ s❡❣♠❡♥t♦ AP ✳

  ✹✳ ABC é ✉♠ tr✐â♥❣✉❧♦ ✐sós❝❡❧❡s ❞❡ ❜❛s❡ BC ❡ D P AB✱ E P AC sã♦ ♣♦♥t♦s t❛✐s q✉❡ ÐÑ DE k ÐÑ BC

  ✳ ❙❡♥❞♦ F ♦ ♣♦♥t♦ ❞❡ ✐♥tr❡s❡❝çã♦ ❞♦s s❡❣♠❡♥t♦s CD ❡ BE✱ ♠♦str❡ q✉❡ BF

  ✏ CF ✳ ✺✳ ◆❛ ✜❣✉r❛ ❛❜❛✐①♦✱ s❡ r k s✱ ♣r♦✈❡ q✉❡ α β ✏ γ✳ s

  β γ

  α r

  1 BC ✻✳ ❊♠ ✉♠ tr✐â♥❣✉❧♦ ABC✱ s❡❥❛ M ♦ ♣♦♥t♦ ♠é❞✐♦ ❞♦ ❧❛❞♦ BC✳ ❙❡ AM ✏ ✱ ♠♦str❡

  2

  ✆

  BAC q✉❡ ③ ✏ 90 ✳ ✼✳ ❙❡ I é ♦ ♣♦♥t♦ ❞❡ ✐♥t❡rs❡❝çã♦ ❞❛s ❜✐ss❡tr✐③❡s ✐♥t❡r♥❛s tr❛ç❛❞❛s ❛ ♣❛rt✐r ❞♦s ✈ért✐❝❡s

  1

  ✆

  ③ B BIC BAC

  ❡ C ❞❡ ✉♠ tr✐â♥❣✉❧♦ ABC✱ ♣r♦✈❡ q✉❡ ③ ✏ 90 ✳ a

  2 ✽✳ ❊♠ ✉♠ tr✐â♥❣✉❧♦ ABC✱ s❡❥❛ I ♦ ♣♦♥t♦ ❞❡ ✐♥t❡rs❡❝çã♦ ❞❛s ❜✐ss❡tr✐③❡s ❡①t❡r♥❛s

  1

  ✆

  ③ BI C BAC a r❡❧❛t✐✈❛s ❛♦s ✈ért✐❝❡s B ❡ C✳ Pr♦✈❡ q✉❡ ④ ✏ 90 ✁ ✳

  2 ✾✳ ◆♦ tr✐â♥❣✉❧♦ ABC✱ ♦ ♣♦♥t♦ D P BC✱ é ♦ ♣é ❞❛ ❜✐ss❡tr✐③ ✐♥t❡r♥❛ r❡❧❛t✐✈✈❛ ❛ A✳ Pr♦✈❡

  ADC ADB B C q✉❡ ④ ✁ ④ ✏ ♣ ✁ ♣ ✳

  ✸✳✶✶ Pr♦❜❧❡♠❛s ✾✸

  ✶✵✳ ❈❛❧❝✉❧❡ ❛ s♦♠❛ ❞♦s â♥❣✉❧♦s ♥♦s ✈ért✐❝❡s A, B, C, D, e E ❞❛ ❡str❡❧❛ ❞❡ ❝✐♥❝♦ ♣♦♥t❛s ❞❛ ✜❣✉r❛ ❛❜❛✐①♦✳

  A D C B E ④ ④ BAC ACD ABD. ✶✶✳ ◆❛ ✜❣✉r❛ ❛❜❛✐①♦✱ ♣r♦✈❡ q✉❡ α ✏ ③

  C B α

  D A ✶✷✳ ◆❛ ✜❣✉r❛✱ s❛❜❡♥❞♦ q✉❡ C é ♣♦♥t♦ ♠é❞✐♦ ❞❡ BE✱ ♣r♦✈❡ q✉❡ ♦s tr✐â♥❣✉❧♦s ABC ❡

  DEC sã♦ ❝♦♥❣r✉❡♥t❡s✳ D C B E A

  ✶✸✳ ◆❛ ✜❣✉r❛✱ s❛❜❡♥❞♦ q✉❡ α ✑ β ❡ γ ✑ δ✱ ♣r♦✈❡ q✉❡ ♦s tr✐â♥❣✉❧♦s ABC ❡ CDA sã♦ ❝♦♥❣r✉❡♥t❡s✳

  ✸✳✶✶ Pr♦❜❧❡♠❛s ✾✹

  C B δ

  β α

  γ 1 2 A D ✶✹✳ ❙❡❥❛♠ P ❡ P ♣♦♥t♦s ♥♦ ✐♥t❡r✐♦r ❞❡ ✉♠ s❡❣♠❡♥t♦ AB✱ t❛✐s q✉❡

  AP 1 AP 21 2 P 1 B P 2 B

  Pr♦✈❡ q✉❡ ♦s ♣♦♥t♦s P ❡ P ❝♦✐♥❝✐❞❡♠✳ ✶✺✳ ❉❛❞♦s s❡❣♠❡♥t♦s ❞❡ ❝♦♠♣r✐♠❡♥t♦s a ❡ b✱ ❞✐③❡♠♦s q✉❡ ✉♠ s❡❣♠❡♥t♦ ❞❡ ❝♦♠♣r✐♠❡♥t♦ x

  é ❛ t❡r❝❡✐r❛ ♣r♦♣♦r❝✐♦♥❛❧ ❞❡ a ❡ b ✭♥❡ss❛ ♦r❞❡♠✮ s❡ a b ✏ b c

  ▼♦str❡ ❝♦♠♦ ❝♦♥str✉✐r✱ ❝♦♠ ré❣✉❛ ❡ ❝♦♠♣❛ss♦✱ ✉♠ s❡❣♠❡♥t♦ ❞❡ t❛❧ ❝♦♠♣r✐♠❡♥t♦ x✳ ✶✻✳ ◆❛ ✜❣✉r❛✱ ♦s ❞♦✐s tr✐â♥❣✉❧♦s ABC ❡ CDE sã♦ ❡q✉✐❧át❡r♦s✳ ◗✉❛❧ é ♦ ✈❛❧♦r ❞♦ â♥❣✉❧♦ x

  ❄ F E B H

  ✆

  C x G

  ✆ ✆

  75

  65 A D ✶✼✳ ❆ ✜❣✉r❛ ♠♦str❛ ✉♠ r❡tâ♥❣✉❧♦ ❡ s✉❛s ❞✐❛❣♦♥❛✐s✳ ◗✉❛❧ ❛ r❡❧❛çã♦ ❡♥tr❡ ♦s â♠❣✉❧♦s x

  ❡ y ❞❛ ✜❣✉r❛❄ y x

  ✸✳✶✶ Pr♦❜❧❡♠❛s ✾✺

  ✶✽✳ ◆❛ ✜❣✉r❛ ∆ABC é ✉♠ tr✐â♥❣✉❧♦ ❡q✉✐❧át❡r♦ ❞❡ 3 cm ❞❡ ❧❛❞♦ ❡ ♦ tr✐â♥❣✉❧♦ r❡tâ♥❣✉❧♦ ∆BCD t❡♠ ❧❛❞♦s 3 cm, 4 cm ❡ 5 cm✳ ❈❛❧❝✉❧❡ ❛ ❞✐stâ♥❝✐❛ ❡♥tr❡ ♦s ♣♦♥t♦s A ❡ D✳

  C

  3

  5

  3 A

  3

  4 D B ✶✾✳ ❯♠ tr✐â♥❣✉❧♦ ✐sós❝❡❧❡s t❡♠ ❜❛s❡ ♠❡❞✐♥❞♦ 10 cm ❡ ❞♦✐s ❧❛❞♦s ✐❣✉❛✐s ❛ 13 cm✳ ➱

  ♣♦ssí✈❡❧ ♠✉❞❛r ❛ ❜❛s❡ ❞♦ tr✐â♥❣✉❧♦ ❡ ♦❜t❡r ♦✉tr♦ tr✐â♥❣✉❧♦ ✐sós❝❡❧❡s ❝♦♠ ♠❡s♠❛ ár❡❛❄

  13

  13

  10 ✷✵✳ ◆♦ q✉❛❞r✐❧át❡r♦ ABCD✱ t❡♠✲s❡✿ AB ✏ 5, BC ✏ 17, CD ✏ 5, DA ✏ 9✱ ❡ ❛ ♠❡❞✐❞❛

  ❞♦ s❡❣♠❡♥t♦ DB é ✉♠ ✐♥t❡✐r♦✳ ❉❡t❡r♠✐♥❡ DB✳ D A B C

  ✶✶

  ✷✶✳ ❖ tr✐â♥❣✉❧♦ ❞❡ ❘❡✉❧❡❛✉① é ✉♠ ❞✐s❝♦ ❢♦r♠❛❞♦ ❛ ♣❛rt✐r ❞❡ ✉♠ tr✐â♥❣✉❧♦ ❡q✉✐❧át❡r♦✱ ❛❣r❡❣❛♥❞♦ ❛r❝♦s ❞❡ ❝✐r❝✉♥❢❡rê♥❝✐❛ ❝♦♠ ❝❡♥tr♦s ♥♦s ✈ért✐❝❡s ❞♦ tr✐â♥❣✉❧♦ ❡ r❛✐♦s ✐❣✉❛✐s ✶✶ ❛♦ ❧❛❞♦ ❞❡ ✉♠ tr✐â♥❣✉❧♦✳

  ❋r❛♥③ ❘❡✉❧❡❛✉① ✭✸✵ ❞❡ s❡t❡♠❜r♦ ❞❡ ✶✽✷✾ ✲ ✷✵ ❞❡ ❛❣♦st♦ ❞❡ ✶✾✵✺✮ ❢♦✐ ✉♠ ❊♥❣❡♥❤❡✐r♦ ▼❡❝â♥✐❝♦ ❆❧❡♠ã♦

❝♦♥❤❡❝✐❞♦ ❝♦♠♦ P❛✐s ❞❛ ❈✐♥❡♠át✐❝❛✳ ❖ tr✐â♥❣✉❧♦ q✉❡ ❧❡✈❛ s❡✉ ♥♦♠❡ é ✉♠❛ ❝✉r✈❛ ❞❡ ❧❛r❣✉r❛ ❝♦♥st❛♥t❡ ❡

❝♦♠ ❜❛s❡ ❡♠ ✉♠ tr✐â♥❣✉❧♦ ❡q✉✐❧át❡r♦✳ ❚♦❞♦s ♦s ♣♦♥t♦s ❞❡ ✉♠ ❧❛❞♦ ❡stã♦ ❡q✉✐❞✐st❛♥t❡s ❞♦ ✈ért✐❝❡ ♦♣♦st♦✳

  ✸✳✶✶ Pr♦❜❧❡♠❛s ✾✻

  ◗✉❛❧ é ❛ ár❡❛ ❞❡ ✉♠ tr✐â♥❣✉❧♦ ❞❡ ❘❡✉❧❡❛✉①✱ s❡ ♦ tr✐â♥❣✉❧♦ ❡q✉✐❧át❡r♦ t❡♠ ❧❛❞♦ ❞❡ ♠❡❞✐❞❛ 1 cm❄

  ✷✷✳ ◆❛ ✜❣✉r❛✱ ABCD é ✉♠ ♣❛r❛❧❡❧♦❣r❛♠♦✱ ❡ M é ♦ ♣♦♥t♦ ♠é❞✐♦ ❞♦ s❡❣♠❡♥t♦ AD✳ B C O A M 2 D

  ❙❡ ❛ ár❡❛ ❞♦ q✉❛❞r✐❧át❡r♦ MOCD é ✐❣✉❛❧ ❛ 5 cm ✱ ❝❛❧❝✉❧❡ ❛ ár❡❛ ❞♦ tr✐â♥❣✉❧♦ AOM✳ ✷✸✳ ◆❛ ✜❣✉r❛ ❛❜❛✐①♦✱ ABCD é ✉♠ q✉❛❞r❛❞♦✱ ❡ ❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ❞❡ ❝❡♥tr♦ O é t❛♥❣❡♥t❡

  ❛♦s s❡❣♠❡♥t♦s DE ❡ CD✳ C D O A B 1 2 E

  ✭❛✮ ▼♦str❡ q✉❡ s❡ L é ❛ r❡t❛ q✉❡ ♣❛ss❛ ♣♦r AC ❡ L é ❛ r❡t❛ q✉❡ ♣❛ss❛ ♣♦r BO✱ ❡♥tã♦ L 1 2

  ❡ L sã♦ ♣❛r❛❧❡❧❛s✳ 2 ✭❜✮ ❙❛❜❡♥❞♦ q✉❡ ❛ ár❡❛ ❞♦ q✉❛❞r❛❞♦ ABCD é ✐❣✉❛❧ ❛ 36 cm ✱ ❝❛❧❝✉❧❡ ❛ ár❡❛ ❞♦ tr✐â♥❣✉❧♦

  ACO ✳

  ✸✳✶✶ Pr♦❜❧❡♠❛s ✾✼

  ✷✹✳ ◆❛ ✜❣✉r❛✱ P QRS é ✉♠ q✉❛❞r❛❞♦✱ QE ✏ 17✱ ❡ P H ✏ 12✳ Q P R H S E

  ❈❛❧❝✉❧❡ SE✳ ✷✺✳ ◆❛ ✜❣✉r❛✱ OC ✏ 12 ❡ DM ✏ 10✳

  B D O A M C ❈❛❧❝✉❧❡ BO✳

  ✷✻✳ ❆ ✜❣✉r❛ ♠♦str❛ ✉♠ r❡tâ♥❣✉❧♦ ABCD ❞❡ ❧❛❞♦ AD ✏ 4 ❡ ♦♥❞❡ ♦ ♣♦♥t♦ M é ♦ ♣♦♥t♦ ♠é❞✐♦ ❞♦ s❡❣♠❡♥t♦ AB✳

  A M B D C ❙❛❜❡♥❞♦ q✉❡ ♦s s❡❣♠❡♥t♦s AC ❡ DM sã♦ ♦rt♦❣♦♥❛✐s✱ ❝❛❧❝✉❧❡ ❛ ár❡❛ ❞♦ r❡tâ♥❣✉❧♦ ABCD

  ✳ ✷✼✳ ✭❊◆◗ ✷✵✶✺✳✶✮ ❆s ❞✐❛❣♦♥❛✐s ❆❉ ❡ ❈❊ ❞♦ ♣❡♥tá❣♦♥♦ r❡❣✉❧❛r ABCDE ❞❡ ❧❛❞♦s ❞❡

  ♠❡❞✐❞❛ a✱ ✐♥t❡rs❡❝t❛♠✲s❡ ♥♦ ♣♦♥t♦ P ✳ ❉❡t❡r♠✐♥❡ AP ❡ P D ❡♠ ❢✉♥çã♦ ❞❡ a✳

  ✸✳✶✶ Pr♦❜❧❡♠❛s ✾✽

  D E C P A B a b c ✷✽✳ ❈♦♥s✐❞❡r❡ ♦ tr✐â♥❣✉❧♦ ABC ❞❡ ❧❛❞♦s a✱ b✱ c ❡ ❛❧t✉r❛s h ✱ h ❡ h r❡❧❛t✐✈✈❛s r❡s♣❡❝t✐✲

  ✈❛♠❡♥t❡ ❛♦s ❧❛❞♦s a✱ b ❡ c✳ Pr♦✈❡ q✉❡ ♦ tr✐â♥❣✉❧♦ ABC é s❡♠❡❧❤❛♥t❡ ❛ ✉♠ tr✐â♥❣✉❧♦

  1

  1

  1 ❞❡ ❧❛❞♦s ✱ ❡ ✳ h h h a b c

  ✷✾✳ ◆❛ ✜❣✉r❛ ❛❜❛✐①♦ t❡♠♦s ✉♠ s❡♠✐❝ír❝✉❧♦ ❞❡ r❛✐♦ ✶ ✐♥s❝r✐t♦ ❡♠ ✉♠ q✉❛❞r❛❞♦ ❞❡ ♠♦❞♦ q✉❡ s❡✉ ❝❡♥tr♦ ♣❛ss❡ ♣♦r ✉♠❛ ❞❛s ❞✐❛❣♦♥❛✐s ❞♦ q✉❛❞r❛❞♦✳ ◗✉❛❧ é ❛ ár❡❛ ❞♦ q✉❛❞r❛❞♦❄

  ❈❛♣ít✉❧♦

  4 Pr✐♥❝í♣✐♦ ❋✉♥❞❛♠❡♥t❛❧ ❞❡ ❈♦♥t❛❣❡♠

  Pr♦❜❧❡♠❛s ❞❡ ❝♦♥t❛❣❡♠ ❡stã♦ ❡♥tr❡ ♦s ❝♦♥s✐❞❡r❛❞♦s ♠❛✐s ❞✐❢í❝❡✐s ♣❡❧♦s ♣r♦❢❡ss♦r❡s ❞♦ ❊♥s✐♥♦ ▼é❞✐♦❬✺❪✳ ❯♠ ❞♦s ♠♦t✐✈♦s é q✉❡ ❡ss❡ ❛ss✉♥t♦ só é ✐♥tr♦❞✉③✐❞♦ ♥❛ ♠❡t❛❞❡ ❞♦ ❊♥s✐♥♦ ▼é❞✐♦✱ ❛♣❡s❛r ❞❡ s❡r❡♠ ✉t✐❧✐③❛❞❛s té❝♥✐❝❛s ♠❛t❡♠át✐❝❛s ❡❧❡♠❡♥t❛r❡s ❞❡ ♦♣❡r❛çõ❡s ❛r✐t♠ét✐❝❛s✳ ➱ ❢á❝✐❧ ✈❡r q✉❡ ♦ ♣r♦❢❡ss♦r ♣♦❞❡ ❞❡s❡♥✈♦❧✈❡r ✉♠❛ ✐♥tr♦❞✉çã♦ ❞❡ss❡ ❛ss✉♥t♦ ♥♦ ❊♥s✐♥♦ ❋✉♥❞❛♠❡♥t❛❧✱ ❛✉①✐❧✐❛❞♦ ♣❡❧♦s ❝♦♥t❡ú❞♦s ♣r♦❣r❛♠át✐❝♦s ♣r❡s❡♥t❡s ❡♠ ❝❛❞❛ sé✲ r✐❡✳ ❆✜♥❛❧ ❞❡ ❝♦♥t❛s✱ ♦s ♣r♦❜❧❡♠❛s ❞❡ss❡s ❛ss✉♥t♦✱ ❡♠ s✉❛ ❣r❛♥❞❡ ♠❛✐♦r✐❛✱ ♣♦❞❡♠ s❡r r❡s♦❧✈✐❞♦s ❝♦♠ r❛❝✐♦❝í♥✐♦ s✐♠♣❧❡s s❡♠ ❡①✐❣✐r ♦ ✉s♦ ❞❡ ❢ór♠✉❧❛s ❝♦♠♣❧✐❝❛❞❛s✳ ❆ ♣r♦♣♦st❛ ♥❡ss❡ ❝❛♣ít✉❧♦ é ✉t✐❧✐③❛r té❝♥✐❝❛s ❞❡ r❡s♦❧✉çã♦ ❛tr❛✈és ❞♦ r❛❝✐♦❝í♥✐♦ s❡♠ ♥♦s ♣r❡♦❝✉♣❛r♠♦s ❝♦♠ ❞❡t❡r♠✐♥❛❞❛s ❢ór♠✉❧❛s✳

  ❱❡❥❛♠♦s ✉♠ ❡①❡♠♣❧♦ s✐♠♣❧❡s ♣❛r❛ ✐♥✐❝✐❛r♠♦s ❡ss❡ ❛ss✉♥t♦✳ ❊①❡♠♣❧♦ ✹✳✵✳✶✳ ❊♠ ✉♠ ❞❡t❡r♠✐♥❛❞♦ ♣❛ís✱ ♦ ❣♦✈❡r♥♦ ❞❡❝✐❞✐✉ r❡❛❧✐③❛r ✉♠ ♣❧❡❜✐s❝✐t♦ ♣❛r❛ ❞❡✜♥✐r ❛s ❝♦r❡s ❞❛ ❇❛♥❞❡✐r❛ ◆❛❝✐♦♥❛❧✳ ❖ ❢♦r♠❛t♦ ❞❛ ❜❛♥❞❡✐r❛ é r❡t❛♥❣✉❧❛r ❝♦♠ ✉♠ ❧♦s❛♥❣♦ ❞❡s❡♥❤❛❞♦✱ ❝♦♥❢♦r♠❡ ❛ ✜❣✉r❛✱ ❡ ❛s ❝♦r❡s ♣❛r❛ s❡r❡♠ ❡s❝♦❧❤✐❞❛s ❞✉❛s sã♦ ✈❡r❞❡✱ ❛♠❛r❡❧♦✱ ❛③✉❧ ❡ ❝✐♥③❛✳

  ✶✵✵ ▲✐st❡ t♦❞❛s ❛s ♣♦ssí✈❡✐s ❜❛♥❞❡✐r❛s✳ ◗✉❡ q✉❛♥t✐❞❛❞❡ é ❡ss❛❄ ❖❜s❡r✈❡ ✉♠❛ ❡str❛té❣✐❛ ❞❡ s♦❧✉çã♦ ❞❡ss❡ ❡①❡♠♣❧♦✳ ➱ ✐♠♣♦rt❛♥t❡ t❡r ✉♠ ♣r♦❝❡❞✐♠❡♥t♦ s✐st❡♠át✐❝♦ ♣❛r❛ ❧✐st❛r t♦❞❛s ❛s ♣♦ssí✈❡✐s ❜❛♥❞❡✐r❛s✱ s❡♠ r❡♣❡tí✲❧❛s✳ P❛r❛ t❛❧✱ ❞❡✈❡♠♦s ✐❞❡♥t✐✜❝❛r ❛s ❞✐❢❡r❡♥t❡s ❞❡❝✐sõ❡s ❛ s❡r❡♠ t♦♠❛❞❛s ❡

  ❡①❛♠✐♥❛r t♦❞❛s ❛s ♣♦ss✐❜✐❧✐❞❛❞❡s ♣❛r❛ ❝❛❞❛ ✉♠❛ ❞❡❧❛s✳ ◆♦ ❝❛s♦ ❞❡st❡ ♣r♦❜❧❡♠❛✱ ✉♠❛ ❢♦r♠❛ ♥❛t✉r❛❧ ♣❛r❛ ♣❧❛♥❡❥❛r ♦ ♣r❡❡♥❝❤✐♠❡♥t♦ ❞❛ ❜❛♥❞❡✐r❛ é ❡s❝♦❧❤❡r ❛ ❝♦r ❛ s❡r ✉t✐❧✐③❛❞❛ ♣❛r❛ ❛ ♣❛rt❡ ❡①t❡r♥❛ ❡✱ ❞❡♣♦✐s✱ ❡s❝♦❧❤❡r ❛ ❝♦r ❞♦ ❧♦s❛♥❣♦✳

  ❆ ♣r✐♠❡✐r❛ ❞❡❝✐sã♦ ♣♦❞❡ s❡r ❢❡✐t❛ ❞❡ q✉❛tr♦ ♠♦❞♦s ❞✐❢❡r❡♥t❡s✱ ❥á q✉❡ ❛ ❝♦r ❡①t❡r♥❛ ♣♦❞❡ s❡r q✉❛❧q✉❡r ✉♠❛ ❞❛s ❝♦r❡s ❞✐s♣♦♥í✈❡✐s✳ ❯♠❛ ✈❡③ t♦♠❛❞❛ ❡ss❛ ❞❡❝✐sã♦✱ ♥ã♦ ♣♦❞❡♠♦s ♠❛✐s r❡♣❡t✐r ❡ss❛ ❝♦r✱ r❡st❛♥❞♦ ❡♥tã♦ ❛ ♦♣çã♦ ❞❡ ❡s❝♦❧❤❡r três ❞❛s ❝♦r❡s ❞✐s♣♦♥í✈❡✐s✳

  P♦❞❡♠♦s✱ ❡♥tã♦✱ ❧✐st❛r t♦❞❛s ❛s ♣♦ssí✈❡✐s ❜❛♥❞❡✐r❛s✳ ❈♦♠ ❛ ❝♦r ❡①t❡r♥❛ ✈❡r❞❡ ❈♦♠ ❛ ❝♦r ❡①t❡r♥❛ ❛♠❛r❡❧❛

  ✶✵✶ ❈♦♠ ❛ ❝♦r ❡①t❡r♥❛ ❛③✉❧ ❈♦♠ ❛ ❝♦r ❡①t❡r♥❛ ❝✐♥③❛ P♦rt❛♥t♦✱ t❡♠♦s ✶✷ t✐♣♦s ❞❡ ❜❛♥❞❡✐r❛ ♣❛r❛ s❡r ❡s❝♦❧❤✐❞❛ ✉♠❛✳ ◆♦t❡ q✉❡ ❛ ❡s❝♦❧❤❛ ❞❛ ❝♦r ♣❛r❛ ♣✐♥t❛r ♦ ❧♦s❛♥❣♦ ♠✉❞❛ ❞❡ ❛❝♦r❞♦ ❝♦♠ ❛ ❞❡❝✐sã♦ ❞❛ ❝♦r

  ✉s❛❞❛ ♣❛r❛ ❛ ♣❛rt❡ ❡①t❡r♥❛✱ ♣♦ré♠ ❛ q✉❛♥t✐❞❛❞❡ ❝♦♥t✐♥✉❛ s❡♥❞♦ ❛ ♠❡s♠❛✱ ❥á q✉❡✱ q✉❛❧q✉❡r q✉❡ s❡❥❛ ❛ ❝♦r ❡①t❡r♥❛ ❡s❝♦❧❤✐❞❛✱ ❤á s❡♠♣r❡ três ❝♦r❡s r❡st❛♥t❡s ♣❛r❛ ♦ ❧♦s❛♥❣♦✳ ▲♦❣♦✱ ♣♦❞❡rí❛♠♦s t❡r ✉s❛❞♦ ♦ s❡❣✉✐♥t❡ r❛❝✐♦❝í♥✐♦ ♣❛r❛ ❝♦♥t❛r ♦ ♥ú♠❡r♦ ❞❡ ♣♦ssí✈❡✐s ❜❛♥❞❡✐r❛s✱ s❡♠ ❧✐stá✲❧❛s✿

  ❆ ❝♦r ❡①t❡r♥❛ ♣♦❞❡ s❡r ❡s❝♦❧❤✐❞❛ ❞❡ q✉❛tr♦ ♠♦❞♦s ❞✐❢❡r❡♥t❡s✳ ◗✉❛❧q✉❡r q✉❡ s❡❥❛ ❡st❛ ❡s❝♦❧❤❛✱ ❛ ❝♦r ❞♦ ❧♦s❛♥❣♦ ♣♦❞❡ s❡r ❡s❝♦❧❤✐❞❛ ❞❡ três ♠♦❞♦s✳ ▲♦❣♦✱ ♦ ♥ú♠❡r♦ t♦t❛❧ é ❞❡

  3

  3

  3

  3 ✏ 4 ☎ 3 ✏ 12 ♣♦ss✐❜✐❧✐❞❛❞❡s✳ ❊ss❡ ♠ét♦❞♦ é ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ Pr✐♥❝í♣✐♦

  ▼✉❧t✐♣❧✐❝❛t✐✈♦ ♦✉ Pr✐♥❝í♣✐♦ ❋✉♥❞❛♠❡♥t❛❧ ❞❛ ❈♦♥t❛❣❡♠✳ P♦❞❡♠♦s✱ ❡♥tã♦✱ ❡♥✉♥❝✐❛r ♦ Pr✐♥❝í♣✐♦ ❋✉♥❞❛♠❡♥t❛❧ ❞❛ ❈♦♥t❛❣❡♠ ❛ss✐♠✿ 1 1

  ❙❡ ✉♠❛ ❞❡❝✐sã♦ D ♣♦❞❡ s❡r t♦♠❛❞❛ ❞❡ m ♠♦❞♦s✱ ❡ q✉❛❧q✉❡r q✉❡ s❡❥❛ ❡ss❛ ❡s❝♦❧❤❛✱ 2 2 n ❛ ❞❡❝✐sã♦ D ♣♦❞❡ s❡r t♦♠❛❞❛ ❞❡ m ♠♦❞♦s✱ ❡ ❛ss✐♠✱ ❝♦♥t✐♥✉❛♠❡♥t❡✱ ❛té ❛ ❞❡❝✐sã♦ D ✱ n q✉❡ ♣♦❞❡ s❡r t♦♠❛❞❛ ❞❡ m ♠♦❞♦s✱ ❡♥tã♦ ♦ ♥ú♠❡r♦ ❞❡ ♠❛♥❡✐r❛s ❞❡ s❡ t♦♠❛r❡♠ ❝♦♥s❡✲ 1 , D 2 , n 1 2 n

  ❝✉t✐✈❛♠❡♥t❡ ❛s ❞❡❝✐sõ❡s D ☎ ☎ ☎ , D é ✐❣✉❛❧ ❛ m ☎ m ☎ ☎ ☎ m ♠♦❞♦s ❞❡ s❡ t♦♠❛r ❛ ❞❡❝✐sã♦✳

  ❱❡❥❛♠♦s ✉♠ ❡①❡♠♣❧♦ ♣❛r❛ ✐❧✉str❛r ❡ss❡ ❡♥✉♥❝✐❛❞♦✳ ❊①❡♠♣❧♦ ✹✳✵✳✷✳ ❊♠ ✉♠❛ s❛❧❛ ❤á ❞❡③ ♣❡ss♦❛s ❡ ❛♣❡♥❛s ❝✐♥❝♦ ❝❛❞❡✐r❛s ♣❛r❛ s❡♥t❛r✳ ❉❡ q✉❛♥t❛s ♠❛♥❡✐r❛s ♣♦❞❡♠♦s ❛❝♦♠♦❞❛r ❝✐♥❝♦ ♣❡ss♦❛s s❡♥t❛❞❛s ❞❛s ❞❡③ ♣❡ss♦❛s ❡①✐st❡♥t❡s ♥❛

  ✶✵✷ ❯♠❛ s♦❧✉çã♦ s✐♠♣❧❡s ♣♦❞❡ s❡r ❢❡✐t❛ ♣❛r❛ ❞❡♠♦♥str❛r ♦ ❡♥✉♥❝✐❛❞♦ é ❛ s❡❣✉✐♥t❡✿ ❊♥✉♠❡r❛♠♦s ❛s ♣❡ss♦❛s ❞❡ 1 ❛ 10 ❡ ♥♦♠❡❛♠♦s ❛s ❝❛❞❡✐r❛s ❞❡ A ❛ E✳ ❆ ♥♦ss❛ ♣r✐♠❡✐r❛ 1

  ❞❡❝✐sã♦ D é ❡s❝♦❧❤❡r q✉❛❧ ♣❡ss♦❛ ✐rá s❡ s❡♥t❛r ♥❛ ❝❛❞❡✐r❛ A✱ ♦ q✉❡ ♣♦❞❡ s❡r ❢❡✐t♦ ❞❡ 2 ✶✵ ♠❛♥❡✐r❛s✱ ♣♦✐s ❤á 10 ♣❡ss♦❛s ♥❛ s❛❧❛✳ ❊s❝♦❧❤✐❞❛ ❡ss❛ ♣❡ss♦❛✱ ❡♥tã♦ ❛ ❞❡❝✐sã♦ D s❡r✐❛ ❡s❝♦❧❤❡r q✉❛❧ ❞❛s ♥♦✈❡ ♣❡ss♦❛s r❡st❛♥t❡s s❡♥t❛r✐❛ ♥❛ ❝❛❞❡✐r❛ B✱ ♦ q✉❡ ♣♦❞❡ s❡r ❢❡✐t♦ 5

  ❞❡ ♥♦✈❡ ♠❛♥❡✐r❛s✳ ❋❛③❡♥❞♦✱ ❛ss✐♠✱ ❛té ❛ ❞❡❝✐sã♦ D ✱ q✉❡ s❡r✐❛ ❡s❝♦❧❤❡r ❞❡♥tr❡ ❛s s❡✐s ♣❡ss♦❛s r❡st❛♥t❡s q✉❛❧ s❡♥t❛r✐❛ ♥❛ ❝❛❞❡✐r❛ E✱ ♣♦✐s ❥á ❛❝♦♠♦❞❛rí❛♠♦s ♥❛s ❝❛❞❡✐r❛s C ❡ D

  ❛s ♦✐t♦ ❡ s❡t❡ ♣❡ss♦❛s✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ q✉❡ s❡ ❡♥❝♦♥tr❛r✐❛♠ ♥❛ s❛❧❛✳ P♦rt❛♥t♦✱ ❤á

  10 ☎ 9 ☎ 8 ☎ 7 ☎ 6 ✏ 30.240 ♠♦❞♦s ❞✐❢❡r❡♥t❡s ❞❡ ❡s❝♦❧❤❡r ❝✐♥❝♦ ❞❡♥tr❡ ❛s ❞❡③ ♣❡ss♦❛s ♣❛r❛ s❡ s❡♥t❛r❡♠ ♥❛s ❝❛❞❡✐r❛s ♣r❡s❡♥t❡s ♥❡ss❛ s❛❧❛✳

  ❱❡❥❛♠♦s ✉♠ ♦✉tr♦ ❡①❡♠♣❧♦ ❝♦♠ ✉♠ ♥í✈❡❧ ✉♠ ♣♦✉❝♦ ♠❛✐s ❡❧❡✈❛❞♦✳ ❊①❡♠♣❧♦ ✹✳✵✳✸✳ P❛r❛ ♣✐♥t❛r ❛ ❜❛♥❞❡✐r❛ ❞❛ ✜❣✉r❛✱ ❤á ❝✐♥❝♦ ❝♦r❡s ❞✐s♣♦♥í✈❡✐s✳

  ❉❡ q✉❛♥t♦s ♠♦❞♦s ❡❧❛ ♣♦❞❡ s❡r ♣✐♥t❛❞❛ ❞❡ ♠♦❞♦ q✉❡ ❢❛✐①❛s ❛❞❥❛❝❡♥t❡s t❡♥❤❛♠ ❝♦r❡s ❞✐❢❡r❡♥t❡s❄

  ➱ ♥❡ss❛ ♣❛rt❡ ❞♦ ❛ss✉♥t♦ q✉❡ ❛❝❤♦ q✉❡ ❡❧❡ ❝♦♠❡ç❛ ❛ ✜❝❛r ✐♥t❡r❡ss❛♥t❡✱ ♣♦✐s ♣❛ss❛♠♦s ♣♦r ✐ss♦ t♦❞♦s ♦s ❞✐❛s ❞❛ ♥♦ss❛ ✈✐❞❛✱ q✉❡ é ❛ t♦♠❛❞❛ ❞❡ ❞❡❝✐sã♦✳ ❙❡ t♦♠❛r♠♦s ✉♠❛ ❞❡❝✐sã♦ ❡rr❛❞❛✱ ♣♦❞❡r❡♠♦s t❡r ❞✐✜❝✉❧❞❛❞❡ ♥❛ r❡s♦❧✉çã♦ ❞♦ ♣r♦❜❧❡♠❛✱ ♦✉ ❛té✱ ❡♥tã♦✱ ♥ã♦ ❝♦♥s❡❣✉✐r❡♠♦s r❡s♦❧✈ê✲❧♦✳

  ❱❡❥❛♠♦s ✉♠❛ s♦❧✉çã♦✳ ❖ ♣r✐♠❡✐r♦ ♣❛ss♦ é ❡s❝♦❧❤❡r♠♦s ❛ ♦r❞❡♠ ❡♠ q✉❡ ✐r❡♠♦s ♣✐♥t❛r ❛ ❜❛♥❞❡✐r❛✳ ❉ê ♣r❡❢❡rê♥❝✐❛ ❛ ✉♠❛ s❡q✉ê♥❝✐❛ ❧ó❣✐❝❛ ❡ ♥❛t✉r❛❧✳ P♦r ❡①❡♠♣❧♦✱ ✐r❡♠♦s ♣✐♥tá✲❧❛ ❞❡ ❜❛✐①♦ ♣❛r❛ ❝✐♠❛✳ ❆♣ós ❡ss❛ ❡s❝♦❧❤❛✱ t❡♠♦s q✉❡ ❤á ❝✐♥❝♦ ❝♦r❡s ♣❛r❛ ❛ ♣r✐♠❡✐r❛ ❢❛✐①❛ ❞❛ ♣❛rt❡ ❞❡ ❜❛✐①♦ ♣❛r❛ ♣✐♥tá✲❧❛✳ ❊s❝♦❧❤✐❞❛ ❡ss❛ ❝♦r✱ ♥ã♦ ♣♦❞❡r❡♠♦s ♠❛✐s ✉sá✲❧❛ ♥❛ ❢❛✐①❛ s✉♣❡r✐♦r ❛ ❡❧❛✱ ♣♦rt❛♥t♦ t❡r❡♠♦s q✉❛tr♦ ❝♦r❡s ❞✐s♣♦♥í✈❡✐s ♣❛r❛ ♣✐♥t❛r ❛ s❡❣✉♥❞❛ ❢❛✐①❛✳ P❛r❛ ❛ t❡r❝❡✐r❛ ❢❛✐①❛✱ ♥ã♦ ♣♦❞❡r❡♠♦s ✉t✐❧✐③❛r ❛ ❝♦r ✉s❛❞❛ ♥❛ s❡❣✉♥❞❛ ❢❛✐①❛✱ ♠❛s ♣♦❞❡r❡♠♦s ✉s❛r ❛ ❝♦r q✉❡ ❢♦✐ ♣✐♥t❛❞❛ ❛ ♣r✐♠❡✐r❛✱ ❡ ♣♦rt❛♥t♦✱ t❡♠♦s q✉❛tr♦ ❝♦r❡s ❞✐s♣♦♥í✈❡✐s ♣❛r❛ ♣✐♥t❛r

  ✶✵✸ ❛ t❡r❝❡✐r❛ ❢❛✐①❛✱ ❛ss✐♠ ❝♦♠♦✱ t❡r❡♠♦s q✉❛tr♦ ❝♦r❡s ♣❛r❛ ♣✐♥t❛r♠♦s ❛ q✉❛rt❛ ❢❛✐①❛✳ ▲♦❣♦✱ t❡♠♦s 5 ☎ 4 ☎ 4 ☎ 4 ✏ 320 ♠❛♥❡✐r❛s ❞❡ ♣✐♥t❛r♠♦s ❡ss❛ ❜❛♥❞❡✐r❛✳

  ❱❡❥❛ ♦✉tr♦ ❡①❡♠♣❧♦ ❡♠ q✉❡ t❡♠♦s q✉❡ t❡r ♦ ❝✉✐❞❛❞♦ ♥❛ ❤♦r❛ ❞❡ t♦♠❛r ❛ ❞❡❝✐sã♦✳ ❊①❡♠♣❧♦ ✹✳✵✳✹✳ ◗✉❛♥t♦s sã♦ ♦s ♥ú♠❡r♦s ❞❡ q✉❛tr♦ ❞í❣✐t♦s ❞✐st✐♥t♦s❄

  ❊ss❡ é ✉♠ ❡①❡♠♣❧♦ ❝❧áss✐❝♦ ❡♠ q✉❡ ❞❡✈❡♠♦s ✜❝❛r ❛t❡♥t♦s ❛♦ ❢❛t♦ ❞❡ q✉❡ ♥ã♦ ♣♦❞❡♠♦s ❝♦❧♦❝❛r ♦ ❞í❣✐t♦ ③❡r♦ ♥❛ ❢r❡♥t❡ ❞♦s ♦✉tr♦s três ♥ú♠❡r♦s✳ ❊ss❛ é ✉♠❛ ❞✐✜❝✉❧❞❛❞❡ s✐♠♣❧❡s✱ ♠❛s q✉❡ ♥ã♦ ♣♦❞❡♠♦s ❞❡✐①á✲❧❛ ♣❛r❛ ❞❡♣♦✐s ❡♠ ♥♦ss❛ ❞❡❝✐sã♦ ❞❡ r❡s♦❧✉çã♦ ❞♦ ♣r♦❜❧❡♠❛✳ P♦rt❛♥t♦✱ ❞❡✈❡♠♦s ❡♥✉♠❡r❛r ♦s ❞í❣✐t♦s ❞❛ ❡sq✉❡r❞❛ ♣❛r❛ ❛ ❞✐r❡✐t❛✱ ♦♥❞❡ ♥♦ ♣r✐♠❡✐r♦ ❞í❣✐t♦ ♣♦❞❡ s❡r ❡s❝♦❧❤✐❞♦ ❞❡ ♥♦✈❡ ♠♦❞♦s✳ ❈♦♠♦ ♥ã♦ ♣♦❞❡♠♦s r❡♣❡t✐r ♦ ❛❧❣❛r✐s♠♦✱ ❡♥tã♦ ♦ q✉❡ ❢♦✐ ✉s❛❞♦ ♥♦ ♣r✐♠❡✐r♦ ❞í❣✐t♦ ♥ã♦ ♣♦❞❡rá s❡r ♠❛✐s ✉s❛❞♦✱ ♣♦ré♠ ♦ ❛❧❣❛r✐s♠♦ ③❡r♦ ❥á ♣♦❞❡ s❡r ✉s❛❞♦✱ ♦ q✉❡ ♥♦s ❞á ♥♦✈❡ ♠♦❞♦s ❞❡ ♣r❡❡♥❝❤❡r ♦ s❡❣✉♥❞♦ ❞í❣✐t♦✳ ❆ ♣❛rt✐r ❞❛í✱ t❡♠♦s ♦✐t♦ ♠♦❞♦s ❞❡ ♣r❡❡♥❝❤❡r♠♦s ♦ t❡r❝❡✐r♦ ❞í❣✐t♦ ❡ s❡t❡ ❞❡ ♣r❡❡♥❝❤❡r♠♦s ♦ q✉❛rt♦ ❞í❣✐t♦✱ ♥♦ t♦t❛❧ ❞❡ 9 ☎ 9 ☎ 8 ☎ 7 ✏ 4.536 ♥ú♠❡r♦s ❞❡ q✉❛tr♦ ❛❧❣❛r✐s♠♦s ❞✐st✐♥t♦s✳

  ❯♠ ❡①❡♠♣❧♦ q✉❡ ♠❡ ❢❛s❝✐♥❛✱ ♦♥❞❡ ❥á ✈✐ ❞✉❛s ❢♦r♠❛s ❞❡ s❡ r❡s♦❧✈❡r ❡ ♦ q✉❡ ❝♦♠♣r♦✈❛ q✉❡ ❛ ❞❡❝✐sã♦ ❡rr❛❞❛ ♥♦s ❞á ❛ ❞✐✜❝✉❧❞❛❞❡ ❞❡ r❡s♦❧✈❡r ♦ ♣r♦❜❧❡♠❛ é ❡ss❡ q✉❡ s❡❣✉❡✳ ❊①❡♠♣❧♦ ✹✳✵✳✺✳ ◆✉♠❛ ❡s❝♦❧❛ ♦s ♣r♦❢❡ss♦r❡s ❞❛s ❞✐s❝✐♣❧✐♥❛s ❞❡ ❇✐♦❧♦❣✐❛✱ ❋ís✐❝❛✱ ▼❛t❡♠á✲ t✐❝❛ ❡ ◗✉í♠✐❝❛ ❡s❝♦❧❤❡r❛♠ ✉♠❛ s❡♠❛♥❛ ❞♦ ♠ês ❞❡ s❡t❡♠❜r♦ ♣❛r❛ ❛♣❧✐❝❛r s✉❛s ♣r♦✈❛s s❡♠ ❝♦♠❜✐♥❛r ♦ ❞✐❛ ✉♠ ❝♦♠ ♦ ♦✉tr♦✳ ❉❡ q✉❛♥t❛s ❢♦r♠❛s ♣♦❞❡♠ s❡r r❡❛❧✐③❛❞❛s ❡ss❛s ♣r♦✈❛s ♥❡ss❛ s❡♠❛♥❛❄

  ❈♦♠♦ ♣r♦❢❡ss♦r❡s✱ ♣♦❞❡rí❛♠♦s t❡r ❛ s❡♥s✐❜✐❧✐❞❛❞❡ ❞❡ ♥♦t❛r q✉❡ ♣♦❞❡rí❛♠♦s t❡r ❛s q✉❛tr♦ ♣r♦✈❛s ❡♠ ✉♠ ❞✐❛ só✱ ❜❡♠ ❝♦♠♦ t❡r ✉♠❛ ♣r♦✈❛ ❛ ❝❛❞❛ ❞✐❛✱ ❝♦♠ ✉♠ ❞✐❛ s❡♠ ♣r♦✈❛✳ ❊ss❛ ❢♦r♠❛ ❞❡ r❛❝✐♦❝✐♥❛r ♣❛r❛ r❡s♦❧✈❡r ♦ ❡①❡♠♣❧♦ ❞á ❝❡rt♦✱ ♣♦ré♠ é ✉♠❛ ❞❡❝✐sã♦ q✉❡ ♥♦s ♣r♦♣♦r❝✐♦♥❛ ♠✉✐t❛ ❝♦♥t❛✱ ♦ q✉❡ t♦r♥❛r✐❛ ❝❛♥s❛t✐✈♦ s✉❛ r❡s♦❧✉çã♦✳ ❱❡❥❛ ✉♠❛ s✐♠♣❧❡s s♦❧✉çã♦ ♣❛r❛ ❡ss❡ ❡①❡♠♣❧♦✳

  ❈❛❞❛ ♣r♦❢❡ss♦r ❞❡ ✉♠❛ ❞❡t❡r♠✐♥❛❞❛ ❞✐s❝✐♣❧✐♥❛ ♣♦❞❡ ❡s❝♦❧❤❡r ✉♠ ❞❡♥tr❡ ♦s ❝✐♥❝♦ ❞✐❛s ❡①✐st❡♥t❡s ♥❛q✉❡❧❛ s❡♠❛♥❛ ♣❛r❛ ❡s❝♦❧❤❡r ❛♣❧✐❝❛r s✉❛ ♣r♦✈❛✱ s❡♠ s❡ ♣r❡♦❝✉♣❛r ❝♦♠ ♦ ❞✐❛ ❞❛s ♣r♦✈❛s ❞❛s ♦✉tr❛s ❞✐s❝✐♣❧✐♥❛s✳ ❈♦♠♦ sã♦ q✉❛tr♦ ❞✐s❝✐♣❧✐♥❛s✱ t❡r❡♠♦s 5 ☎ 5 ☎ 5 ☎ 5 ✏ 625 ♠❛♥❡✐r❛s ❞❡ s❡r❡♠ ❛♣❧✐❝❛❞❛s ❛s ♣r♦✈❛s ♥❡ss❛ s❡♠❛♥❛✳

  ❯♠ ❡①❡♠♣❧♦ q✉❡ é ❞♦ ♥♦ss♦ ❝♦t✐❞✐❛♥♦ ❊①❡♠♣❧♦ ✹✳✵✳✻✳ ◗✉❛♥t❛s ♣❧❛❝❛s ❛✉t♦♠♦t✐✈❛s sã♦ ♣♦ssí✈❡✐s✱ s❛❜❡♥❞♦ q✉❡ ❡♠ ❞❡t❡r♠✐♥❛❞♦ ❡st❛❞♦ ❛s ♣❧❛❝❛s ♣♦❞❡♠ ✐♥✐❝✐❛r ❝♦♠ ❛s ❧❡tr❛s N ♦✉ M❄ ❖❜s✿ ❍á ✷✻ ❧❡tr❛s ❡♠ ♥♦ss♦ ❛❧❢❛❜❡t♦ ❡ ♣♦❞❡♠♦s tr❛❜❛❧❤❛r ❝♦♠ ✶✵ ❞í❣✐t♦s✳ ❆s ♣❧❛❝❛s ❛✉t♦♠♦t✐✈❛s sã♦ ❝♦♠♣♦st❛s ♣♦r

  ✶✵✹ ❉❡✈❡♠♦s ❝♦♠❡ç❛r ♣❡❧❛ ♥♦ss❛ ♠❛✐♦r r❡str✐çã♦✱ q✉❡ ♥❡ss❡ ❝❛s♦ é ❛ ♣r✐♠❡✐r❛ ❧❡tr❛ ❞❛ ♣❧❛❝❛

  ❥á q✉❡ ❡❧❛ só ❛❞♠✐t❡ ❞✉❛s ❧❡tr❛s ♥❡❧❛✳ ❈♦♠♦ ♥ã♦ ❤á r❡str✐çã♦ ❞❡ r❡♣❡t✐çã♦ ❡♥tã♦ ♣♦❞❡♠♦s ✉s❛r ❛s 26 ❧❡tr❛s ♣❛r❛ ❛ s❡❣✉♥❞❛ ❡ ♣❛r❛ ❛ t❡r❝❡✐r❛ ❧❡tr❛✱ ❛ss✐♠ ❝♦♠♦ ♣❛r❛ ❝❛❞❛ ❞í❣✐t♦ ❞♦s ♥ú♠❡r♦s ❤á 10 ♣♦ss✐❜✐❧✐❞❛❞❡s✱ ❞❛♥❞♦ ✉♠ t♦t❛❧ ❞❡ 2 ☎ 26 ☎ 26 ☎ 10 ☎ 10 ☎ 10 ☎ 10 ✏ 13.520.000 ♣❧❛❝❛s ❛✉t♦♠♦t✐✈❛s ♥❡ss❡ ❡st❛❞♦✳

  ❱❡❥❛♠♦s ✉♠ ❡①❡♠♣❧♦ ✐♥t❡r❡ss❛♥t❡✱ ♦♥❞❡ ♣r❡❝✐s❛♠♦s t❡r ♣♦st✉r❛ ♣❛r❛ t♦♠❛r ❛ ❞❡❝✐sã♦ ❝♦rr❡t❛ ♥♦ ♣r♦❜❧❡♠❛ ❡ ♥ã♦ ❛❞✐❛r ❛s ❞✐✜❝✉❧❞❛❞❡s ♣❛r❛ r❡s♦❧✈ê✲❧♦✳ ❊①❡♠♣❧♦ ✹✳✵✳✼✳ ◗✉❛♥t♦s sã♦ ♦s ♥ú♠❡r♦s ♣❛r❡s ❞❡ q✉❛tr♦ ❞í❣✐t♦s ❞✐st✐♥t♦s❄

  ■r❡♠♦s s❡♣❛r❛r ❡♠ ❞♦✐s ❝❛s♦s ❞✐s❥✉♥t♦s✳ ❈♦♠❡ç❛♥❞♦ ❝♦♠ ♦s ♥ú♠❡r♦s q✉❡ t❡r♠✐♥❛♠ ❡♠ ③❡r♦✱ ♥❡ss❡ ❝❛s♦✱ ❛ ❝❛s❛ ❞❛s ✉♥✐❞❛❞❡s ❡stá ❞❡✜♥✐❞❛ ❝♦♠ ♦ ❛❧❣❛r✐s♠♦ ③❡r♦✳ P♦rt❛♥t♦✱ t❡♠♦s 9 ☎ 8 ☎ 7 ☎ 1 ✏ 504 ♥ú♠❡r♦s✳

  ❆❣♦r❛✱ ❛q✉❡❧❡s ♥ú♠❡r♦s q✉❡ sã♦ ♣❛r❡s✱ ♠❛s ♥ã♦ t❡r♠✐♥❡♠✱ ♥❡♠ ❝♦♠❡ç❡♠ ❝♦♠ ③❡r♦✳ ❆ss✐♠✱ t❡r❡♠♦s 8 ☎ 8 ☎ 7 ☎ 4 ✏ 1792 ♥ú♠❡r♦s✳

  ❉❡ss❛ ❢♦r♠❛✱ ❝♦♥❝❧✉✐✲s❡ q✉❡ ❤á 504 1792 ✏ 2296 ♥ú♠❡r♦s ♣❛r❡s ❝♦♠ q✉❛tr♦ ❞í❣✐t♦s ❞✐st✐♥t♦s✳

  ✶✵✺

Pr♦❜❧❡♠❛s

  ✶✳ ❊♠ ✉♠❛ s❛❧❛ ❤á ✉♠❛ t✉r♠❛ ❞❡ ❞❡③ ❛❧✉♥♦s✳ Pr❡❝✐s❛✲s❡ ❡s❝♦❧❤❡r ✉♠❛ ❝♦♠✐ssã♦ ❞❡ três ❛❧✉♥♦s ♣❛r❛ r❡♣r❡s❡♥t❛r ❡ss❛ t✉r♠❛✱ s❡♥❞♦ ❛ ❝♦♠✐ssã♦ ❝♦♠♣♦st❛ ♣♦r✿ ✉♠ ♣♦rt❛✲✈♦③✱ ✉♠ ❞✐r❡t♦r ❞❡ ❛rt❡s ❡ ✉♠ ❛ss❡ss♦r té❝♥✐❝♦✳ ◆❡♥❤✉♠ ❛❧✉♥♦ ♣♦❞❡ ❛❝✉♠✉❧❛r ❝❛r❣♦s✳

  ✭❛✮ ❉❡ q✉❛♥t❛s ♠❛♥❡✐r❛s ❡st❛ ❝♦♠✐ssã♦ ♣♦❞❡ s❡r ❢♦r♠❛❞❛❄ ✭❜✮ ◗✉❛♥t❛s ❝♦♠✐ssõ❡s ❞✐❢❡r❡♥t❡s ♣♦❞❡♠ s❡r ❞♦r♠❛❞❛s ❝♦♠ ♦s ❛❧✉♥♦s ▲❡❛♥❞r♦✱

  ❘❡♥❛t♦ ❡ ▼❛r❝❡❧♦❄ ✭❝✮ ❈♦♥s✐❞❡r❡ ❛❣♦r❛ ❝♦♠✐ssõ❡s s❡♠ ❝❛r❣♦s ❡s♣❡❝í✜❝♦s✳ ❯s❡ ♦s ✐t❡♥s ♣aq ❡ ♣bq ❛♥✲ t❡r✐♦r❡s ♣❛r❛ ❞❡s❝♦❜r✐r q✉❛♥t❛s ❝♦♠✐ssõ❡s s❡♠ ❝❛r❣♦s ❡s♣❡❝í✜❝♦s ♣♦❞❡♠ s❡r

  ❢♦r♠❛❞❛s✳ ✷✳ ❆ s❡❣✉✐♥t❡ ✜❣✉r❛ ♠♦str❛ ✉♠ ❞♦❞❡❝á❣♦♥♦ r❡❣✉❧❛r✳

  ❘❡s♣♦♥❞❛✿ ✭❛✮ ◗✉❛♥t♦s tr✐â♥❣✉❧♦s ❡q✉✐❧át❡r♦s ♣♦❞❡♠ s❡r ❢♦r♠❛❞♦s ❞❡ ♠♦❞♦ q✉❡ s❡✉s três

  ✈ért✐❝❡s s❡❥❛♠ ✈ért✐❝❡s ❞♦ ❞♦❞❡❝á❣♦♥♦❄ ✭❜✮ ◗✉❛♥t♦s r❡tâ♥❣✉❧♦s ❡s❝❛❧❡♥♦s ♣♦❞❡♠ s❡r ❢♦r♠❛❞♦s ❞❡ ♠♦❞♦ q✉❡ s❡✉s três ✈ér✲ t✐❝❡s s❡❥❛♠ ✈ért✐❝❡s ❞♦ ❞♦❞❡❝á❣♦♥♦❄

  ✸✳ ◆✉♠❛ ❧✐st❛ ❞❡ t♦❞♦s ♦s ♥ú♠❡r♦s ❞❡ 5 ❛❧❣❛r✐s♠♦s ❞✐st✐♥t♦s q✉❡ s❡ ❢♦r♠❛♠ ❝♦♠ ♦s ❞í❣✐t♦s 1, 2, 3, 4, 5✱ ❡ss❡s ♥ú♠❡r♦s ❡stã♦ ♦r❞❡♥❛❞♦s ❡♠ ♦r❞❡♠ ❝r❡s❝❡♥t❡✳

  ✭❛✮ ◗✉❛❧ ♦ ♥ú♠❡r♦ q✉❡ ♦❝✉♣❛ ❛ ♣♦s✐çã♦ 10 ❞❛ ❧✐st❛❄ ✭❜✮ ◗✉❛❧ ♦ ♥ú♠❡r♦ q✉❡ ♦❝✉♣❛ ❛ ♣♦s✐çã♦ 85 ❞❛ ❧✐st❛❄

  ✹✳ ◆✉♠ ❣r✉♣♦ ❞❡ ✷✵ ♣❡ss♦❛s✱ ❛❧❣✉♠❛s ♣❡ss♦❛s tr♦❝❛♠ ❛♣❡rt♦s ❞❡ ♠ã♦✳ ✭❛✮ ❈♦♥t❛♠♦s q✉❛♥t♦s ❛♣❡rt♦s ❞❡ ♠ã♦ ❝❛❞❛ ♣❡ss♦❛ ❞❡✉ ❡ s♦♠❛♠♦s t♦❞♦s ❡ss❡s

  ✶✵✻ ✭❜✮ ➱ ♣♦ssí✈❡❧ q✉❡ ♥✉♠ ❣r✉♣♦ ❞❡ ✾✾ ♣❡ss♦❛s ❝❛❞❛ ♣❡ss♦❛ t❡♥❤❛ ❞❛❞♦ ❡①❛t❛♠❡♥t❡ três ❛♣❡rt♦s ❞❡ ♠ã♦❄

  ✺✳ ❖ ♣❛✐ ❞❡ ❱✐❝t♦r ❡ ▼❛②❛r❛ ❝❤❡❣♦✉ à ❝❛s❛ ❝❛rr❡❣❛♥❞♦ ❞❡③ ❜r✐♥q✉❡❞♦s ❞✐st✐♥t♦s ❡ ❡♥✉♠❡r❛❞♦s ❞❡ ✶ ❛ ✶✵ ❡ ❞✐ss❡ ❛ ❡❧❡s q✉❡ ♦ ❜r✐♥q✉❡❞♦ ♥ú♠❡r♦ ✶ é ♣❛r❛ ❱✐❝t♦r ❡ ♦ ❞❡ ♥ú♠❡r♦ ✷ é ♣❛r❛ ▼❛②❛r❛✳ P♦ré♠✱ s❡✉s ✜❧❤♦s ♣♦❞❡rã♦ ✜❝❛r ❝♦♠ ♠❛✐s ❜r✐♥q✉❡❞♦✱ ❝♦♥t❛♥t♦ q✉❡ ❞❡✐①❡♠ ❛♦ ♠❡♥♦s ✉♠ ♣❛r❛ s❡✉ ♣❛✐✳ ❉❡ q✉❛♥t♦s ♠♦❞♦s ❱✐❝t♦r ❡ ▼❛②❛r❛ ♣♦❞❡rã♦ ❞✐✈✐❞✐r ❡♥tr❡ ❡❧❡s ♦ r❡st❛♥t❡ ❞♦s ❜r✐♥q✉❡❞♦s❄

  ✻✳ ❯♠ ❛❧✉♥♦ ❡s❝r❡✈❡✉ t♦❞♦s ♦s ♥ú♠❡r♦ ✐♥t❡✐r♦s ❝♦♠♣r❡❡♥❞✐❞♦s ❡♥tr❡ ✶✵✵ ❡ ✾✾✾ ❝✉❥❛ s♦♠❛ ❞♦s ❛❧❣❛r✐s♠♦s é ✶✷✳ P♦r ❡①❡♠♣❧♦✱ ♦s ♥ú♠❡r♦s ✶✷✾ ❡ ✼✺✵ ❛♣❛r❡❝❡♠ ❡♥tr❡ ♦s ♥ú♠❡r♦s ❡s❝r✐t♦s✳

  ✭❛✮ ◗✉❛♥t♦s ♥ú♠❡r♦s ❡s❝r✐t♦s tê♠ ❛♣❡♥❛s ❞♦✐s ❛❧❣❛r✐s♠♦s ✐❣✉❛✐s❄ ✭❜✮ ◗✉❛♥t♦s ♥ú♠❡r♦s ❡s❝r✐t♦s sã♦ ❢♦r♠❛❞♦s ❛♣❡♥❛s ♣♦r ❛❧❣❛r✐s♠♦s í♠♣❛r❡s❄

  ✼✳ ◗✉❛tr♦ t✐♠❡s ❞✐s♣✉t❛r❛♠ ✉♠ t♦r♥❡✐♦ ❞❡ ❢✉t❡❜♦❧ ❡♠ q✉❡ ❝❛❞❛ ✉♠ ❥♦❣♦✉ ✉♠❛ ✈❡③ ❝♦♥tr❛ ❝❛❞❛ ✉♠ ❞♦s ♦✉tr♦s✳ ◗✉❛♥❞♦ ✉♠❛ ♣❛rt✐❞❛ t❡r♠✐♥❛✈❛ ❡♠♣❛t❛❞❛✱ ❝❛❞❛ t✐♠❡ ❣❛♥❤❛✈❛ ✉♠ ♣♦♥t♦✱ ❝❛s♦ ❝♦♥trár✐♦✱ ♦ ✈❡♥❝❡❞♦r ❣❛♥❤❛✈❛ três ♣♦♥t♦s ❡ ♦ ♣❡r❞❡❞♦r ③❡r♦✳ ❆ t❛❜❡❧❛ ♠♦str❛ ❛ ♣♦♥t✉❛çã♦ ✜♥❛❧ ❞♦ t♦r♥❡✐♦✳ ◗✉❛♥t♦s ❢♦r❛♠ ♦s ❡♠♣❛t❡s❄

  ❚✐♠❡ P♦♥t♦s ❆ ✺ ❇ ✸ ❈ ✸ ❉ ✷

  ✽✳ ❉❡s❡❥❛✲s❡ ❝♦❧♦r✐r ♦s ❝ír❝✉❧♦s ❞❛s ❋✐❣✉r❛s ✶✱ ✷ ❡ ✸ ❞❡ ❛③✉❧ ✭❆✮✱ ♣r❡t♦ ✭P✮ ♦✉ ✈❡r♠❡❧❤♦ ✭❱✮ ❞❡ ♠♦❞♦ q✉❡ ❜♦❧✐♥❤❛s ❧✐❣❛❞❛s ♣♦r ✉♠ s❡❣♠❡♥t♦ t❡♥❤❛♠ ❝♦r❡s ❞✐❢❡r❡♥t❡s✳

  F igura 1 F igura 2

  F igura 3

  ✶✵✼ ✭❛✮ ❉❡ q✉❛♥t❛s ♠❛♥❡✐r❛s ❞✐❢❡r❡♥t❡ ♣♦❞❡♠♦s ❝♦❧♦r✐r ❛ ❋✐❣✉r❛ ✶❄

  ✭❜✮ ❉❡ q✉❛♥t❛s ♠❛♥❡✐r❛s ❞✐❢❡r❡♥t❡ ♣♦❞❡♠♦s ❝♦❧♦r✐r ❛ ❋✐❣✉r❛ ✷❄ ✭❝✮ ❉❡ q✉❛♥t❛s ♠❛♥❡✐r❛s ❞✐❢❡r❡♥t❡ ♣♦❞❡♠♦s ❝♦❧♦r✐r ❛ ❋✐❣✉r❛ ✸❄

  ✾✳ ✭❊◆◗ ✷✵✶✹✳✷✮ ❈♦♥s✐❞❡r❡ q✉❡ ❢♦r❛♠ ❡❢❡t✉❛❞❛s t♦❞❛s ❛s ♣❡r♠✉t❛çõ❡s ♣♦ssí✈❡✐s ❞♦s ❛❧❣❛r✐s♠♦s q✉❡ ❝♦♠♣õ❡♠ ♦ ♥ú♠❡r♦ ✼✽✺✷✸✱ ❧✐st❛♥❞♦ ♦s ♥ú♠❡r♦s ♦❜t✐❞♦s ❡♠ ♦r❞❡♠ ❝r❡s❝❡♥t❡✳

  ✭❛✮ ❉❡t❡r♠✐♥❡ ❛ ♣♦s✐çã♦ ♦❝✉♣❛❞❛ ♣❡❧♦ ♥ú♠❡r♦ ✼✽✺✷✸✳ ✭❜✮ ❈❛❧❝✉❧❡ ❛ s♦♠❛ ❞❡ t♦❞♦s ♦s ♥ú♠❡r♦s ❧✐st❛❞♦s✳

  ✶✵✳ ◆❛ t❛❜❡❧❛ ❛❜❛✐①♦✱ ♦ ♦❜❥❡t✐✈♦ é ♣r❡❡♥❝❤❡r ✉♠ t❛❜✉❧❡✐r♦ ❞❡ ❞✉❛s ❧✐♥❤❛s ❡ ✷✵✶✺ ❝♦❧✉♥❛s

  ✶ ✶

  s s ❝♦♠ ❛❧❣❛r✐s♠♦s 0 ❡ 1 ❞❡ ♠♦❞♦ q✉❡ ❞♦✐s ♥ú♠❡r♦s ✈✐③✐♥❤♦s ✐❣✉❛✐s ❞❡ ✉♠❛ ♠❡s♠❛ ❧✐♥❤❛ ✐♠♣❡❞❡♠ q✉❡ s❡ ♣r❡❡♥❝❤❛ ❝♦♠ ♥ú♠❡r♦ ✐❣✉❛✐s ❛s ❝❛s❛s ❝♦rr❡s♣♦♥❞❡♥t❡s ❞❛ ♦✉tr❛ ❧✐♥❤❛✳ P♦r ❡①❡♠♣❧♦✱ ♥❛ t❛❜❡❧❛ ❛❜❛✐①♦✱ ♦s ✈❛❧♦r❡s ❞❡ A ❡ B ♥ã♦ ♣♦❞❡♠ s❡r ✐❣✉❛✐s✳

  ✵ ✶ ✵ ☎ ☎ ☎ ✶ ✶ ☎ ☎ ☎ ✶ ✶ ✵ ☎ ☎ ☎ A B ☎ ☎ ☎

  ❉❡t❡r♠✐♥❡ ♦ ♥ú♠❡r♦ ❞❡ ♣♦ssí✈❡✐s ♣r❡❡♥❝❤✐♠❡♥t♦s ❞✐st✐♥t♦s ❞❡ t❛❧ t❛❜✉❧❡✐r♦ s❡❣✉✐♥❞♦ ❛s r❡❣r❛s✳

  ✶✶✳ ✭❊◆❆ P❘❖❋▼❆❚ ✲ ✷✵✶✹✮ ❯♠ ❝✉❜♦ t❡♠ ❛s ❢❛❝❡s ♥✉♠❡r❛❞❛s ❞❡ ✶ ❛ ✻ ✭❢❛❝❡s ♦♣♦st❛s ♥ã♦ s♦♠❛♠ ✼ ♥❡❝❡ss❛r✐❛♠❡♥t❡✮✳ ❙ã♦ ❢❡✐t♦s ✷ ❧❛♥ç❛♠❡♥t♦s✳ ◆♦ ♣r✐♠❡✐r♦ ❧❛♥ç❛♠❡♥t♦✱ ❛ s♦♠❛ ❞♦s ♥ú♠❡r♦s ❣r❛✈❛❞♦s ♥❛s ✹ ❢❛❝❡s ❧❛t❡r❛✐s é ✶✺✳ ◆✉♠ s❡❣✉♥❞♦ ❧❛♥ç❛♠❡♥t♦✱ ❛ s♦♠❛ ❞♦s ♥ú♠❡r♦s ❣r❛✈❛❞♦s ♥❛s ✹ ❢❛❝❡s ❧❛t❡r❛✐s é ✶✷✳ ◗✉❛❧ ♦ ♥ú♠❡r♦ q✉❡ ❡stá ♥❛ ❢❛❝❡ ♦♣♦st❛ à ❢❛❝❡ ❞♦ ✻❄

  ✭❛✮ ✶ ✭❜✮ ✷

  ✭❝✮ ✸ ✭❞✮ ✹

  ✭❡✮ ✺

  ✶✵✽ ✶✷✳ ✭❊◆❆ P❘❖❋▼❆❚ ✲ ✷✵✶✹✮ ❯♠❛ ♣❡ss♦❛ ♣r❡❝✐s❛ ❛t✐✈❛r ♦ ❛❧❛r♠❡ ❞❡ s✉❛ r❡s✐❞ê♥❝✐❛

  ❝✉❥❛ s❡♥❤❛ é ❢♦r♠❛❞❛ ♣♦r q✉❛tr♦ ❛❧❣❛r✐s♠♦s ❞✐st✐♥t♦s ❡ ❡❧❛ ❧❡♠❜r❛ ❛♣❡♥❛s q✉❡ ♦ ♣r✐♠❡✐r♦ ❛❧❣❛r✐s♠♦ é ✺ ❡ q✉❡ ♦ ú❧t✐♠♦ é ✸✳ ❆ ❝❡♥tr❛❧ ❞♦ ❛❧❛r♠❡ ♣❡r♠✐t❡ ❛♣❡♥❛s ✉♠❛ t❡♥t❛t✐✈❛ ❛ ❝❛❞❛ ✷ ♠✐♥✉t♦s✳ ◆❛ ♣✐♦r s✐t✉❛çã♦ ♣♦ssí✈❡❧✱ q✉❛❧ é ❛ ♠❛✐♦r ❞✉r❛çã♦ ❞❡ t❡♠♣♦ q✉❡ ❡❧❛ ♣♦❞❡rá ❧❡✈❛r ♣❛r❛ ❛t✐✈❛r ♦ ❛❧❛r♠❡ ❞✐❣✐t❛♥❞♦ s❡♥❤❛ ❛♣ós s❡♥❤❛❄

  ✭❛✮ 1 h 22 min ✭❜✮ 1 h 50 min

  ✭❝✮ 2 h 02 min ✭❞✮ 2 h 10 min

  ✭❡✮ 2 h 20 min ✶✸✳ ✭❊◆❆ P❘❖❋▼❆❚ ✲ ✷✵✶✹✮ ◗✉❛♥t❛s ♣❛❧❛✈r❛s ♣♦❞❡♠♦s ❡s❝r❡✈❡r ❝♦♠ ❛s s❡✐s ❧❡tr❛s

  ❛✱ ❜✱ ❝✱ ❞✱ ❡✱ ❢✱ s❡♠ r❡♣❡t✐r ❧❡tr❛s✱ ❞❡ ♠♦❞♦ q✉❡ ❛s ❧❡tr❛s ❛✱ ❜✱ ❝ s❡♠♣r❡ ❛♣❛r❡ç❛♠ ♥❛ ♦r❞❡♠ ❛❧❢❛❜ét✐❝❛❄

  ✭❛✮ ✶✷✵ ✭❜✮ ✷✹✵

  ✭❝✮ ✸✻✵ ✭❞✮ ✼✶✺

  ✭❡✮ ✼✷✵ ✶✹✳ ✭❊◆❆ P❘❖❋▼❆❚ ✲ ✷✵✶✺✮ ❯♠❛ ❡s❝♦❧❛ ❞❡ ❡❞✉❝❛çã♦ ❜ás✐❝❛ ♣♦ss✉✐ ✶✷ ♣r♦❢❡ss♦✲ r❡s ❞❡ ♠❛t❡♠át✐❝❛✱ s❡♥❞♦ q✉❡ ♦✐t♦ ❛t✉❛♠ ❡①❝❧✉s✐✈❛♠❡♥t❡ ♥♦ ❊♥s✐♥♦ ❋✉♥❞❛♠❡♥t❛❧

  ❛

  ❡ q✉❛tr♦ ❛t✉❛♠ ❡①❝❧✉s✐✈❛♠❡♥t❡ ♥♦ ❊♥s✐♥♦ ▼é❞✐♦✳ P❛r❛ ❛ ♦r❣❛♥✐③❛çã♦ ❞❛ ✶ ❖❧✐♠✲ ♣í❛❞❛ ❞❡ ▼❛t❡♠át✐❝❛ ❞❛ ❡s❝♦❧❛✱ s❡rá ❢♦r♠❛❞❛ ✉♠❛ ❝♦♠✐ssã♦ ❞❡ ❝✐♥❝♦ ♣r♦❢❡ss♦r❡s ❞❡ ♠❛t❡♠át✐❝❛✱ ❞❡ ♠♦❞♦ q✉❡ ♣❡❧♦ ♠❡♥♦s ✉♠ ❞❡❧❡s s❡❥❛ ♣r♦❢❡ss♦r ❞♦ ❊♥s✐♥♦ ▼é❞✐♦✳ ❉❡ q✉❛♥t❛s ♠❛♥❡✐r❛s ❡ss❛ ❝♦♠✐ssã♦ ♣♦❞❡rá s❡r ❢♦r♠❛❞❛❄

  ✶✺✳ ✭❊◆❆ P❘❖❋▼❆❚ ✲ ✷✵✶✺✮ ❊♠ ✉♠ ❝❡rt♦ ♣❛ís✱ ♦s ♣♦ssí✈❡✐s ♥ú♠❡r♦s ❞❡ t❡❧❡❢♦♥❡s ❝❡❧✉❧❛r❡s ❡r❛♠ ❢♦r♠❛❞♦s ♣♦r ♦✐t♦ ❛❧❣❛r✐s♠♦s✱ ✉t✐❧✐③❛♥❞♦✲s❡ ❞í❣✐t♦s ❞❡ ✵ ❛ ✾✱ ✐♥✐❝✐❛✲ ❞♦s✱ ♦❜r✐❣❛t♦r✐❛♠❡♥t❡✱ ❝♦♠ ✾✱ ✽ ♦✉ ✼✳ ❈♦♠ ♦ ❝r❡s❝✐♠❡♥t♦ ❞❛ ♣♦♣✉❧❛çã♦✱ ❤♦✉✈❡ ❛ ♥❡❝❡ss✐❞❛❞❡ ❞❡ s❡ ❝r✐❛r ♥♦✈♦s ♥ú♠❡r♦s✳ ❖s ♥ú♠❡r♦s ❛♥t✐❣♦s ❢♦r❛♠ ♠❛♥t✐❞♦s✱ ❛♣❡♥❛s r❡❝❡❜❡♥❞♦ ✉♠ ❛❧❣❛r✐s♠♦ ✾ ❡♠ s❡✉ ✐♥í❝✐♦✱ ♣❛ss❛♥❞♦ ❛ss✐♠ ❛ t❡r ✾ ❛❧❣❛r✐s♠♦s✳ ❏á ♦s

  ✶✵✾ ♥ú♠❡r♦s ♥♦✈♦s✱ ❢♦r♠❛❞♦s t❛♠❜é♠ ❝♦♠ ♥♦✈❡ ❛❧❣❛r✐s♠♦s✱ tê♠ ❛ ú♥✐❝❛ r❡str✐çã♦ ❞❡ ❝♦♠❡ç❛r ❝♦♠ ♦ ❞í❣✐t♦ ✾✳ ❉❡st❛ ♠❛♥❡✐r❛✱ q✉❛♥t♦s ♥ú♠❡r♦s ❛ ♠❛✐s ❢♦r❛♠ ❝r✐❛❞♦s❄

  ✶✻✳ ✭❖❇▼ ✷✵✵✺✮ ◆✉♠ r❡❧ó❣✐♦ ❞✐❣✐t❛❧✱ ❛s ❤♦r❛s sã♦ ❡①✐❜✐❞❛s ♣♦r ♠❡✐♦ ❞❡ q✉❛tr♦ ❛❧❣❛✲ r✐s♠♦s✳ ❖ r❡❧ó❣✐♦ ✈❛r✐❛ ❞❛s ✵✵✿✵✵ às ✷✸✿✵✵ ❤♦r❛s✳ ◗✉❛♥t❛s ✈❡③❡s ♣♦r ❞✐❛ ♦s q✉❛tr♦ ❛❧❣❛r✐s♠♦s ♠♦str❛❞♦s sã♦ t♦❞♦s ♣❛r❡s❄

  ❈❛♣ít✉❧♦

  5

❙♦❧✉çõ❡s ❞♦s Pr♦❜❧❡♠❛s

  ✺✳✶ ❚ó♣✐❝♦s ❞❡ ▼❛t❡♠át✐❝❛ ❊❧❡♠❡♥t❛r ✺✳✶✳✶ Pr✐♥❝í♣✐♦ ❞❛ ■♥❞✉çã♦ ❋✐♥✐t❛

  ✶✳ ✭❛✮ ◗✉❡r❡♠♦s ✈❛❧✐❞❛r ❛ ❢ór♠✉❧❛ 2 2 2 n

  ☎ ♣n 1q ☎ ♣2n 1q P 2 .

  ♣nq : 1 ☎ ☎ ☎ n ✏

  6 ◆♦t❡ q✉❡ 2 ☎ ♣1 1q ☎ ♣2 ☎ 1 1q

  1 P ♣1q : 1 ✏

  6 é ✈❡r❞❛❞❡✐r❛✳ ❙✉♣♦♥❤❛ q✉❡ ♣❛r❛ ❛❧❣✉♠ n P N✱ s❡ t❡♥❤❛ q✉❡ P ♣nq é ✈❡r❞❛❞❡✐r❛✱ ✐st♦ é✱ 2 2 2 n

  ☎ ♣n 1q ☎ ♣2n 1q

  1 2 .

  ☎ ☎ ☎ n ✏

  6 ❆ ♣❛rt✐r ❞❛ ❤✐♣ót❡s❡ ❞❡ ✐♥❞✉çã♦✱ ✐r❡♠♦s ♣r♦✈❛r q✉❡ P ♣n 1q t❛♠❜é♠ é ✈❡r❞❛❞❡✐r❛✳

  ✺✳✶ ❚ó♣✐❝♦s ❞❡ ▼❛t❡♠át✐❝❛ ❊❧❡♠❡♥t❛r ✶✶✶

  ♣n 1q ☎ r2 ☎ ♣n 1q 2 ♣n 1q 2♣n 1q 1s

  ♣n 1q

  2 ☎ ☎ ☎ n ✏ n

  1

  ♣1q : 1 3 ✏ 1 2 é ✈❡r❞❛❞❡✐r❛ ❡ q✉❡ ❛ ✐❣✉❛❧❞❛❞❡

  P❡r❝❡❜❛ q✉❡ P

  2 3 ☎ ☎ ☎ n 3 ✏ ♣1 2 ☎ ☎ ☎ nq 2 .

  1 3

  ✭❜✮ ❉❛ ♠❡s♠❛ ❢♦r♠❛✱ q✉❡r❡♠♦s ✈❛❧✐❞❛r ❛ ❢ór♠✉❧❛

  ❡st❛❜❡❧❡❝❡♥❞♦✱ ❛ss✐♠✱ ❛ ✈❡r❛❝✐❞❛❞❡ ❞❡ P ♣n 1q✳ P♦rt❛♥t♦✱ ♣❡❧♦ Pr✐♥❝í♣✐♦ ❞❛ ■♥❞✉çã♦ ❋✐♥✐t❛✱ ❛ ❢ór♠✉❧❛ é ✈á❧✐❞❛ ♣❛r❛ t♦❞♦ n P N✳

  6 ,

  ♣n 1q ☎ r♣n 1q 1s ☎ r2♣n 1q 1s

  6 ✏

  6 ✏

  ❉❡ ❢❛t♦✱ s♦♠❛♥❞♦ ♣n 1q 2 ❛ ❛♠❜♦s ♦s ❧❛❞♦s ❞❛ ✐❣✉❛❧❞❛❞❡ ❛❝✐♠❛✱ t❡♠♦s q✉❡

  2 q ♣n 1q ♣2n 2q 1s

  ♣n 1q ☎ r♣2n 2 4n

  6 ✏

  ♣n 1q ☎ r♣2n 2 n 6n 6 qs

  6 ✏

  ♣n 1q ☎ rn ☎ ♣2n 1q 6 ☎ ♣n 1qs

  6 ✏

  ✏ n ☎ ♣n 1q ☎ ♣2n 1q 6 ☎ ♣n 1q 2

  6 ♣n 1q 2

  ☎ ♣n 1q ☎ ♣2n 1q

  2 2 ☎ ☎ ☎ n 2 ♣n 1q 2 ✏ n

  1 2

  2 é ✈á❧✐❞❛✳ ❙✉♣♦♥❤❛✱ ❡♥tã♦ q✉❡ ♣❛r❛ ❛❧❣✉♠ n P N ❛ ❡①♣r❡ssã♦ P ♣nq s❡❥❛ ✈❡r❞❛❞❡✐r❛✳ ❙♦♠❛♥❞♦

  ✺✳✶ ❚ó♣✐❝♦s ❞❡ ▼❛t❡♠át✐❝❛ ❊❧❡♠❡♥t❛r ✶✶✷

  ☎

  ☎ n 2 ✏ ♣✁1q n

  ✁1

  ☎ n ☎ ♣n 1q

  2 . ◆♦t❡ q✉❡

  P ♣1q : 1 ✏ ♣✁1q 1

  ✁1

  1 ☎ ♣1 1q

  3 2 ✁ ☎ ☎ ☎ ♣✁1q n

  2 é ✈❡r❞❛❞❡✐r❛✳ ❙✉♣♦♥❤❛ q✉❡ ♣❛r❛ ❛❧❣✉♠ n P N✱ s❡ t❡♥❤❛ q✉❡ P ♣nq é ✈❡r❞❛❞❡✐r❛✱ ✐st♦ é✱

  1 ✁ 2 2

  3 2 ✁ ☎ ☎ ☎ ♣✁1q n

  

✁1

  ☎ n 2 ✏ ♣✁1q n

  ✁1

  ☎ n ☎ ♣n 1q

  ✁1

  ♣nq : 1 ✁ 2 2

  ♣n 1q 3 ❡♠ ❛♠❜♦s ♠❡♠❜r♦s ❞❛ ✐❣✉❛❧❞❛❞❡✱ t❡♠♦s q✉❡

  4 ✏

  1 3

  2 3 ☎ ☎ ☎ n 3 ♣n 1q 3

  ✒ n ☎ ♣n 1q

  2 ✚ 2

  ♣n 1q 3 ✏ n 2

  ☎ ♣n 1q 2

  4 ♣n 1q 3

  ♣n 1q 2 ☎ rn 2

  ✭❝✮ ◗✉❡r❡♠♦s ✈❛❧✐❞❛r ❛ ❢ór♠✉❧❛ P

  4 ☎ ♣n 1qs

  4 ✏

  ♣n 1q 2 ☎ ♣n 2q 2

  4 ✏

  ✒ ♣n 1q ☎ ♣n 2q

  2 ✚ 2

  , s❡♥❞♦ ✈❡r❞❛❞❡ ♣❛r❛ P ♣n 1q✳ ▲♦❣♦✱ ❛ ✐❣✉❛❧❞❛❞❡ é ✈á❧✐❞❛ ♣❛r❛ t♦❞♦ n P N✳

  2 . ❆ ♣❛rt✐r ❞❛ ❤✐♣ót❡s❡ ❞❡ ✐♥❞✉çã♦ ✐r❡♠♦s ♣r♦✈❛r q✉❡ P ♣n 1q t❛♠❜é♠ é ✈❡r❞❛❞❡✐r❛✳

  ✺✳✶ ❚ó♣✐❝♦s ❞❡ ▼❛t❡♠át✐❝❛ ❊❧❡♠❡♥t❛r n 2 ✶✶✸

  ❉❡ ❢❛t♦✱ s♦♠❛♥❞♦ ♣✁1q ☎ ♣n 1q ❛ ❛♠❜♦s ♦s ♠❡♠❜r♦s ❞❛ ✐❣✉❛❧❞❛❞❡✱ t❡♠♦s q✉❡ 2 n n ☎ ♣n 1q n 2 n 2

  ✁1

  1 ✁ 2 ☎ ☎ ☎ ♣✁1q ☎ ♣n 1q ✏ ♣✁1q ☎ ♣✁1q ☎ ♣n 1q n

  2 n 2 ♣✁1q ☎ ♣n 1q n

  ✏ ☎ ♣✁1q ☎ ♣n 1q

  2 ♣✁1q

  ✒ ✚ n 2 n ☎ ♣n 1q

  ✏ ✁♣✁1q ☎ ♣n 1q ☎ ✁

  2

  2 n ♣n 2q ✒ ✚ ✏ ✁♣✁1q ☎ ♣n 1q ☎ ✁

  2 n ♣n 1q ☎ ♣n 2q ,

  ✏ ♣✁1q ☎

  2 ❡st❛❜❡❧❡❝❡♥❞♦ ❛ ✈❡r❛❝✐❞❛❞❡ ♣❛r❛ P ♣n 1q✳ P♦rt❛♥t♦✱ ♣❡❧♦ Pr✐♥❝í♣✐♦ ❞❛ ■♥❞✉çã♦ ❋✐♥✐t❛✱ ❛ ❢ór♠✉❧❛ é ✈á❧✐❞❛ ♣❛r❛ t♦❞♦ n P N✳

  ✷✳ ✭❛✮ ◗✉❡r❡♠♦s ♣r♦✈❛r q✉❡

  P ♣nq : n ✁ n ✏ 6k, ♣❛r❛ ❛❧❣✉♠ k P Z. ◆♦t❡ q✉❡ 3 P

  ♣1q : 1 ✁ 1 ✏ 0 ✏ 6 ☎ 0 é ✈❡r❞❛❞❡✐r❛✳ ❙✉♣♦♥❤❛ q✉❡ ♣❛r❛ ❛❧❣✉♠ n P N✱ s❡ t❡♥❤❛ q✉❡ P ♣nq é ✈❡r❞❛❞❡✐r❛✱ ✐st♦ é✱ 3 n

  ✁ n ✏ 6k. ❆ ♣❛rt✐r ❞❛ ❤✐♣ót❡s❡ ❞❡ ✐♥❞✉çã♦ ✐r❡♠♦s ♣r♦✈❛r q✉❡ P ♣n 1q t❛♠❜é♠ é ✈❡r❞❛❞❡✐r❛✳

  ✺✳✶ ❚ó♣✐❝♦s ❞❡ ▼❛t❡♠át✐❝❛ ❊❧❡♠❡♥t❛r ✶✶✹

  ❉❡ ❢❛t♦✱ s✉❜st✐t✉✐♥❞♦ n ♣♦r n 1✱ t❡♠♦s q✉❡ 3 2 ♣n 1q ✁ ♣n 1q ✏ ♣n 1q ☎ r♣n 1q ✁ 1s 2

  2n ✏ ♣n 1q ☎ rn s 3 2 2

  2n n 2n ✏ n 3 2 2 2

  3n n 3n ✏ n ✁ n ✁ n 3

  ✏ ♣n ✁ nq 3n ☎ ♣n 1q ✏ 6k 3 ☎ 2t ✏ 6 ☎ ♣k tq.

  ❊st❛❜❡❧❡❝❡♥❞♦ ❛ ✈❡r❛❝✐❞❛❞❡ ♣❛r❛ P ♣n 1q✳ 3 P♦rt❛♥t♦✱ ♣❡❧♦ Pr✐♥❝í♣✐♦ ❞❛ ■♥❞✉çã♦ ❋✐♥✐t❛✱ n ✁ n é ✉♠ ♠ú❧t✐♣❧♦ ❞❡ 6✱ ♣❛r❛ t♦❞♦ n P N✳

  ✭❜✮ ◗✉❡r❡♠♦s ♣r♦✈❛r q✉❡ n P ♣nq : 5 ✁ 1 ✏ 24 ☎ t, ♣❛r❛ t♦❞♦ n P N, n é ♣❛r. ❯s❛♥❞♦ q✉❡ n ✏ 2k✱ ❝♦♠ k P N✱ ❛♣❧✐❝❛r❡♠♦s ♦ Pr✐♥❝í♣✐♦ ❞❛ ■♥❞✉çã♦ ❋✐♥✐t❛ s♦❜r❡ k

  P N✳ ◆♦t❡ q✉❡ 2

  ☎1

  P ♣2 ☎ 1q : 5 ✁ 1 ✏ 24

  é ✈❡r❞❛❞❡✐r❛ ♣❛r❛ k ✏ 1✳ ❙✉♣♦♥❤❛ q✉❡ ♣❛r❛ ❛❧❣✉♠ k P N✱ s❡ t❡♥❤❛ q✉❡ P ♣2kq é ✈❡r❞❛❞❡✐r❛✱ ✐st♦ é✱ n k 2

  5 ✁ 1 ✏ 5 ✁ 1 ✏ 24 ☎ t. ❆ ♣❛rt✐r ❞❛ ❤✐♣ót❡s❡ ❞❡ ✐♥❞✉çã♦ ✐r❡♠♦s ♣r♦✈❛r q✉❡ P ♣2☎♣k 1qq t❛♠❜é♠ é ✈❡r❞❛❞❡✐r❛✳ 2

  ❉❡ ❢❛t♦✱ ♠✉❧t✐♣❧✐❝❛♥❞♦ ❛♠❜♦s ♦s ♠❡♠❜r♦s ♣♦r 5 ✱ ♦❜t❡♠♦s 2 k 2 2 2

  5 2 k 2 ☎ 5 ✁ 1 ☎ 5 ✏ 24 ☎ t ☎ 5

  5 2 k 2 ✁ ♣24 1q ✏ 24 ☎ t ☎ 25

  5 ✁ 1 ✏ 24 ☎ ♣25 ☎ t 1q. ❚❡♠♦s✱ ❡♥tã♦ ❛ ✈❡r❛❝✐❞❛❞❡ ❞❡ P ♣2 ☎ ♣k 1qq ♣❛r❛ t♦❞♦ k P N✳ Pr♦✈❛♥❞♦✱ ❛ss✐♠✱ ♣❡❧♦ n Pr✐♥❝í♣✐♦ ❞❛ ■♥❞✉çã♦ ❋✐♥✐t❛ q✉❡ 5 ✁ 1 é ✉♠ ♠ú❧t✐♣❧♦ ❞❡ 24✱ ♣❛r❛ t♦❞♦ n P N✱ ❝♦♠

  ✺✳✶ ❚ó♣✐❝♦s ❞❡ ▼❛t❡♠át✐❝❛ ❊❧❡♠❡♥t❛r ✶✶✺ n

  ♣❛r✳ ✭❝✮ ◗✉❡r❡♠♦s ♣r♦✈❛r q✉❡ n

  P

  1 ♣nq : 2 ✏ 3 ☎ t, ♣❛r❛ t♦❞♦ n P N, n é í♠♣❛r. ♦✉ 2 k

  ✁1

  P

  1 ♣kq : 2 ✏ 3 ☎ t, ♣❛r❛ t♦❞♦ k P N. ❯s❛♥❞♦ q✉❡ n ✏ 2k ✁ 1✱ ❝♦♠ k P N✱ ♥♦t❡ q✉❡ 2

  ☎1✁1

  P

  1 ♣2 ☎ 1 ✁ 1q : 2 ✏ 3

  é ✈❡r❞❛❞❡✐r❛✳ ❙✉♣♦♥❤❛ q✉❡ ♣❛r❛ ❛❧❣✉♠ k P N✱ s❡ t❡♥❤❛ q✉❡ P ♣nq ✏ P ♣2k ✁ 1q é ✈❡r❞❛❞❡✐r❛✱ ✐st♦ é n k 2

  ✁1

  2

  1

  1 ✏ 2 ✏ 3 ☎ t. ❆ ♣❛rt✐r ❞❛ ❤✐♣ót❡s❡ ❞❡ ✐♥❞✉çã♦ ✐r❡♠♦s ♣r♦✈❛r q✉❡ P ♣2☎k 1q t❛♠❜é♠ é ✈❡r❞❛❞❡✐r❛✳ 2

  ❉❡ ❢❛t♦✱ ♠✉❧t✐♣❧✐❝❛♥❞♦ ❛♠❜♦s ♦s ♠❡♠❜r♦s ♣♦r 2 ✱ ♦❜t❡♠♦s 2 k 2 2 2

  ✁1

  2

  1 2 k 1 ☎ 2 ☎ 2 ✏ 3 ☎ t ☎ 2

  2 2 k 1 ♣3 1q ✏ 3 ☎ t ☎ 4

  2

  1 ✏ 3 ☎ ♣4 ☎ t ✁ 1q. ❚❡♠♦s✱ ❡♥tã♦ ❛ ✈❡r❛❝✐❞❛❞❡ ❞❡ P ♣2k 1q✳ Pr♦✈❛♥❞♦✱ ❛ss✐♠✱ ♣❡❧♦ Pr✐♥❝í♣✐♦ ❞❛ ■♥❞✉çã♦ n

  1 ❋✐♥✐t❛ q✉❡ 2 é ✉♠ ♠ú❧t✐♣❧♦ ❞❡ 3✱ ♣❛r❛ t♦❞♦ n P N✱ ❝♦♠ n í♠♣❛r✳

  ✸✳ ❖❜s❡r✈❡ q✉❡ P ♣1q é ✈❡r❞❛❞❡✐r❛✱ ♣♦✐s 2 1 . 2 ✏ 1 ❙✉♣♦♥❤❛ P ♣nq ✏ n ✈❡r❞❛❞❡ ♣❛r❛ ❛❧❣✉♠ n P N✱ ✐st♦ é✱ 2

  1

  3 ☎ ☎ ☎ ♣2n ✁ 1q ✏ n 2 s❡❥❛ ✈á❧✐❞❛ ❡ ❞❡✈❡♠♦s ♠♦str❛r q✉❡ P ♣n 1q ✏ ♣n 1q é ✈❡r❞❛❞❡✐r❛✳ ❙♦♠❛♥❞♦ ❛

  1 ❛♠❜♦s ♦s ♠❡♠❜r♦s 2n ✱ ♣♦✐s ❡ss❛ é ❛ ❡①♣r❡ssã♦ q✉❡ ✐♥❞✐❝❛ ♦ ♣ró①✐♠♦ ♥ú♠❡r♦

  ✺✳✶ ❚ó♣✐❝♦s ❞❡ ▼❛t❡♠át✐❝❛ ❊❧❡♠❡♥t❛r ✶✶✻

  í♠♣❛r ❛♣ós 2n ✁ 1✱ t❡♠♦s q✉❡ 2

  1

  3 ☎ ☎ ☎ ♣2n ✁ 1q ♣2n 1q ✏ n ♣2n 1q 2

  ✏ ♣n 1q t❛♠❜é♠ é ✈❡rí❞✐❝❛✳ ▲♦❣♦✱ ❛ ✐❣✉❛❧❞❛❞❡ é ✈á❧✐❞❛ ♣❛r❛ t♦❞♦ ♦ ♥ú♠❡r♦ n P N✳ ✹✳ ❖❜s❡r✈❡ q✉❡ P ♣1q é ✈❡r❞❛❞❡✐r❛✱ ♣♦✐s

  P ♣1q ✏ 1 ☎ ♣1 1q ✏ 2. ❙✉♣♦♥❞♦ P ♣nq ✈❡r❞❛❞❡ ♣❛r❛ ❛❧❣✉♠ n P N✱ ♦✉ s❡❥❛

  2

  4 ☎ ☎ ☎ 2n ✏ n ☎ ♣n 1q

  é ✈á❧✐❞❛✱ ❞❡✈❡♠♦s ♠♦str❛r q✉❡ P ♣n 1q ✏ ♣n 1q ☎ ♣n 2q é ✈❡r❞❛❞❡✐r❛✳ ❙♦♠❛♥❞♦ ❛ ❛♠❜♦s ♦s ♠❡♠❜r♦s 2n 2✱ ♣♦✐s ❡ss❛ é ❛ ❡①♣r❡ssã♦ q✉❡ ✐♥❞✐❝❛ ♦ ♣ró①✐♠♦ ♥ú♠❡r♦ ♣❛r ❛♣ós 2n✱ t❡♠♦s q✉❡

  2

  4 ☎ ☎ ☎ 2n ♣2n 2q ✏ n ☎ ♣n 1q ♣2n 2q

  ✏ n ☎ ♣n 1q 2 ☎ ♣n 1q ✏ ♣n 1q ☎ ♣n 2q. P♦rt❛♥t♦✱ ❛ ✐❣✉❛❧❞❛❞❡ é ✈á❧✐❞❛ ♣❛r❛ P ♣n 1q✳ ▲♦❣♦✱ ❛ ✐❣✉❛❧❞❛❞❡ 2 4 ☎ ☎ ☎ 2n ✏ n

  ☎ ♣n 1q é ✈á❧✐❞❛ ♣❛r❛ t♦❞♦ n P N✳ ✺✳ ❖❜s❡r✈❡ q✉❡ P ♣1q é ✈❡r❞❛❞❡✱ ♣♦✐s

  10

  1 1 ✟ P .

  ♣1q : 1 ✏ ☎ 10 ✁ 1 ✁

  81

  9 ❙✉♣♦♥❞♦ P ♣nq ✈❡r❞❛❞❡ ♣❛r❛ ❛❧❣✉♠ n P N✱ ♦✉ s❡❥❛ 10 n n

  1

  11

  1 ☎ ☎ ☎ ☎ ☎ ☎ 1 ✏ ☎ ♣10 ✁ 1q ✁

  ❧♦♦♠♦♦♥ n uns

  81

  9 é ✈á❧✐❞❛✱ ❞❡✈❡♠♦s ♠♦str❛r q✉❡ P ♣n 1q é ✈❡r❞❛❞❡✐r❛✳

  ✺✳✶ ❚ó♣✐❝♦s ❞❡ ▼❛t❡♠át✐❝❛ ❊❧❡♠❡♥t❛r ✶✶✼

  9 ☎

  81 ☎ ♣10 n ✁ 1q ✁ n

  10

  ✏

  9 ✡

  10 ☎ 10 n ✁ 1

  ✂

  9

  10

  9

  81 ☎ ♣10 n ✁ 1q ✁ n

  10

  ✏

  ♣n 1q uns

  ❧♦♦♠♦♦♥

  9

  81 ☎ 9 ☎ 10 n

  1 ☎ ☎ ☎

  1

  2 ✏ 0,

  3 ☎ ♣3 ✁ 3q

  ♣3q ✏

  2 é ✈❡r❞❛❞❡✐r❛ ♣❛r❛ n ✏ 3✱ ♦♥❞❡ t❡♠♦s q✉❡ d

  ✻✳ ◆♦t❡ q✉❡ ❛ ❢ór♠✉❧❛ d♣nq ✏ n ☎ ♣n ✁ 3q

  9 ❉♦♥❞❡ s❡❣✉❡ ♦ r❡s✉❧t❛❞♦✳

  81 ♣10 n 1 ✁ 1q ✁ n

  1

  10

  9 ✏

  1

  81 ♣10 n ✁ 1 9 ☎ 10 n q ✁ n

  10

  9 ✏

  11 ☎ ☎ ☎ 1

  ♣♦rt❛♥t♦

  ❆❝r❡s❝❡♥❞♦ 11 ☎ ☎ ☎ 1 ❧♦♦♠♦♦♥ n 1 uns

  ✏

  ♣n 1q uns

  ❧♦♦♠♦♦♥

  11 ☎ ☎ ☎ 1

  9

  81 ☎ ♣10 n ✁ 1q ✁ n

  10

  ♣n 1q uns

  10

  ❧♦♦♠♦♦♥

  11 ☎ ☎ ☎ 1

  ❧♦♦♠♦♦♥ n uns

  1 ☎ ☎ ☎ 1

  1 ☎ ☎ ☎

  ❛ ❛♠❜♦s ♦s ♠❡♠❜r♦s✱ t❡♠♦s q✉❡

  ✏

  81 ☎ ♣10 n ✁ 1q ✁ n

  9 ,

  ❱✐st♦ q✉❡

  10 n 1 ✁ 1

  S ✏

  ✏ 1 10 ☎ ☎ ☎ 10 n é ✉♠❛ ♣r♦❣r❡ssã♦ ❣❡♦♠étr✐❝❛ ❞❡ ♣r✐♠❡✐r♦ t❡r♠♦ ✐❣✉❛❧ ❛ ✶ ❡ r❛③ã♦ ✐❣✉❛❧ ❛ ✶✵✱ t❡♠♦s q✉❡ s✉❛ s♦♠❛ é ✐❣✉❛❧ ❛

  ♣n 1q uns

  ❧♦♦♠♦♦♥

  11 ☎ ☎ ☎ 1

  ☞ ✌.

  9

  ☞ ✌

  ♣n 1q uns

  ❧♦♦♠♦♦♥

  ☎ ✆ 11 ☎ ☎ ☎ 1

  ☎ ✆9 ☎

  9 ☎

  1

  ♣♦✐s✱ s❡♥❞♦ d♣nq ❞❡✜♥✐❞♦ ❝♦♠♦ ♦ ♥ú♠❡r♦ ❞❡ ❞✐❛❣♦♥❛✐s ❞❡ ✉♠ ♣♦❧í❣♦♥♦ ❞❡ n ❧❛❞♦s ❡ ✉♠ tr✐â♥❣✉❧♦ ♥ã♦ ❤á ❞✐❛❣♦♥❛✐s✱ t❡♠♦s d♣3q ✏ 0✳ ❙✉♣♦♥❞♦✱ ❛❣♦r❛✱ q✉❡ ❛ ❢ór♠✉❧❛ s❡❥❛ ✈❡r❞❛❞❡ ♣❛r❛ ❛❧❣✉♠ n P N✱ ♠♦str❛r❡♠♦s q✉❡

  ✺✳✶ ❚ó♣✐❝♦s ❞❡ ▼❛t❡♠át✐❝❛ ❊❧❡♠❡♥t❛r ✶✶✽ d

  ♣n 1q t❛♠❜é♠ é ✈❡r❞❛❞❡✱ ♦✉ s❡❥❛✱ ♣n 1q ☎ ♣n ✁ 2q d . ♣n 1q ✏

  2 ◗✉❛♥❞♦ ❛❝r❡s❝❡♥t❛♠♦s ✉♠ ✈ért✐❝❡ ♣❛r❛ ❞❡t❡r♠✐♥❛r♠♦s ❛ ♥♦✈❛ q✉❛♥t✐❞❛❞❡ ❞❡ ❞✐❛✲ ❣♦♥❛✐s✱ t❡r❡♠♦s ❛ q✉❛♥t✐❞❛❞❡ ❞❛s ❛♥t✐❣❛s ❞✐❛❣♦♥❛✐s✱ ❛❝r❡s❝✐❞♦ ❞❡ ✉♠❛ ❞✐❛❣♦♥❛❧ q✉❡ ❡r❛ ❧❛❞♦ ❞♦ ♣♦❧í❣♦♥♦ ❞❡ n ❧❛❞♦s ❡ s✉r❣❡ ♣n ✁ 2q ♥♦✈❛s ❞✐❛❣♦♥❛✐s✱ ❛ss✐♠✱ t❡♠♦s✿ d

  ♣n 1q ✏ d♣nq 1 ♣n ✁ 2q n ☎ ♣n ✁ 3q n

  ✏ ✁ 1 2

  2 n ✁ 3n 2n ✁ 2

  ✏ 2

  2 n ✁ n ✁ 2

  ✏

  2 ♣n 1q ☎ ♣n ✁ 2q . ✏ n 1

  2 ✼✳ ◆♦t❡ q✉❡ ❛ ❢ór♠✉❧❛ a ✏ a ♣n ✁ 1q ☎ r é ✈❡r❞❛❞❡✐r❛ ♣❛r❛ n ✏ 1✱ ♦♥❞❡ t❡♠♦s q✉❡ a 1 1 ✏ a ♣1 ✁ 1q ☎ r.

  ❙✉♣♦♥❞♦✱ ❛❣♦r❛✱ q✉❡ ❛ ❢ór♠✉❧❛ s❡❥❛ ✈❡r❞❛❞❡ ♣❛r❛ ❛❧❣✉♠ n P N✱ s♦♠❛♥❞♦ r ❛ ❛♠❜♦s ♦s ❧❛❞♦s ❞❛ ✐❣✉❛❧❞❛❞❡✱ s❡❣✉❡ 1 1 a n r n 1 n

  ✏ a ✏ a ♣n ✁ 1q ☎ r r ✏ a ☎ r, ♦ q✉❡ ♠♦str❛ q✉❡ ❛ ❢ór♠✉❧❛ é ✈❡r❞❛❞❡✐r❛ ♣❛r❛ n 1✳ P♦rt❛♥t♦✱ é ✈á❧✐❞❛ ♣❛r❛ t♦❞♦ n

  P N✳ 1 a n n ♣a q ☎ n ✽✳ ❖❜s❡r✈❡ q✉❡ P ♣1q é ✈❡r❞❛❞❡✐r❛ ♣❛r❛ S ✏ ✱ ♣♦✐s 1 a 1

  2 ♣a q ☎ 1

  P 1 1 . ♣1q : S ✏ ✏ a

  2 1 a n n ♣a q ☎ n ❙✉♣♦♥❤❛ ✈❡r❞❛❞❡ ♣❛r❛ ❛❧❣✉♠ n P N✱ ✐st♦ é✱ P ♣nq : S ✏ ✳ P❛r❛ ♦❜t❡r

  ✺✳✶ ❚ó♣✐❝♦s ❞❡ ▼❛t❡♠át✐❝❛ ❊❧❡♠❡♥t❛r ✶✶✾

  ❛ s♦♠❛ ❞♦s n 1 ❡❧❡♠❡♥t♦s ❞❡ss❛ s❡q✉ê♥❝✐❛ ❜❛st❛ ❛❝r❡s❝❡r♠♦s ♦ ú❧t✐♠♦ ❡❧❡♠❡♥t♦ à s♦♠❛ ❞♦s n ❡❧❡♠❡♥t♦s✳ P♦rt❛♥t♦✱ t❡♠♦s q✉❡ S a . n 1 n n 1

  ✏ S ❉♦♥❞❡✱ 1 a n

  ♣a q ☎ n S a n 1 n 1

  ✏

  2 a 1 n n 1 ☎ n a ☎ n 2 ☎ a . ✏ n n 1 1 1 n

  2 ❙❡♥❞♦✱ a ✏ a ♣n ✁ 1q ☎ r ❡ a ✏ a ☎ r✱ s❡❣✉❡ q✉❡ 2 a 1 1 1

  2 ☎ n a ☎ n n ☎ r ✁ r ☎ n 2 ☎ a ☎ n ☎ r

  S n 12

  2

  2 1 1 n ☎ a ☎ n 2 ☎ a ☎ r n ☎ r

  ✏

  2

  2 1 ☎ a ☎ ♣n 1q n ☎ r ☎ ♣n 1q

  ✏ 1 a 1 n

  2 ♣a ☎ rq ☎ ♣n 1q

  ✏ 1 a n 1 ♣a q ☎ ♣n 1q . ✏

  2

  1 ❖ q✉❡ ♠♦str❛ q✉❡ ❛ ♣r♦♣r✐❡❞❛❞❡ P ♣n q t❛♠❜é♠ é ✈á❧✐❞❛✳ P♦rt❛♥t♦ é ✈❡r❞❛❞❡ ♣❛r❛ t♦❞♦ n P N✳ n n 1 ✁1

  ✾✳ ❖❜s❡r✈❡ q✉❡ P ♣1q é ✈❡r❞❛❞❡✐r❛ ♣❛r❛ a ✏ a ☎ q ✱ ♣♦✐s 1

  ✁1

  a 1 1 .

  ✏ a ☎ q n n 1 ✁1 ❙✉♣♦♥❤❛ ✈❡r❞❛❞❡ ♣❛r❛ ❛❧❣✉♠ n P N✱ ✐st♦ é✱ a ✏ a ☎ q ✳ ▼✉❧t✐♣❧✐❝❛♥❞♦ ♣♦r q ❛♠❜♦s ♦s ♠❡♠❜r♦s ❞❡ss❛ ✐❣✉❛❧❞❛❞❡✱ t❡♠♦s q✉❡ n n

  ✁1

  a n n 1 1 1 . ☎ q ✏ a ✏ a ☎ q ☎ q ✏ a ☎ q

  ❖ q✉❡ ♠♦str❛ q✉❡ ❛ ❢ór♠✉❧❛ é ❝♦rr❡t❛ ♣❛r❛ n 1✳ P♦rt❛♥t♦ é ✈á❧✐❞❛ ♣❛r❛ ♣❛r❛ t♦❞♦

  ✺✳✶ ❚ó♣✐❝♦s ❞❡ ▼❛t❡♠át✐❝❛ ❊❧❡♠❡♥t❛r ✶✷✵ n

  P N✳ n a 1 n ☎ ♣q ✁ 1q ✶✵✳ ❖❜s❡r✈❡ q✉❡ P ♣1q é ✈❡r❞❛❞❡✐r❛ ♣❛r❛ S ✏ ✱ ♣♦✐s q 1 ✁ 1 a 1

  ☎ ♣q ✁ 1q S 1 1 .

  ✏ ✏ a q n 1 1 n ✁ 1

  1 ❖❜s❡r✈❡ q✉❡ a ✏ a ☎ q ❡ q✉❡ ♣❛r❛ ♦❜t❡r ❛ s♦♠❛ ❞♦s ♣n q t❡r♠♦s✱ t❡♠♦s S n 1 n a n 1

  ✏ S ✳ ▲♦❣♦✱ n a 1 ☎ ♣q ✁ 1q n

  S a n 1 1 ✏ ☎ q q ✁ 1 n n n a 1 1 a 1 1

  ☎ q ✁ a ☎ q ☎ q ✁ a ☎ q ✏ q n 1 ✁ 1 a 1

  ☎ ♣q ✁ 1q . ✏ q

  ✁ 1 ❖ q✉❡ ♠♦str❛ q✉❡ ❛ ♣r♦♣r✐❡❞❛❞❡ P ♣n 1q é ✈❡r❞❛❞❡✐r❛ ❡ ❝♦♥❝❧✉✐✲s❡ q✉❡ ❛ ♣r♦♣r✐❡❞❛❞❡ P

  ♣nq é ✈❡r❞❛❞❡✐r❛ ♣❛r❛ t♦❞♦ n P N✳ ✶✶✳ ✭❆ ▼♦❡❞❛ ❋❛❧s❛ ❞♦ ❘❡✐✮ P❛r❛ n ✏ 1 t❡♠♦s três ♠♦❡❞❛s✳ ❈♦❧♦❝❛✲s❡ ✉♠❛ ❡♠ ❝❛❞❛

  ♣r❛t♦ ❡ ❞❡✐①❛✲s❡ ❛ ♦✉tr❛ ❞❡ ❢♦r❛✳ ❆ss✐♠✱ ❞❡s❝♦❜r❡✲s❡ ❢❛❝✐❧♠❡♥t❡ ❛ ❢❛❧s❛✱ ✉♠❛ ✈❡③ q✉❡ s❡ ♦s ♣r❛t♦s ✜❝❛♠ ❡q✉✐❧✐❜r❛❞♦s ❛ ♠♦❡❞❛ ❢❛❧s❛ ❡stá ❞❡ ❢♦r❛✱ ❝❛s♦ ❝♦♥trár✐♦✱ ❛ ♠♦❡❞❛ ❢❛❧s❛ é ❛ q✉❡ ✜❝❛ ♥♦ ♣r❛t♦ ♠❛✐s ❛❧t♦✳ ❙✉♣♦♥❤❛ q✉❡✱ ♣❛r❛ ❛❧❣✉♠ n✱ ❝♦♠ n ♣❡s❛❣❡♥s n n 1 ♣♦ss❛ ❞❡s❝♦❜r✐r ❛ ♠♦❡❞❛ ❢❛❧s❛ ❞❡♥tr❡ ❛s 3 ♠♦❡❞❛s✳ ❈♦♥s✐❞❡r❛♥❞♦ 3 ♠♦❡❞❛s✱ n n 1 n ❜❛st❛ ❞✐✈✐❞í✲❧❛s ❡♠ três ❣r✉♣♦s ❞❡ 3 ♠♦❡❞❛s✱ ✈✐st♦ q✉❡ 3 ✏ 3 ☎ 3✱ ❡ ❝♦♠ ✉♠ ♠ét♦❞♦ ♣❛r❡❝✐❞♦ ♣❛r❛ ♦ ❝❛s♦ n ✏ 1 ❞❡s❝♦❜r❡✲s❡ ❝♦♠ ✉♠❛ só ♣❡s❛❣❡♠ ❡♠ q✉❛❧ ❞❡st❡s três ❣r✉♣♦s ❛ ♠♦❡❞❛ ❡stá✳ ❆ss✐♠ s❡♥❞♦✱ r❡❛❧✐③❛✲s❡ ❛s n ♣❡s❛❣❡♥s ♥❡❝❡ssár✐❛s ♣❛r❛ s❡ ❞❡s❝♦❜r✐r ❛ ♠♦❡❞❛ ❢❛❧s❛ ♥♦ ❣r✉♣♦ q✉❡ ❛ ❝♦♥té♠✳ ❈♦♠ ✐ss♦ ♦ ♥ú♠❡r♦ ❞❡ ♣❡s❛❣❡♥s s❡rá n 1✳ ❆ ❞❡♠♦s♥tr❛çã♦ ❡stá ❝♦♥❝❧✉í❞❛✳

  ✺✳✶✳✷ Pr✐♥❝í♣✐♦ ❞❛ ❈❛s❛ ❞♦s P♦♠❜♦s

  ✶✳ ❙❡♥❞♦ ♦s ♠❡s❡s ❞♦ ❛♥♦ ❛s ❝❛s❛s ❡ ❛s ♣❡ss♦❛s ♦s ♣♦♠❜♦s✱ ❡♥tã♦ ♣❡❧♦ Pr✐♥❝í♣✐♦ ❞❛ ❈❛s❛ ❞♦s P♦♠❜♦s✱ t❡r❡♠♦s ♥♦ ♠í♥✐♠♦ tr❡③❡ ♣❡ss♦❛s✱ ♣♦✐s ❞❡✈❡♠♦s ❛❧♦❝❛r ✉♠❛ ♣❡ss♦❛ ❡♠ ❝❛❞❛ ♠ês ❡ s♦❜r❛r ✉♠❛ ❞❛s tr❡③❡ ♣❛r❛ ❣❛r❛♥t✐r♠♦s ❛ ♣r❡s❡♥ç❛ ❞❡ ❞✉❛s ♣❡ss♦❛s ♥♦ ♠❡s♠♦ ♠ês✳

  ✺✳✶ ❚ó♣✐❝♦s ❞❡ ▼❛t❡♠át✐❝❛ ❊❧❡♠❡♥t❛r ✶✷✶

  ✷✳ ❚❡♥❞♦ q✉❛tr♦ ♣❡ss♦❛s ❡♠ ❝❛❞❛ ♠ês✱ t♦t❛❧✐③❛♥❞♦ 48 ♣❡ss♦❛s✱ ❡♥tã♦ s♦❜r❛ ❞✉❛s ♣❛r❛ ❝♦❧♦❝❛r ❡♠ q✉❛❧q✉❡r ♠ês ♣❛r❛ ❣❛r❛♥t✐r♠♦s ♣❡❧♦ P❈P q✉❡ t❡r❡♠♦s ♠ês ❝♦♠ ♣❡❧♦ ♠❡♥♦s ❝✐♥❝♦ ♣❡ss♦❛s ❢❛③❡♥❞♦ ❛♥✐✈❡rsár✐♦✳

  ✸✳ ❈♦♥s✐❞❡r❡ ❛s n ♣❡ss♦❛s ❝♦♠♦ ♦s ♣♦♠❜♦s ❡ ❝♦♥s✐❞❡r❡ ♦ ♥ú♠❡r♦ ❞❡ ❝♦♥❤❡❝✐❞♦s ❝♦♠♦ ❛s ❝❛s❛s✳ ❊♠❜♦r❛ ❤❛❥❛ n ❝❛s❛s✱ ♣♦✐s ♦ ♥ú♠❡r♦ ❞❡ ❝♦♥❤❡❝✐❞♦s ♣♦❞❡ ✈❛r✐❛ ❞❡ 0 ❛ n

  ✁ 1✱ ♥♦ ♠á①✐♠♦ n ✁ 1 ❝❛s❛s sã♦ ✉s❛❞❛s✱ ♣♦✐s é ✐♠♣♦ssí✈❡❧ ❤❛✈❡r✱ ❛♦ ♠❡s♠♦ t❡♠♣♦✱ ♣❡ss♦❛s ❝♦♠ 0 ❝♦♥❤❡❝✐❞♦ ❡ ❝♦♠ n ✁ 1 ❝♦♥❤❡❝✐❞♦s✳ P♦rt❛♥t♦✱ ♣❡❧♦ Pr✐♥❝í♣✐♦ ❞❛ ❈❛s❛ ❞♦s P♦♠❜♦s ❤❛✈❡rá ❞✉❛s ♣❡ss♦❛s ❝♦♠ ♦ ♠❡s♠♦ ♥ú♠❡r♦ ❞❡ ❝♦♥❤❡❝✐❞♦s✳

  ✹✳ ❉❡ ✐♠❡❞✐❛t♦✳ ❯s❛♥❞♦ ♦ ♣r✐♥❝í♣✐♦ ❞❛ ✐♥❞✉çã♦✱ ♣♦❞❡♠♦s ❝♦♠♣r♦✈❛r ✐ss♦✳ ✺✳ ❉✐✈✐❞✐♥❞♦ ❡ss❡ q✉❛❞r❛❞♦ ❡♠ q✉❛tr♦ q✉❛❞r❛❞♦s ✐❣✉❛✐s ❞❡ ❧❛❞♦ 1 ❡ ❝♦❧♦❝❛♥❞♦ q✉❛tr♦

  ❞❡ss❡s ♣♦♥t♦s ♥♦s ✈ért✐❝❡s ❞❡ss❡ q✉❛❞r❛❞♦✱ r❡st❛✲♥♦s ❝♦❧♦❝❛r ♦ q✉✐♥t♦ ♣♦♥t♦ ❞❡ ♠♦❞♦ q✉❡ s❡❥❛ ❛ ♠❛✐♦r ❞✐stâ♥❝✐❛ ♣♦ssí✈❡❧ ♣❛r❛ ♦s ♣♦♥t♦s ❧♦❝❛❧✐③❛❞♦s ♥♦s ✈ért✐❝❡s✳ ❙❡♥❞♦ ❛ss✐♠✱ ❡ss❡ ♣♦♥t♦ s❡rá ❝♦❧♦❝❛❞♦ ♥♦ ❡♥❝♦♥tr♦ ❞❛ ❞✐❛❣♦♥❛❧ ❞♦ q✉❛❞r❛❞♦ ♠❛✐♦r ❞❡ ♠♦❞♦

  ❄

  2 q✉❡ ❛ ❞✐stâ♥❝✐❛ ❞❡st❡ ♣❛r❛ ♦s ♦✉tr♦s ♣♦♥t♦s é ❞❡ ✱ ❡ q✉❛❧q✉❡r ♦✉tr♦ ❧♦❝❛❧ ♣❛r❛ ❄

  2 ❡ss❡ ♣♦♥t♦ ❞❡♥tr♦ ❞♦ q✉❛❞r❛❞♦ s❡rá ♠❡♥♦r q✉❡ ♣❛r❛ ✉♠ ❞♦s ♣♦♥t♦s ❧♦❝❛❧✐③❛❞♦ ♥♦ ✈ért✐❝❡✳ 1 , p 2 , 8

  ✻✳ ❈♦♥s✐❞❡r❡ ✉♠ ❝♦♥❥✉♥t♦ P ✏ tp ☎ ☎ ☎ , p ✉ ❞❡ ♦✐t♦ ♥ú♠❡r♦s ✐♥t❡✐r♦s q✉❛✐sq✉❡r✱ ♦♥❞❡ ✐r❡♠♦s ❝♦♥s✐❞❡r❛r ❝♦♠♦ ♦s ♣♦♠❜♦s✱ ✉s❛♥❞♦ ♦ ♣r✐♥❝í♣✐♦ ❞❛ ❝❛s❛ ❞♦s ♣♦♠❜♦s✳ ❆♦ ❞✐✈✐❞✐r♠♦s q✉❛❧q✉❡r ♥ú♠❡r♦ ♣♦r s❡t❡✱ ❡st❡ ♣♦❞❡ ❞❡✐①❛r r❡st♦s 0, 1, 2, 3, 4, 5, ou 6✳

  , 6 ❈♦♥s✐❞❡r❡✱ ❛ss✐♠✱ ♦ ❝♦♥❥✉♥t♦ C ✏ tc ☎ ☎ ☎ , c ✉ ❝♦♠♦ ♦ ❝♦♥❥✉♥t♦ q✉❡ ❞❡✐①❛ r❡st♦s 0 ❛

  6 q✉❛♥❞♦ ❞✐✈✐❞✐❞♦ ♣♦r s❡t❡✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ❙❡❥❛ ❝❛❞❛ ❡❧❡♠❡♥t♦ ❞❡ C ❛s ❝❛s❛s ♦♥❞❡ ✐r❡♠♦s ❝♦❧♦❝❛r ❝❛❞❛ ❡❧❡♠❡♥t♦ ❞♦ ❝♦♥❥✉♥t♦ P ✳ ❉❡ss❛ ❢♦r♠❛✱ t❡r❡♠♦s ♣❡❧♦ ♠❡♥♦s ❞♦✐s ❡❧❡♠❡♥t♦s ❞♦ ❝♦♥❥✉♥t♦ P q✉❡ ❞❡✐①❛♠ ♦ ♠❡s♠♦ r❡st♦ q✉❛♥❞♦ ❞✐✈✐❞✐❞♦ ♣♦r 7✳ ▲♦❣♦✱ ❛ ❞✐❢❡r❡♥ç❛ ❡♥tr❡ ❡ss❡s ❞♦✐s ♥ú♠❡r♦s s❡rá ✉♠ ♠ú❧t✐♣❧♦ ❞❡ 7✳

  ✼✳ ❙❡♠♣r❡ ❡①✐st❡ ✉♠❛ ❞✐❢❡r❡♥ç❛ q✉❡ é ♠ú❧t✐♣❧♦ ❞❡ ✾✳ ❉❡ ❢❛t♦✱ q✉❛♥❞♦ ❞✐✈✐❞✐♠♦s ✉♠ ♥ú♠❡r♦ ♣♦r ✾✱ ♣♦❞❡♠♦s ❡♥❝♦♥tr❛r ♥♦✈❡ r❡st♦s ❞✐❢❡r❡♥t❡s✳ ▲♦❣♦✱ ❡♥tr❡ ♦s ✶✵ ♥ú♠❡r♦s ❞♦ ❝♦♥❥✉♥t♦✱ ♣❡❧♦ ♠❡♥♦s ❞♦✐s ❞❡❧❡s tê♠ ♠❡s♠♦ r❡st♦ q✉❛♥❞♦ ❞✐✈✐❞✐❞♦ ♣♦r ✾✳ ◗✉❛♥❞♦ ❢❛③❡♠♦s ❛ ❞✐❢❡r❡♥ç❛ ❞❡ss❡s ❞♦✐s ♥ú♠❡r♦s q✉❡ tê♠ ♦ ♠❡s♠♦ r❡st♦✱ ♦❜t❡♠♦s ✉♠ ♥ú♠❡r♦ ❝♦♠ r❡st♦ ③❡r♦✱ ♦✉ s❡❥❛✱ ❞✐✈✐sí✈❡❧ ♣♦r ✾✳

  ✽✳ ❙✉♣♦♥❤❛ q✉❡ ❡♠ ✉♠ ❝❛♠♣❡♦♥❛t♦ ❤❛❥❛ n t✐♠❡s✳ ❈♦♠♦ ❝❛❞❛ t✐♠❡ ❥♦❣❛ ❛♣❡♥❛s ✉♠❛ ✈❡③ ❝♦♠ ♦✉tr♦ t✐♠❡✱ ❡♥tã♦ ❝❛❞❛ t✐♠❡ ❥♦❣❛rá ♣n ✁ 1q ✈❡③❡s✳ ❈♦♥s✐❞❡r❡ ♦ ♥ú♠❡r♦ ❞❡ ❥♦❣♦s ❞❡ ✉♠ t✐♠❡ ❝♦♠♦ ❛ ❝❛s❛ ❞♦s ♣♦♠❜♦s ❡ ♦s t✐♠❡s q✉❡ ♣❛rt✐❝✐♣❛♠ ❞❡ss❡ ❝❛♠♣❡♦♥❛t♦ ❝♦♠♦ ♦s ♣♦♠❜♦s✳ P❡❧♦ Pr✐♥❝í♣✐♦ ❞❛ ❈❛s❛ ❞♦s P♦♠❜♦s✱ t❡r❡♠♦s n t✐♠❡s

  ✺✳✶ ❚ó♣✐❝♦s ❞❡ ▼❛t❡♠át✐❝❛ ❊❧❡♠❡♥t❛r ✶✷✷

  ♣❛r❛ ❛❧♦❝❛r ♥❛s ♣n ✁ 1q ❝❛s❛s ❞♦s ♣♦♠❜♦s✳ ▲♦❣♦✱ t❡r❡♠♦s s❡♠♣r❡✱ ♣❡❧♦ ♠❡♥♦s ❞♦✐s t✐♠❡s✱ q✉❡ t❡rã♦ ❥♦❣❛❞♦s ♦ ♠❡s♠♦ ♥ú♠❡r♦ ❞❡ ♣❛rt✐❞❛s✳ ✾✳ P❛r❛ ♣r♦✈❛r ✐st♦✱ ❡s❝♦❧❤❛♠♦s ❣❛✈❡t❛s ❛❞❡q✉❛❞❛s ❛♦ ♣r♦❜❧❡♠❛✳ ❉✐str✐❜✉í♠♦s ♦s ♥ú✲

  ♠❡r♦s ❞♦ ❝♦♥❥✉♥t♦ A ❡♠ ✺✵ ❣❛✈❡t❛s ❛ss✐♠ ❝♦♥str✉í❞❛s✿ t1, 2✉; t3, 4✉; ☎ ☎ ☎ ; t99, 100✉.

  ❈♦♠♦ ❤á ✺✵ ❣❛✈❡t❛s ❞❛s q✉❛✐s r❡t✐r❛♠♦s ✺✶ ♥ú♠❡r♦s✱ s❡♠♣r❡ ❡①✐st✐rá ✉♠❛ ❣❛✈❡t❛ ❞❛ q✉❛❧ ❡s❝♦❧❤❡♠♦s ❞♦✐s ♥ú♠❡r♦s ❡ ❡st❡s✱ ❣r❛ç❛s à ♥♦ss❛ ❝♦♥str✉çã♦✱ s❡rã♦ ❝♦♥s❡❝✉t✐✈♦s✳ ✶✵✳ ❉✐✈✐❞❛ ♦ r❡tâ♥❣✉❧♦ ❝♦♠♦ ❛ ✉♥✐ã♦ ❞❡ s❡✐s q✉❛❞r❛❞♦s ❞❡ ❧❛❞♦ ✶✱ ❡ ♦❜s❡r✈❡ q✉❡ ❞♦✐s ❞♦s s❡t❡ ♣♦♥t♦s ❞❡✈❡♠ ❡st❛r ♥✉♠ ♠❡s♠♦ q✉❛❞r❛❞♦✳ ✶✶✳ ❈♦♥s✐❞❡r❡ ✉♠❛ ❞❛s ♣❡ss♦❛s✱ q✉❡ ❝❤❛♠❛r❡♠♦s ❞❡ P ✳ ❈♦♠♦ ❤á ♦✉tr❛s ❝✐♥❝♦ ♣❡ss♦❛s

  ♥❛ ❢❡st❛✱ ♦✉ ❤á três ♣❡ss♦❛s q✉❡ P ❝♦♥❤❡❝❡✱ ♦✉ ❤á três ♣❡ss♦❛s q✉❡ P ♥ã♦ ❝♦♥❤❡❝❡✳ ❙✉♣♦♥❤❛ ❛ ♣r✐♠❡✐r❛ ❛❧t❡r♥❛t✐✈❛✱ s❡♠ ♣❡r❞❛ ❞❡ ❣❡♥❡r❛❧✐❞❛❞❡✳ ❙❡ ❤á ❞✉❛s ❞❡ss❛s três ♣❡ss♦❛s q✉❡ P ❝♦♥❤❡❝❡ q✉❡ s❡ ❝♦♥❤❡❝❡♠✱ ❥✉♥t♦ ❝♦♠ P t❡rí❛♠♦s três ♣❡ss♦❛s q✉❡ s❡ ❝♦♥❤❡❝❡♠ ♠✉t✉❛♠❡♥t❡✳ ❈❛s♦ ❝♦♥trár✐♦✱ ❡ss❛s três ♣❡ss♦❛s q✉❡ P ❝♦♥❤❡❝❡ sã♦ três ♣❡ss♦❛s q✉❡ s❡ ❞❡s❝♦♥❤❡❝❡♠ ♠✉t✉❛♠❡♥t❡✳

  ✶✷✳ ❖s ♣♦ssí✈❡✐s ❢❛t♦r❡s ♣r✐♠♦s ❞♦s ❡❧❡♠❡♥t♦s ❞❡ M sã♦ ♦s s❡❣✉✐♥t❡s ♥♦✈❡ ♣r✐♠♦s✿ ✷✱ 9 ✸✱ ✺✱ ✼✱ ✶✶✱ ✶✸✱ ✶✼✱ ✶✾✱ ✷✸✳ ❉❛❞♦ ✉♠ s✉❜❝♦♥❥✉♥t♦ ❞❡ M ❝♦♠ ♠❛✐s ❞❡ 512 ✏ 2 ❡❧❡♠❡♥t♦s✱ ❤❛✈❡rá ❞♦✐s ❞❡❧❡s ❡♠ ❝✉❥❛ ❛ ❢❛t♦r❛çã♦ ♣r✐♠❛ ❛s ♣❛r✐❞❛❞❡s ❞♦s ❡①♣♦❡♥t❡s ❞❡ ❝❛❞❛ ♣r✐♠♦ ❝♦✐♥❝✐❞❡♠ ❡ ♣♦rt❛♥t♦ ♦ s❡✉ ♣r♦❞✉t♦ s❡rá ✉♠ q✉❛❞r❛❞♦ ♣❡r❢❡✐t♦✳ ❯s❛♥❞♦ r❡♣❡t✐❞❛♠❡♥t❡ ❡ss❡ ❛r❣✉♠❡♥t♦✱ ♦❜t❡♠♦s ✼✸✼ ♣❛r❡s ❞✐s❥✉♥t♦s ❞❡ ❡❧❡♠❡♥t♦s ❞❡ M t❛✐s q✉❡ ♦ ♣r♦❞✉t♦ ❞♦s ❡❧❡♠❡♥t♦s ❞❡ ❝❛❞❛ ♣❛r é ✉♠ q✉❛❞r❛❞♦ ♣❡r❢❡✐t♦ ✭❞❡ ❢❛t♦✱ s❡ r❡t✐r❛r♠♦s ❞❡ M ♠❡♥♦s ❞❡ ✼✸✼ ♣❛r❡s ❞❡ ❡❧❡♠❡♥t♦s ❞✐st✐♥t♦s✱ s♦❜r❛♠ ♣❡❧♦ ♠❡♥♦s 1985

  ✁2☎736 ✏ 513 ❡❧❡♠❡♥t♦s✱ ❞♦s q✉❛✐s ♣♦❞❡♠♦s ♦❜t❡r ✉♠ ♥♦✈♦ ♣❛r✮✳ ❆❣♦r❛✱ ❝♦♠♦ ❤á ♠❛✐s ❞❡ ✺✶✷ ❡❧❡♠❡♥t♦s ❡♠ q✉❡ ♦ ♣r♦❞✉t♦ ❞♦s ❡❧❡♠❡♥t♦s é ✉♠ q✉❛❞r❛❞♦ ♣❡r❢❡✐t♦✱ ❡ ♦s ♣r✐♠♦s q✉❡ ❞✐✈✐❞❡♠ ❝❛❞❛ ✉♠ ❞❡ss❡s ♣r♦❞✉t♦s ❡stã♦ ❡♥tr❡ ♦s ♥♦✈❡ ♣r✐♠♦s ❧✐st❛❞♦s ❛❝✐♠❛✱ ❤❛✈❡rá ❞♦✐s ❞❡ss❡s ♣❛r❡s ♣❛r❛ ♦s q✉❛✐s✱ s❡ t♦♠❛r♠♦s ♦ ❡①♣♦❡♥t❡ ❞❡ ❝❛❞❛ ✉♠ ❞❡ss❡s ♣r✐♠♦s ♥♦ ♣r♦❞✉t♦ ❞♦s t❡r♠♦s ❞♦ ♣❛r ❡ ♦s ❞✐✈✐❞✐r♠♦s ♣♦r ✷✱ ❛s ♣❛r✐❞❛❞❡s ❞♦ r❡s✉❧t❛❞♦ ❝♦✐♥❝✐❞✐rã♦✱ ❡ ♣♦rt❛♥t♦ ♦ ♣r♦❞✉t♦ ❞♦s q✉❛tr♦ t❡r♠♦s ❞❡ss❡s ❞♦✐s ♣❛r❡s s❡rá ✉♠❛ q✉❛rt❛ ♣♦tê♥❝✐❛✳

  ✶✸✳ Pr❡❝✐s❛♠♦s s❛❜❡r q✉❛❧ ♦ ♥ú♠❡r♦ ❞❡ r❡s♣♦st❛s ❞✐❢❡r❡♥t❡s q✉❡ ♣♦❞❡♠ s✉r❣✐r ♥❛ ♣❡s✲ q✉✐s❛✳ ❈♦♠♦ ♣❛r❛ ❝❛❞❛ ♣❡r❣✉♥t❛ ❤á ❞✉❛s r❡s♣♦st❛s ❞✐❢❡r❡♥t❡s✱ ♣♦rt❛♥t♦ ❤á ✶✻ ♠❛✲

  ✺✳✶ ❚ó♣✐❝♦s ❞❡ ▼❛t❡♠át✐❝❛ ❊❧❡♠❡♥t❛r ✶✷✸

  ✈✐st❛❞♦s ♣❛r❛ ❣❛r❛♥t✐r♠♦s q✉❡ ❤❛❥❛ ❞✉❛s ♣❡ss♦❛s ❝♦♠ ❛ ♠❡s♠❛ r❡s♣♦st❛✳ ◆♦t❡ q✉❡ ❛s r❡s♣♦st❛s ❢❛③❡♠ ♦ ♣❛♣❡❧ ❞❛s ❝❛s❛s ❡ ♦s ❡♥tr❡✈✐st❛❞♦s ♦ ♣❛♣❡❧ ❞♦s ♣♦♠❜♦s✳

  ✶✹✳ ◆♦ ❝♦♥❥✉♥t♦ A ✏ tm 1, m 2, ☎ ☎ ☎ , m n✉ t❡♠♦s n ♥ú♠❡r♦s ❞✐❢❡r❡♥t❡s✳ ❙❡ ♥ã♦ ❡①✐st✐ss❡ ♥❡♥❤✉♠ ♠ú❧t✐♣❧♦ ❞❡ n✱ q✉❛♥❞♦ ❞✐✈✐❞íss❡♠♦s ♦s ♥ú♠❡r♦s ❞♦ ❝♦♥❥✉♥t♦ A ♣♦r n

  ✱ ♦s r❡st♦s ♣❡rt❡♥❝❡r✐❛♠ ❛♦ ❝♦♥❥✉♥t♦ B ✏ t1, 2, ☎ ☎ ☎ , n ✁ 1✉✱ q✉❡ ♣♦ss✉✐ n ✁ 1 ❡❧❡♠❡♥t♦s✳ ▲♦❣♦✱ ❞❡✈❡♠ ❡①✐st✐r ❞♦✐s ♥ú♠❡r♦s m i ❡ m j✱ ❝♦♠ 1 ↕ i ➔ j ↕ n t❛✐s q✉❡ ♦ r❡st♦ ❞❛ ❞✐✈✐sã♦ ❞❡ m i ♣♦r n é ♦ ♠❡s♠♦ q✉❡ ♦ r❡st♦ ❞❛ ❞✐✈✐sã♦ ❞❡ m j ♣♦r n✳ ▲♦❣♦✱ m j ✁ ♣m iq é ✉♠ ♠ú❧t✐♣❧♦ ❞❡ n✱ ♦ q✉❡ ✐♠♣❧✐❝❛ q✉❡ n → j ✁ i ➙ 1 é ✉♠ ♠ú❧t✐♣❧♦ ❞❡ n ♠❡♥♦r q✉❡ n ❛❜s✉r❞♦✦✮✳ ▲♦❣♦ ❞❡✈❡ ❡①✐st✐r ❛❧❣✉♠ ♠ú❧t✐♣❧♦ ❞❡ n ♥♦ ❝♦♥❥✉♥t♦ A✳

  ✶✺✳ ❙❡ m ❡ n sã♦ ✐♥t❡✐r♦s ♣♦s✐t✐✈♦s✱ ♣❡❧♦ ❡①❡♠♣❧♦ ❛♥t❡r✐♦r ✉♠ ❞♦s ♥ú♠❡r♦s m 1, m 2,

  ☎ ☎ ☎ , m n é ✉♠ ♠ú❧t✐♣❧♦ ❞❡ n✳ ❆ss✐♠✱ ❞❛❞♦ n ✉♠ ✐♥t❡✐r♦ q✉❛❧q✉❡r✱ ❡s❝♦❧❤❡✲s❡ n 1 m ✏ 1234567890 ☎ 10 ✳ ❉❡st❡ ♠♦❞♦✱ t♦❞♦s ♦s ✐♥t❡✐r♦s m 1, m 2, ☎ ☎ ☎ , m n

  ❝♦♠❡ç❛♠ ❝♦♠ ✶✷✸✹✺✻✼✽✾✵ ❡ ❛❧❣✉♠ ❞❡❧❡s é ♠ú❧t✐♣❧♦ ❞❡ n✳ ✶✻✳ ❈♦♥s✐❞❡r❛♠♦s ♦s n 1 ♥ú♠❡r♦s

  1, 11, 111, ☎ ☎ ☎ , 111 ☎ ☎ ☎ 1 ✭✺✳✶✮

  ❧♦♦♦♠♦♦♦♥

  ♣n 1q vezes

  ❝♦♠♦ s❡♥❞♦ ♦s ♣♦♠❜♦s ❡ n ❝❛s❛s ❡♥✉♠❡r❛❞❛s ❝♦♠ ♦s ♥ú♠❡r♦s 0, 1, 2, 3,

  ☎ ☎ ☎ , n ✁ 1, ♦✉ s❡❥❛✱ ❝♦♠ ♦s ♣♦ssí✈❡✐s r❡st♦s ♥❛ ❞✐✈✐sã♦ ♣♦r n✳ ❙✐♠✐❧❛r♠❡♥t❡ ❛♦ ❡①❡♠♣❧♦ ❛♥t❡r✐♦r ❡①✐st❡♠ ❞♦✐s ♥ú♠❡r♦s ♥❛ ❧✐st❛ ✺✳✶ q✉❡ ❞❡✐①❛♠ ♦ ♠❡s♠♦ r❡st♦ ♥❛ ❞✐✈✐sã♦ ♣♦r n ❡✱ ♣♦rt❛♥t♦✱ ❛ ❞✐❢❡r❡♥ç❛ ❡♥tr❡ ♦ ♠❛✐♦r ❡ ♦ ♠❡♥♦r é ♠ú❧t✐♣❧♦ ❞❡ n✳ ❖❜✈✐❛♠❡♥t❡ ❛ ❞✐❢❡r❡♥ç❛ ❡♥tr❡ ❞♦✐s ♥ú♠❡r♦s q✉❛✐sq✉❡r ❞❛ ❧✐st❛ ✺✳✶ r❡s✉❧t❛ ❡♠ ✉♠ ♥ú♠❡r♦ ❢♦r♥❛❞♦ ❛♣❡♥❛s ♣❡❧♦s ❛❧❣❛r✐s♠♦s ✵ ❡ ✶✳

  ✶✼✳ ❖s ♣♦♠❜♦s r❡♣r❡s❡♥t❛♠ ♦s n 1 ♥ú♠❡r♦s ❡s❝♦❧❤✐❞♦s ❞♦ ❝♦♥❥✉♥t♦ t1, 2, 3, ☎ ☎ ☎ , 2n✉ ❡ ❛s ❝❛s❛s sã♦ ❡s❝♦❧❤✐❞❛s ❝♦♠♦ s❡♥❞♦ ♦s n ❝♦♥❥✉♥t♦s✿

  C j 1 ✏ t2j ✁ 1, 2j✉, 1 ↕ j ↕ n, 2 ✐st♦ é✱ ❡♠ C só ❝♦❧♦❝❛r❡♠♦s ♦ ✶ ❡ ♦ ✷✱ ❡♠ C ♦ ✷ ❡ ♦ ✸✱ ❡ ❛ss✐♠ s✉❝❡ss✐✈❛♠❡♥t❡✳ ▲♦❣♦✱ ♣❡❧♦ Pr✐♥❝í♣✐♦ ❞❛ ❈❛s❛ ❞♦s P♦♠❜♦s✱ q✉❛♥❞♦ ❞✐str✐❜✉í♠♦s ♦s n 1 ♥ú♠❡r♦s

  ✺✳✷ ❆❧❣✉♥s ❆s♣❡❝t♦s ❞❛ ❉✐✈✐s✐❜✐❧✐❞❛❞❡ ♥❛ ➪❧❣❡❜r❛ ✶✷✹ j , 1 j ♥♦s n ❝♦♥❥✉♥t♦s C ↕ j ↕ n✱ ❞♦✐s ❞❡❧❡s ✜❝❛rã♦ ❥✉♥t♦s ❡♠ ❛❧❣✉♠ ❝♦♥❥✉♥t♦ C ✱ ♦✉ s❡❥❛✱ ❡ss❡s ♥ú♠❡r♦s s❡rã♦ ❝♦♥s❡❝✉t✐✈♦s ❡ ♣♦rt❛♥t♦ ♣r✐♠♦s ❡♥tr❡ s✐✳

  ✺✳✷ ❆❧❣✉♥s ❆s♣❡❝t♦s ❞❛ ❉✐✈✐s✐❜✐❧✐❞❛❞❡ ♥❛ ➪❧❣❡❜r❛

  ✶✳ ❋❛r❡♠♦s ❛♣❡♥❛s ♦ ✐t❡♠ ♣aq✱ ❞❡✐①❛♥❞♦ ♦s ♦✉tr♦s ✐t❡♥s ❝♦♠♦ ❡①❡r❝í❝✐♦ ♣❛r❛ ♦ ❧❡✐t♦r✳ ✭❛✮ mdc

  ♣637, 3887q ✏ mdc ♣637, 3887 ✁ 6 ☎ 637q ✏ mdc ♣637, 65q ✏ mdc ♣637 ✁ 9 ☎ 65, 65q ✏ mdc ♣52, 65q ✏ mdc ♣52, 65 ✁ 52 ☎ 1q ✏ mdc ♣52, 13q ✏ 13.

  ❉❡t❡r♠✐♥❛♥❞♦ m ❡ n✱ t❡♠♦s✿

  13 ✏ 65 ✁ ♣637 ✁ 9 ☎ 65q ✏ 10 ☎ 65 ✁ 637 ✏ 10 ☎ ♣3887 ✁ 6 ☎ 637q ✁ 637 ✏ 10 ☎ 3887 ✁ 61 ☎ 637.

  ▲♦❣♦✱ m ✏ ✁61 ❡ n ✏ 10✳ ✷✳ ❋❛r❡♠♦s ❛♣❡♥❛s ♦ ✐t❡♠ ♣aq✱ ❞❡✐①❛♥❞♦ ♦s ♦✉tr♦s ✐t❡♥s ❝♦♠♦ ❡①❡r❝í❝✐♦ ♣❛r❛ ♦ ❧❡✐t♦r✳

  ✭❛✮ mdc ♣n, 2n 1q ✏ mdc ♣n, 2n 1 ✁ 2 ☎ nq

  ✏ mdc ♣n, 1q ✏ 1. ✸✳

  ✭❛✮ ❚♦❞♦ ♥ú♠❡r♦ q✉❛♥❞♦ ❞✐✈✐❞✐❞♦ ♣♦r três✱ ❞❡✐①❛ r❡st♦ 0, 1✱ ♦✉ 2✱ ♦✉ s❡❥❛✱ ♣♦❞❡♠ s❡r ❡s❝r✐t♦s ♥❛ ❢♦r♠❛ 3k, 3k 1✱ ♦✉ 3k 2✳ ❉❡✈❡♠♦s ❡❧❡✈❛r ❛♦ q✉❛❞r❛❞♦ ❝❛❞❛ ❡①♣r❡ssã♦ ❞❡ss❛ ❡ ✈❡r ♦ q✉❡ ❛❝♦♥t❡❝❡✳ ❊❧❡✈❛♥❞♦ ❛♦ q✉❛❞r❛❞♦ 3k✱ t❡♠♦s 2 2

  ✶

  , 2 2 ♣3kq ✏ 9k ✏ 3k

  ✶

  ♦♥❞❡ k ✏ 3k ✱ ❡ ♣3kq ❞❡✐①❛ r❡st♦ ③❡r♦ q✉❛♥❞♦ ❞✐✈✐❞✐❞♦ ♣♦r 3✳

  ✺✳✷ ❆❧❣✉♥s ❆s♣❡❝t♦s ❞❛ ❉✐✈✐s✐❜✐❧✐❞❛❞❡ ♥❛ ➪❧❣❡❜r❛ ✶✷✺ ❊❧❡✈❛♥❞♦ ❛♦ q✉❛❞r❛❞♦ 3k 1✱ t❡♠♦s

  ✏ 3k 2 4k

  ❝✮ ❉❛ ♠❡s♠❛ ❢♦r♠❛✱ ❡❧❡✈❛✲s❡ ❛♦ q✉❛❞r❛❞♦ ❛s ❡①♣r❡ssõ❡s 8k, 8k 1, 8k 2, 8k 3, 8k 4, 8k 5, 8k

  ♣4k 3q 2 ✏ 16k 2 24k 9, deixa resto 1 quando dividido por 4.

  ♣4k 2q 2 ✏ 16k 2 16k 4, deixa resto 0 quando dividido por 4.

  ♣4k 1q 2 ✏ 16k 2 8k 1, deixa resto 1 quando dividido por 4.

  ♣4kq 2 ✏ 16k 2 , deixa resto 0 quando dividido por 4.

  ❜✮ ■❞ê♥t✐❝♦ ❛♦ ✐t❡♠ ❛♥t❡r✐♦r✳ ❱❡❥❛ ♦ q✉❡ ❛❝♦♥t❡❝❡ ❝♦♠ ❛s ❡①♣r❡ssõ❡s 4k, 4k 1, 4k 2 ❡ 4k 3 q✉❛♥❞♦ ❡❧❡✈❛❞❛s ❛♦ q✉❛❞r❛❞♦✳

  3 ✳

  P♦rt❛♥t♦✱ t♦❞♦ ♥ú♠❡r♦ q✉❛❞r❛❞♦ ♣❡r❢❡✐t♦ ❞❡✐①❛ r❡st♦ ③❡r♦ ♦✉ 1 q✉❛♥❞♦ ❞✐✈✐❞✐❞♦ ♣♦r

  1 ✱ ❡ ♣3k 2q 2 ❞❡✐①❛ r❡st♦ 1 q✉❛♥❞♦ ❞✐✈✐❞✐❞♦ ♣♦r 3✳

  ✶

  ♣3k 1q 2 ✏ 9k 2 6k

  1, ♦♥❞❡ k

  ✶

  4 ✏ 3k

  ♣3k 2q 2 ✏ 9k 2 12k

  ❡ ♣3k 1q 2 ❞❡✐①❛ r❡st♦ 1 q✉❛♥❞♦ ❞✐✈✐❞✐❞♦ ♣♦r 3✳ ❊❧❡✈❛♥❞♦ ❛♦ q✉❛❞r❛❞♦ 3k 2✱ t❡♠♦s

  ✏ 3k 2 2k

  ✶

  1, ♦♥❞❡ k

  ✶

  1 ✏ 3k

  6 ❡8k 7. ♣8kq 2 ✏ 64k 2 , deixa resto 0 quando dividido por 8. ♣8k 1q 2 ✏ 64k 2 16k 1, deixa resto 1 quando dividido por 8. ♣8k 2q 2 ✏ 64k 2 32k 4, deixa resto 4 quando dividido por 8. ♣8k 3q 2 ✏ 64k 2 48k 9, deixa resto 1 quando dividido por 8. ♣8k 4q 2 ✏ 64k 2 64k 16, deixa resto 0 quando dividido por 8. ♣8k 5q 2 ✏ 64k 2 80k 25, deixa resto 1 quando dividido por 8. ♣8k 6q 2 ✏ 64k 2 96k 36, deixa resto 4 quando dividido por 8. ♣8k 7q 2 ✏ 64k 2 112k 49, deixa resto 1 quando dividido por 8. ▲♦❣♦✱ t♦❞♦ ♥ú♠❡r♦ q✉❛❞r❛❞♦ ♣❡r❢❡✐t♦ ❞❡✐①❛ r❡st♦ 0✱ 1 ♦✉ 4 q✉❛♥❞♦ ❞✐✈✐❞✐❞♦ ♣♦r 8✳

  ✺✳✷ ❆❧❣✉♥s ❆s♣❡❝t♦s ❞❛ ❉✐✈✐s✐❜✐❧✐❞❛❞❡ ♥❛ ➪❧❣❡❜r❛ ✶✷✻ ✹✳ ▼♦str❛r❡♠♦s✱ ♣r✐♠❡✐r♦✱ ❛ ✐❞❛✱ ♦✉ s❡❥❛ b 1 2 1 1 r 1 e a 2 2 r 2 , com r 1 2 . ⑤ ♣a ✁ a q ñ a ✏ b ☎ q ✏ b ☎ q ✏ r 1 1 r 1 2 2 r 2 1 2

  ❙✉♣♦♥❤❛ q✉❡ a ✏ b ☎ q ❡ a ✏ b ☎ q ✱ ❝♦♠ r ➔ b ❡ r ➔ b✱ ♠♦str❛r❡♠♦s q✉❡ r 1 2 ✏ r ✳

  ❙❡♥❞♦ a 1 2 1 r 1 2 2 1 2 1 2 ✁ a ✏ b ☎ q ✁ b ☎ q ✁ r ✏ b ☎ ♣q ✁ q q ♣r ✁ r q,

  ♠❛s b 1 2 1 2 1 2 ⑤ ♣a ✁ a q, b ⑤ b ☎ ♣q ✁ q q ❡ ⑤r ✁ r ⑤ ➔ b,

  ❧♦❣♦ r 1 2 1 2 . 1 2 ✁ r ✏ 0 ñ r ✏ r P♦rt❛♥t♦✱ a ❡ a ❞❡✐①❛♠ r❡st♦s ✐❣✉❛✐s ♥❛ ❞✐✈✐sã♦ ♣♦r b✳ ❘❡❝✐♣r♦❝❛♠❡♥t❡✱ ♠♦str❛r❡♠♦s ❛ ✈♦❧t❛✱ ♦✉ s❡❥❛ a 1 1 r 1 2 2 r 2 , com r 1 2 1 2 ✏ b ☎ q ❡ a ✏ b ☎ q ✏ r ñ b ⑤ ♣a ✁ a q.

  ❙❡♥❞♦✱ a 1 2 1 2 1 2 1 2 , ✁ a ✏ b♣q ✁ q q r ✁ r ❡ r ✏ r

  ❡♥tã♦✱ a 1 2 1 2 1 2 ✁ a ✏ b♣q ✁ q q.

  P♦rt❛♥t♦✱ b ⑤ ♣a ✁ a q✳ ✺✳ ❙❡ 1 n 2 n 2 α n a n a n n .

  ✁1

  ✏ a ☎ ☎ ☎ a ✏ ♣a ☎ 10 q ☎ ☎ ☎ a ☎ 10 ☎ a ☎ ☎ ☎ a 2 ❆ss✐♠✱ t♦❞❛s ❛s ♣❛r❝❡❧❛s t❡r♠✐♥❛♠ ❡♠ ③❡r♦ ❝♦♠ ❡①❝❡çã♦ ❞❡ a q✉❛♥❞♦ a ✘ 0✳ ▲♦❣♦ ❜❛st❛ ♦❜s❡r✈❛r♠♦s ♦ q✉❡ ❛❝♦♥t❡❝❡ ❝♦♠ ♦ ú❧t✐♠♦ ❛❧❣❛r✐s♠♦ ❞❡ ✉♠ ♥ú♠❡r♦ ❛♦ 2 2 2 2 2 2 2

  ♦ ❡❧❡✈❛r♠♦s ❛♦ q✉❛❞r❛❞♦✳ 0 ✏ 0, 1 ✏ 1, 2 ✏ 4, 3 ✏ 9, 4 ✏ 16, 5 ✏ 25, 6 ✏ 2 2 2 36, 7

  ✏ 49, 8 ✏ 64, 9 ✏ 81✳ ▲♦❣♦✱ t♦❞♦ ♥ú♠❡r♦ q✉❛❞r❛❞♦ ♣❡r❢❡✐t♦ t❡r♠✐♥❛ ❡♠ 0, 1, 4, 5, 6

  ✱ ♦✉ 9✳ n n n n

  1 ✻✳ ❉❡✈❡♠♦s ♣r♦✈❛r q✉❡ ♦ ♥ú♠❡r♦ 8 ✁ 3 ✁ 6 é ❞✐✈✐sí✈❡❧ ♣♦r 2 ❡ ♣♦r 5✳ ❯s❡ ♦ ❢❛t♦

  ✺✳✷ ❆❧❣✉♥s ❆s♣❡❝t♦s ❞❛ ❉✐✈✐s✐❜✐❧✐❞❛❞❡ ♥❛ ➪❧❣❡❜r❛ ✶✷✼ ❞❡ q✉❡ n n n n

  8 ✑ 3 ♣mod 5q ñ 8 ✑ 3 ♣mod 5q ñ 8 ✁ 3 ✑ 0 ♣mod 5q,

  ❛ss✐♠ ❝♦♠♦ n n n n

  6 ✑ 1 ♣mod 5q ñ 6 ✑ 1 ♣mod 5q ñ 6 ✁ 1 ✑ 0 ♣mod 5q. ❊♥tã♦✱ n n n n n n n n ♣8 ✁ 3 q ✁ ♣6 ✁ 1 q ✑ 0 ♣mod 5q.

  1 ❊ ♦ ♥ú♠❡r♦ 8 ✁ 3 ✁ 6 é ❞✐✈✐sí✈❡❧ ♣♦r 5✳ ❉❛ ♠❡s♠❛ ❢♦r♠❛✱ t❡♠♦s q✉❡ n n n n

  8 ✑ 6 ♣mod 2q ñ 8 ✑ 6 ♣mod 2q ñ 8 ✁ 6 ✑ 0 ♣mod2q,

  ❡ n n n n

  3 n n n n ✑ 1 ♣mod 2q ñ 3 ✑ 1 ♣mod 2q ñ 3 ✁ 1 ✑ 0 ♣mod 2q.

  1 ❊ ♦ ♥ú♠❡r♦ 8 ✁ 3 ✁ 6 t❛♠❜é♠ é ❞✐✈✐sí✈❡❧ ♣♦r 2✳ P♦rt❛♥t♦✱ ❡ss❡ ♥ú♠❡r♦ é ❞✐✈✐sí✈❡❧ ♣♦r 10✳

  ✼✳ ◆♦t❡ q✉❡ 99

  1 ✑ 1 ♣mod 5q,

  ✂ ✡ 2 2 49 98 mod 5

  2 ✑ ✁1 ♣mod 5q ñ ♣2 q ✑ ✁1 ♣mod 5q ñ 2 ☎ 2 ✑ ✁1 ☎ 2 mdc 99

  ♣5, 2q 2 ñ 2 ✑ ✁2 ♣mod 5q, 98 99

  3 ✑ ✁1 ♣mod 5q ñ 3 ✑ ✁1 ♣mod 5q ñ 3 ✑ ✁3 ♣mod 5q, 98 99

  4 ✑ ✁1 ♣mod 5q ñ 4 ✑ 1 ♣mod 5q ñ 4 ✑ 4 ♣mod 5q

  ❡ 99

  5 ✑ 0 ♣mod 5q. ❆❣♦r❛✱ t❡♠♦s 99 99 99 99 99

  1

  2

  3

  4

  5 99 99 99 ✑ 1 ♣✁2q ♣✁3q 4 0 ♣mod 5q. 99 99

  2

  3

  4

  5 P♦rt❛♥t♦✱ ♦ ♥ú♠❡r♦ 1 é ❞✐✈✐sí✈❡❧ ♣♦r 5✳

  ✺✳✷ ❆❧❣✉♥s ❆s♣❡❝t♦s ❞❛ ❉✐✈✐s✐❜✐❧✐❞❛❞❡ ♥❛ ➪❧❣❡❜r❛ ✶✷✽ ✽✳ ❙❡❥❛ S ✏ 1 10

  1 10 ✑ 6 10 ✑ 14 10 ✑ 17 10 ✑ 36 10 ✑ 1 ♣mod 101q

  ✭❛✮ ◆♦t❡ q✉❡ 2002 é ♠ú❧t✐♣❧♦ ❞❡ 7✱ ♣♦rt❛♥t♦ 2001 ☎ 2002 ☎ 2003 ☎ 2004 t❛♠❜é♠ é✳ ❇❛st❛ s❛❜❡r ♦ r❡st♦ q✉❡ é ❞❡✐①❛❞♦ ❡♠ 2005 2 q✉❛♥❞♦ ❞✐✈✐❞✐❞♦ ♣♦r 7✳ 2005

  ♦✉ s❡❥❛✱ 2S ✑ 0 ♣mod 101q ❡ S ✑ 0 ♣mod 101q✱ ♦ q✉❡ s✐❣♥✐✜❝❛ q✉❡ ✶✵✶ ❞✐✈✐❞❡ S✳ ✾✳

  2S ✑ 2☎5☎♣1✁1q 2☎5☎♣6✁6q 2☎5☎♣14✁14q 2☎5☎♣17✁17q 2☎5☎♣36✁36q ♣mod 101q;

  3 10 ✑ 7 1 ✑ 18 10 ✑ 42 10 ✑ 50 10 ✑ ✁36 ♣mod 101q

  5 10 ✑ 16 10 ✑ 22 10 ✑ 30 10 ✑ 31 10 ✑ 36 ♣mod 101q

  9 10 ✑ 21 10 ✑ 25 10 ✑ 47 10 ✑ 49 10 ✑ ✁17 ♣mod 101q

  8 10 ✑ 11 10 ✑ 15 10 ✑ 35 10 ✑ 48 10 ✑ 17 ♣mod 101q

  13 10 ✑ 19 10 ✑ 20 10 ✑ 23 10 ✑ 37 10 ✑ ✁14 ♣mod 101q

  2 10 ✑ 12 10 ✑ 28 10 ✑ 29 10 ✑ 34 10 ✑ 14 ♣mod 101q

  4 10 ✑ 24 10 ✑ 33 10 ✑ 43 10 ✑ 45 10 ✑ ✁6 ♣mod 101q

  26 10 ✑ 27 10 ✑ 38 10 ✑ 40 10 ✑ 46 10 ✑ 6 ♣mod 101q

  10 10 ✑ 32 10 ✑ 39 10 ✑ 41 10 ✑ 44 10 ✑ ✁1 ♣mod 101q

  ❡ ❛ss✐♠ ❜❛st❛ ♦❜s❡r✈❛♠♦s ❛s ❝♦♥❣r✉ê♥❝✐❛s ❞❡ 1 10 ❛té 1 5 ✳ ❖❜s❡r✈❛♠♦s q✉❡

  2 10 ☎ ☎ ☎ 100 10 ♦✉ S ✏ 100 10

  51 10 ✑ 50 10 ♣mod 101q,

  ☎ ☎ ☎

  99 10 ✑ 2 10 ♣mod 101q

  ✑ 1 10 ♣mod 101q

  ❆♣❧✐❝❛♥❞♦✲s❡ ♦ ❝♦♥❝❡✐t♦ ❞❡ ❝♦♥❣r✉ê♥❝✐❛✱ ♣❡r❝❡❜❡♠♦s q✉❡✿ 100 10

  1 10 q.

  2 10 q ♣100 10

  99 10 q ☎ ☎ ☎ ♣99 10

  100 10 q ♣2 10

  2S ✏ ♣1 10

  ✐❣✉❛❧❞❛❞❡s✱ ♦❜t❡♠♦s✿

  99 10 ☎ ☎ ☎ 1 10 ✳ ❙♦♠❛♥❞♦✲s❡ ❛s ❞✉❛s

  ✑ 3 ♣mod 7q ñ 2005 2 ✑ 9 ♣mod 7q ñ 2005 2 ✑ 2 ♣mod 7q.

  ✺✳✷ ❆❧❣✉♥s ❆s♣❡❝t♦s ❞❛ ❉✐✈✐s✐❜✐❧✐❞❛❞❡ ♥❛ ➪❧❣❡❜r❛ ✶✷✾ 2 P♦rt❛♥t♦ 2001 ☎ 2002 ☎ 2003 ☎ 2004 2005 ❞❡✐①❛ r❡st♦ 2 q✉❛♥❞♦ ❞✐✈✐❞✐❞♦ ♣♦r 7✳ ✭❜✮ ❇❛st❛ s❡❣✉✐r 50

  4 50 ✑ 1 ♣mod 3q ñ 4 ✑ 1 ♣mod 3q.

  ▲♦❣♦✱ 4 ❞❡✐①❛ r❡st♦ 1 q✉❛♥❞♦ ❞✐✈✐❞✐❞♦ ♣♦r 3✳ ✶✵✳ ➱ só ♦❜s❡r✈❛r q✉❡ 2468 2468

  999994 ✑ 1 ♣mod 333331q ñ 999994 ✑ 1 ♣mod 333331q 2468 2468 ñ 999994 ✁ 1 ✑ 0 ♣mod 333331q.

  ▲♦❣♦✱ 999994 ✁ 1 é ❞✐✈✐sí✈❡❧ ♣♦r 333331✳ ✶✶✳ ◆❡ss❡s ❝❛s♦s✱ só é ♥❡❝❡ssár✐♦ ♦❜s❡r✈❛r ♦ q✉❡ ❛❝♦♥t❡❝❡ ❝♦♠ ♦ ú❧t✐♠♦ ❛❧❣❛r✐s♠♦ ❞♦

  ♥ú♠❡r♦ ❛♣ós ❛ ❡❧❡✈❛çã♦ ❞♦ ❡①♣♦❡♥t❡✳ 2 k 2 k 1 ✭❛✮ ◆♦t❡ q✉❡ 9 ✏ ☎ ☎ ☎ 1 ❡ 9 ✏ ☎ ☎ ☎ 9✱ ❝♦♠ k P N✳ 2015

  ❈♦♠♦ 2015 é ❞❛ ❢♦r♠❛ 2k 1✱ ❡♥tã♦ ♦ ú❧t✐♠♦ ❞í❣✐t♦ ❞♦ ♥ú♠❡r♦ 1989 é 9✳ ✭❜✮ ❖❜s❡r✈❛♥❞♦ ❛ s❡q✉ê♥❝✐❛ 1 2 3 4 7 ; 7 5 ✏ 7 ✏ 49 ; 7 ✏ 343 ; 7 ✏ 2401

  7 ✏ 16807 ; ☎ ☎ ☎

  ✳✳✳ ◆♦t❡ q✉❡ ♦s ú❧t✐♠♦s ❞í❣✐t♦s s❡ r❡♣❡t❡♠ ❛ ♣❛rt✐r ❞❛ q✉✐♥t❛ ♣♦tê♥❝✐❛✳ 777 4

  ☎194 1

  ❚❡♠♦s q✉❡ 777 ✏ 4 ☎ 194 1 ❡ ♦ ♥ú♠❡r♦ 777 ✏ 777 t❡♠ 7 ♥♦ ú❧t✐♠♦ ❞í❣✐t♦✳ ❡ 1 2 3 4

  2 5 ✏ 2 ; 2 ✏ 4 ; 2 ✏ 8 ; 2 ✏ 16

  2 ✏ 32 ; ☎ ☎ ☎ 50 ✳✳✳ 4

  ☎12 2

  ❉❛ ♠❡s♠❛ ❢♦r♠❛ 2 ✏ 2 ❡ ❞❡✐①❛ 4 ♥♦ ú❧t✐♠♦ ❞í❣✐t♦ ❞❛ ♣♦tê♥❝✐❛✳ 777 50

  2 ▲♦❣♦✱ 777 t❡♠ ❡♠ s❡✉ ú❧t✐♠♦ ❛❧❣❛r✐s♠♦ ♦ ♥ú♠❡r♦ 1✱ ♣♦✐s 7 4 ✏ 11✳ 2 2

  ✭❝✮ ❋❛ç❛ ❛ ♣♦t❡♥❝✐❛çã♦ ❞❡ 1 ✏ 1 ❛té 20 ✏ 400 ❡ ♣❡r❝❡❜❛ q✉❡ ♦ ú❧t✐♠♦ ❞í❣✐t♦ ❞❛ s♦♠❛ ❞❡ss❛s ♣♦tê♥❝✐❛ é ✐❣✉❛❧ ❛ ③❡r♦✳ P♦rt❛♥t♦ ❛té 2000 q✉❡ é ✉♠ ♠ú❧t✐♣❧♦ ❞❡ 20✱ ♦ ú❧t✐♠♦ ❛❧❣❛r✐s♠♦ ❞❛ s♦♠❛✱ t❛♠❜é♠ é ③❡r♦✳ ❘❡st❛ ✈❡r ❛ s♦♠❛ ❞♦s ú❧t✐♠♦s 15 ♥ú♠❡r♦s✳ 2 2

  ◗✉❛♥❞♦ ✈♦❝ê ❞✐str✐❜✉✐r✱ ♣❡r❝❡❜❡rá q✉❡ ♦ ú❧t✐♠♦ ❛❧❣❛r✐s♠♦ ❞❛ s♦♠❛ ❞❡ 2001 ❛ 2015 2 2

  ✺✳✷ ❆❧❣✉♥s ❆s♣❡❝t♦s ❞❛ ❉✐✈✐s✐❜✐❧✐❞❛❞❡ ♥❛ ➪❧❣❡❜r❛ ✶✸✵ ✭❞✮ ❖❜s❡r✈❡ q✉❡ 1 2

  4 3 ✏ 4 ; 4 ✏ 16 4

  4 ✏ 64 ; 4 ✏ 256 2 k ✳✳✳ 2 k

  ✁1

  ❚♦❞♦ ♥ú♠❡r♦ ♥❛ ❢♦r♠❛ 4 t❡r♠✐♥❛ ❝♦♠ ♦ ❞í❣✐t♦ 4 ❡ ♥❛ ❢♦r♠❛ 4 t❡♠ ú❧t✐♠♦

  ✝

  ❞í❣✐t♦ ✐❣✉❛❧ ❛ 6✱ ❝♦♠ k P N ✳ ✭❚❡st❡ ♣♦r ✐♥❞✉çã♦✳ ➱ ❢á❝✐❧✳✮✳ ✭❡✮ ◗✉❛❧q✉❡r ♣♦tê♥❝✐❛ ❞❡ ✉♠ ♥ú♠❡r♦ ❝✉❥♦ ú❧t✐♠♦ ❞í❣✐t♦ é 5✱ t❡r♠✐♥❛ ❡♠ 5✳ ✶✷✳ ❚♦❞♦s ♦s ♥ú♠❡r♦s ♣♦❞❡♠ s❡ ❡s❝r✐t♦s ♥❛ ❢♦r♠❛ 7n, 7n 1, 7n 2, 7n 3, 7n 4, 7n 5

  ❡ 7n 6✳ ❚❡♠♦s s❡t❡ ❢♦r♠❛s ❞❡ ♥ú♠❡r♦s✱ ✉s❛♥❞♦ ♦ Pr✐♥❝í♣✐♦ ❞❛ ❈❛s❛ ❞♦s P♦♠❜♦s✱ ♦ ♣ró①✐♠♦ ♥ú♠❡r♦ t❡r✐❛ ✉♠ ❞❡ss❡s ❢♦r♠❛t♦s✱ ♦♥❞❡ ❛ ❞✐❢❡r❡♥ç❛ ❡♥tr❡ ❞♦✐s ♥ú♠❡r♦s ❝♦♠ ♦ ♠❡s♠♦ r❡st♦✱ s❡r✐❛ ✉♠ ♠ú❧t✐♣❧♦ ❞❡ 7✳

  ✶✸✳ ✭❛✮ ❚❡♠♦s q✉❡ 1989 ✑ 1 ♣mod 7q, 1990 ✑ 2 ♣mod7q, 1991 ✑ 3 ♣mod7q ❡ 1992 ✑ 3

  4 ♣mod 7q ♦ q✉❡ ✐♠♣❧✐❝❛ ❞✐③❡r q✉❡ 1992 ✑ 1 ♣mod 7q✱ ❧♦❣♦ 3

  1989 ☎ 1990 ☎ 1991 1992 ✑ 1 ☎ 2 ☎ 3 1 ♣mod 7q. 3 P♦rt❛♥t♦✱ 1989 ☎ 1990 ☎ 1991 1992 é ❞✐✈✐sí✈❡❧ ♣♦r 7 ✭❜✮ ◆♦t❡ q✉❡ 100

  9 ✑ 1 ♣mod 8q ñ 9 ✑ 1 ♣mod 8q

  ✭❝✮ ❖❜s❡r✈❡ q✉❡ 2 2 2014

  ✑ 5 ♣mod 7q ñ 2014 ✑ 5 ♣mod 7q 2 ✑ 4 ♣mod 7q 2 2015

  ✑ 6 ♣mod 7q ñ 2015 ✑ 6 ♣mod 7q ✑ 1 ♣mod 7q

  ▲♦❣♦✱ 2 2 2014 2015 ✑ 4 1 ♣mod 7q.

  ✶✹✳ ❙❡❥❛ 5 4 n 4n

  4 ✏ n ☎ ♣n q, 4

  4 s❡ 5 ⑤ n ♥❛❞❛ ❛ ❞❡♠♦♥str❛r✱ ♠❛s s❡ 5 ∤ n✱ ❡♥tã♦ ❞❡✈❡♠♦s ♠♦str❛r q✉❡ 5 ⑤ ♣n q✳

  ✺✳✷ ❆❧❣✉♥s ❆s♣❡❝t♦s ❞❛ ❉✐✈✐s✐❜✐❧✐❞❛❞❡ ♥❛ ➪❧❣❡❜r❛ ✶✸✶ P❡❧♦ P❡q✉❡♥♦ ❚❡♦r❡♠❛ ❞❡ ❋❡r♠❛t✱ t❡♠♦s q✉❡ p 5 4 4

  ✁1 ✁1

  a 4 ✑ 1 ♣mod pq ñ n ✑ 1 ♣mod 5q ñ n ✑ ✁4 ♣mod 5q ñ n ✑ 0 ♣mod 5q. 5

  4n P♦rt❛♥t♦✱ n é ❞✐✈✐sí✈❡❧ ♣♦r 5✱ ♣❛r❛ t♦❞♦ n P N✳ 3 3

  ✶✺✳ ❙❡❣✉✐♥❞♦ ❛ ❞✐❝❛✱ ✐r❡♠♦s ♠♦str❛r q✉❡ n ✁ n é ♠ú❧t✐♣❧♦ ❞❡ 2 ❡ ❞❡ 3✳ ❙✉❜st✐t✉✐♥❞♦ n ♣♦r 2k ✁ 1✱ ♣❛r❛ k P N✱ t❡♠♦s q✉❡ 3 2 n

  ✁ n ✏ n ☎ ♣n ✁ 1q ✏ n ☎ ♣n ✁ 1q ☎ ♣n 1q ✏ ♣2k ✁ 1q ☎ ♣2k ✁ 2q ☎ 2k ✏ 4 ☎ ♣2k ✁ 1q ☎ k ☎ ♣k ✁ 1q

  ◆♦t❡ q✉❡ k ☎ ♣k ✁ 1q sã♦ ♥ú♠❡r♦s ❝♦♥s❡❝✉t✐✈♦s✱ ♣♦rt❛♥t♦ é ✉♠ ♥ú♠❡r♦ ♣❛r✳ ▲♦❣♦

  4 ☎ ♣2k ✁ 1q ☎ k ☎ ♣k ✁ 1q é ✉♠ ♠ú❧t✐♣❧♦ ❞❡ 8✳

  P❡r❝❡❜❛✱ t❛♠❜é♠ q✉❡✱ 3 2 n ✁ n ✏ n ☎ ♣n ✁ 1q ✏ ♣n ✁ 1q ☎ n ☎ ♣n 1q.

  ❖♥❞❡ ♣n ✁ 1q ☎ n ☎ ♣n 1q sã♦ três ♥ú♠❡r♦s ❝♦♥s❡❝✉t✐✈♦s ❡✱ ♣♦rt❛♥t♦✱ é ✉♠ ♠ú❧t✐♣❧♦ 3 ❞❡ 3✳ ▲♦❣♦ n ✁ n é ✉♠ ♠ú❧t✐♣❧♦ ❞❡ ✷✹ ♣❛r❛ q✉❛❧q✉❡r ✐♥t❡✐r♦ n í♠♣❛r✳

  ✶✻✳ ❖❜s❡r✈❡ q✉❡ 3 3 671 671 2013

  3 ✑ ✁1 ♣mod 7q ñ ♣3 q ✑ ♣✁1q ♣mod 7q ñ 3 ✑ ✁1 ♣mod 7q

  ✂ ✡ 2013 2 2 2015 mod 7 ñ 3 ☎ 3 ✑ ✁1 ☎ 3 ñ 3 ✑ 5 ♣mod 7q. 2 mdc , 7 2015 ♣3 q P♦rt❛♥t♦✱ ♦ r❡st♦ ❞❛ ❞✐✈✐sã♦ ❞❡ 3 ♣♦r 7 é 5✳

  ✶✼✳ ❯s❛♥❞♦ ♦ ❝♦♥❤❡❝✐♠❡♥t♦ ❞❡ ❝♦♥❣r✉ê♥❝✐❛✱ t❡♠♦s q✉❡ 3 3 1851 1851 2222

  ✑ 3 ♣mod 7q ñ 2222 ✑ ✁1 ♣mod 7q ñ ♣2222 q ✑ ♣✁1q ♣mod 7q ✂ ✡ 5553 2 2 mod7

  ñ 2222 ☎ 2222 ✑ ♣✁1q ☎ 2222 2 mdc , 7 5555 ♣2222 q ñ 2222 ✑ ✁2 ♣mod 7q.

  ✺✳✷ ❆❧❣✉♥s ❆s♣❡❝t♦s ❞❛ ❉✐✈✐s✐❜✐❧✐❞❛❞❡ ♥❛ ➪❧❣❡❜r❛ ✶✸✷ ❆ss✐♠ ❝♦♠♦✱

  2 2

  1 ✁

  1 225

  ✡ ✏

  ✂

  1 ✁

  1

  ☎ ✂

  9 ✡

  1 ✁

  1

  3 2

  ☎ ☎ ☎ ✂

  1 ✁

  1

  ☎ ☎ ☎ ✂

  1

  5555 ✑ 4 ♣mod 7q ñ 5555 4 ✑ 1 ♣mod 7q ñ ♣5555 4 q 555 ✑ 1 ♣mod 7q

  ✭❞✮ ♣n 1q 2 ✁ n 2 ✏ ♣n 1 nq ☎ ♣n 1 ✁ nq

  ñ 5555 2220 ☎ 5555 2 ✑ 1 ☎ 5555 2 ♣mod 7q ñ 5555 2222 ✑ 2 ♣mod 7q.

  ▲♦❣♦✱ 2222 5555 5555 2222

  ❞❡✐①❛ r❡st♦ ✁2 2 ✏ 0 q✉❛♥❞♦ ❞✐✈✐❞✐❞♦ ♣♦r 7✱ ♣♦rt❛♥t♦ ❡ss❡ ♥ú♠❡r♦ é ❞✐✈✐sí✈❡❧ ♣♦r 7✳

  ✶✽✳ ❘❡s♦❧✉çã♦ ❞♦s ✐t❡♥s ♣cq ❡ ♣dq ✉s❛♥❞♦ ❛ ❢❛t♦r❛çã♦ a 2 ✁ b 2 ✏ ♣a bq ☎ ♣a ✁ bq✳ ✭❝✮

  555.555 2 ✁ 444.444 2 ✏ ♣555.555 444.444q ☎ ♣555.555 ✁ 444.444q

  ✏ 999.999 ☎ 111.111 ✏ ♣1.000.000 ✁ 1q ☎ 111.111 ✏ 111.111.000.000 ✁ 111.111 ✏ 111.110.888.889.

  ✏ ♣2n 1q ☎ 1 ✏ 2n 1. ❯t✐❧✐③❡ ♦ ✐t❡♠ ♣dq ♣❛r❛ r❡s♦❧✈❡r ♦s ✐t❡♥s ♣aq ❡ ♣bq✳

  1 ✁

  ✶✾✳ ❖ ❛❧❣❛r✐s♠♦ ✜♥❛❧ ❞❡ 9867 3 é ♦ ♠❡s♠♦ q✉❡ ♦ ❞❡ 7 3 ✱ ✐st♦ é✱ 3❀ ♦ ❛❧❣❛r✐s♠♦ ✜♥❛❧ ❞❡ 9867 2

  é ♦ ♠❡s♠♦ q✉❡ ♦ ❞❡ 7 2 ✏ 49✱ ✐st♦ é✱ 9✳ ❙❡ ❞❡ ✉♠ ♥ú♠❡r♦ t❡r♠✐♥❛❞♦ ❡♠ 3 s✉❜tr❛í♠♦s ♦✉tr♦ t❡r♠✐♥❛❞♦ ❡♠ 9✱ ♦ ❛❧❣❛r✐s♠♦ ✜♥❛❧ ❞♦ r❡s✉❧t❛❞♦ é 4✳ ✷✵✳ ❈❛❞❛ ✉♠ ❞♦s ❢❛t♦r❡s é ❞❛ ❢♦r♠❛ ❞✐❢❡r❡♥ç❛ ❞❡ q✉❛❞r❛❞♦s ✐st♦ é a 2 ✁ b 2 ✱ ♦♥❞❡ a ✏ 1✳

  ❚❡♠♦s q✉❡ ✂

  1 ✁

  1

  4 ✡

  ☎ ✂

  15 2 ✡ .

  ✺✳✷ ❆❧❣✉♥s ❆s♣❡❝t♦s ❞❛ ❉✐✈✐s✐❜✐❧✐❞❛❞❡ ♥❛ ➪❧❣❡❜r❛ ✶✸✸ 2 2 ❯s❛♥❞♦ ❛ ❢❛t♦r❛çã♦ a ✁ b ✏ ♣a ✁ bq ☎ ♣a bq✱ ♦❜t❡♠♦s✿

  ✂ ✡ ✂ ✡ ✂ ✡ ✂ ✡ ✂ ✡ ✂ ✡ ✂ ✡

  1

  1

  1

  1

  1

  1

  1

  1

  1

  1

  1

  1

  1

  1 ✁ ✁ ☎ ☎ ☎ ✁ ✏ ✁ ☎ ☎ ☎ ✁

  4 9 225

  2

  2

  15

  15

  1

  3

  2

  15

  14

  16 ✏ ☎ ☎ ☎ ☎ ☎ ☎ ☎

  2

  2

  3

  14

  15

  15 ❧♦♦♠♦♦♥ ❧♦♦♠♦♦♥ 1 1

  1

  16 ✏ ☎

  2

  15

  8 . ✏

  15 ✷✶✳ ❆ t❛❜❡❧❛ ❛❜❛✐①♦ ♠♦str❛ ❝♦♠♦ ❛♣❛r❡❝❡ ❡♠ ♦r❞❡♠✱ ❞❡③❡♥❛ ❡ ✉♥✐❞❛❞❡✱ ♦s ❞♦✐s ú❧t✐♠♦s

  ❛❧❣❛r✐s♠♦s ❞❡ ❛❧❣✉♠❛s ♣♦tê♥❝✐❛s ❞❡ 9✳ n n n n ❞♦✐s ú❧t✐♠♦s ❛❧❣❛r✐s♠♦s ❞❡ 9 ❞♦✐s ú❧t✐♠♦s ❛❧❣❛r✐s♠♦s ❞❡ 9

  ✵ ✵✶ ✻ ✹✶ ✶ ✵✾ ✼ ✻✾ ✷ ✽✶ ✽ ✷✶ ✸ ✷✾ ✾ ✽✾ ✹ ✻✶ ✶✵ ✵✶ ✺ ✹✾ ✶✶ ✵✾ 10

  ❖❜s❡r✈❡ q✉❡ ❛ ♣❛rt✐r ❞❡ 9 ♦s ❞♦✐s ú❧t✐♠♦s ❛❧❣❛r✐s♠♦s sã♦ ♦s ♠❡s♠♦s✱ ❢♦r♠❛♥❞♦ ✉♠❛ s❡q✉ê♥❝✐❛ ♣❡r✐ó❞✐❝❛✱ ❞❡ ♣❡rí♦❞♦ ✶✵✳ ❈♦♠♦ 2015 ✏ 10 ☎ 201 5 ❡ ♦s ❞♦✐s ú❧t✐♠♦s 10 2015

  ☎201

  ❛❧❣❛r✐s♠♦s ❞❡ 9 sã♦ ✵✶✱ s❡❣✉❡ q✉❡ ♦s ❞♦✐s ú❧t✐♠♦s ❛❧❣❛r✐s♠♦s ❞❡ 9 sã♦ ♦s ❞♦✐s 5 2015

  1 ú❧t✐♠♦s ❛❧❣❛r✐s♠♦s ❞❡ 9 ✱ ♦✉ s❡❥❛ ✹✾✳ ❉❛í ♦s ❞♦✐s ú❧t✐♠♦s ❛❧❣❛r✐s♠♦s ❞❡ 9 sã♦ 2015

  1 ✐❣✉❛✐s ❛ 49 1 ✏ 50✳ P♦rt❛♥t♦ ❡①✐st❡ ✉♠ ú♥✐❝♦ ③❡r♦ ♥♦ ✜♥❛❧ ❞♦ ♥ú♠❡r♦ 9 ✳

  ✷✷✳ ❙❡❥❛ x ✉♠ ♥ú♠❡r♦ ❞❡ ♦✐t♦ ❛❧❣❛r✐s♠♦s ❞❛ ❢♦r♠❛ x ✏ 9999 ✝ ✝ ✝ ✝.

  ❈♦♠♦ ♦ ♠❡♥♦r ❞❡ss❡s ♥ú♠❡r♦s é 99.990.000 ❡ ♦ ♠❛✐♦r é 99.999.999✱ t❡♠♦s q✉❡✿ 99.990.000 8 ↕ x ↕ 99.999.999. 8

  ❖❜s❡r✈❡♠♦s q✉❡ 10 ✏ 100.000.000 ✏ 99.999.999 1✳ ❊♥tã♦ 99.990.000 ↕ x ➔ 10 ✳

  ✺✳✷ ❆❧❣✉♥s ❆s♣❡❝t♦s ❞❛ ❉✐✈✐s✐❜✐❧✐❞❛❞❡ ♥❛ ➪❧❣❡❜r❛ ✶✸✹ 8 4 2 2 2 ❈♦♠♦ 10 ✏ ♣10 q ✏ 10.000 ✱ t❡♠♦s q✉❡ 99.990.000 ↕ 10.000 ✳ ❆❣♦r❛ ♦ ♠❛✐♦r 2 q✉❛❞r❛❞♦ ♣❡r❢❡✐t♦ ♠❡♥♦r q✉❡ 10.000 é ✐❣✉❛❧ ❛ 2 2 2

  9.999 ✏ ♣10.000✁1q ✏ 10.000 ✁20.000 1 ✏ 100.000.000✁20.000 1 ✏ 99.980.001. 2 2 ❈♦♠♦ 99.980.001 ➔ 99.990.000 ❝♦♥❝❧✉í♠♦s q✉❡ 9.999 ➔ x ➔ 10.000 ✳ ■st♦ ♠♦str❛ q✉❡ x ❡stá ❝♦♠♣r❡❡♥❞✐❞♦ ❡♥tr❡ ❞♦✐s q✉❛❞r❛❞♦s ♣❡r❢❡✐t♦s ❝♦♥s❡❝✉t✐✈♦s✳ P♦rt❛♥t♦✱ x ♥ã♦ ♣♦❞❡ s❡r ✉♠ q✉❛❞r❛❞♦ ♣❡r❢❡✐t♦✳

  ❄ ❄

  ❄ ❄ n n n n ✷✸✳ ❆ ✐♥❡q✉❛çã♦ ✁ ✁ 1 ➔ 0, 01 é ❡q✉✐✈❛❧❡♥t❡ ❛ ➔ 0, 01 ✁ 1✳ ❈♦♠♦ ♦s ❞♦✐s

  ❧❛❞♦s ❞❡st❛ ✐♥❡q✉❛çã♦ sã♦ ♥ú♠❡r♦s ♣♦s✐t✐✈♦s✱ ♣♦❞❡♠♦s ❡❧❡✈❛r ❡ss❡s ❞♦✐s ♠❡♠❜r♦s ❛♦ q✉❛❞r❛❞♦ ♣❛r❛ ♦❜t❡r ❛ ✐♥❡q✉❛çã♦ ❡q✉✐✈❛❧❡♥t❡✿ 2 ❄ ❄ 2 ❄ ✟ ✟ 2 n n 0, 02 n

  ➔ 0, 01 ✁ 1 ô n ➔ 0, 01 ✁ 1 n ✁ 1. ❉❛í✱ ♦❜t❡♠♦s 2

  1 2

  1 ✁ 2

  ❄ 1 100 ✁ 0, 01 ✁ 1

  100 n . ✁ 1 → ✏ ✏

  2 0, 02 200

  100 ❊❧❡✈❛♥❞♦ ♥♦✈❛♠❡♥t❡ ❛♦ q✉❛❞r❛❞♦ ♦s ❞♦✐s ❧❛❞♦s ✭♥ã♦ ♥❡❣❛t✐✈♦s✮ ❞❡st❛ ✐♥❡q✉❛çã♦✱ 2 2 4 2 2

  100 1 100

  1

  1 ♣100 ✁ 1q ✁ 2 ☎ 100 n

  , ✁ 1 → 2 ✏ 2 ✏ ✁ 2

  200

  4

  4

  2

  4 ☎ 100 ☎ 100

  ♦✉ s❡❥❛✱

  1

  1

  1

  1

  1

  1 n 1 . ✁1 → 2.500✁ ô n → 2.500✁ ô n → 2.500 2 40.000

  2 40.000 2 40.000

  1

  1 ❯♠❛ ✈❡③ q✉❡ t❡♠♦s ➔ 1✱ é ❢á❝✐❧ ♦❜s❡r✈❛r q✉❡ ♦ ♠❡♥♦r ♥ú♠❡r♦ ✐♥t❡✐r♦ 2 40.000

  1

  1 ♠❛✐♦r q✉❡ 2.500 é 2.501✳ 2 40.000 ❉❛í ❝♦♥❝❧✉í♠♦s q✉❡ ♦ ♠❡♥♦r ♥ú♠❡r♦ ✐♥t❡✐r♦ ♣♦s✐t✐✈♦ q✉❡ s❛t✐s❢❛③ ❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❞❛❞❛ é ♦ ♥ú♠❡r♦ 2.501✳

  ✺✳✷ ❆❧❣✉♥s ❆s♣❡❝t♦s ❞❛ ❉✐✈✐s✐❜✐❧✐❞❛❞❡ ♥❛ ➪❧❣❡❜r❛ ✶✸✺ ✷✹✳ ■♥✐❝✐❛❧♠❡♥t❡✱ ♦❜s❡r✈❡♠♦s q✉❡✿

  N ✏ ♣n 6mq ☎ ♣2n 5mq ☎ ♣3n 4mq ✏ ♣n 7m ✁ mq ☎ ♣2n 7m ✁ 2mq ☎ ♣3n 7m ✁ 3mq ✏ ♣n ✁ m 7mq ☎ r2 ☎ ♣n ✁ mq 7ms ☎ r3 ☎ ♣n ✁ mq 7ms ✏ ♣k 7mq ☎ ♣2k 7mq ☎ ♣3k 7mq,

  ♦♥❞❡ k ✏ n ✁ m✳ ❆✜r♠❛♠♦s q✉❡ s❡ N é ♠ú❧t✐♣❧♦ ❞❡ 7✱ ❡♥tã♦ k é ♠ú❧t✐♣❧♦ ❞❡ 7✳ ❉❡ ❢❛t♦✱ ❝♦♠♦ 7 é ♣r✐♠♦ ❡ ❞✐✈✐❞❡ N✱ ❡♥tã♦ ✉♠ ❞♦s ❢❛t♦r❡s k 7m, 2k 7m ♦✉ 3k 7m é ♠ú❧t✐♣❧♦ ❞❡

  7 ✳ ❚❡♠♦s✿ k 7m k m

  ✭✐✮ ❙❡ k 7m é ♠ú❧t✐♣❧♦ ❞❡ 7✱ ❡♥tã♦ ✏ é ✐♥t❡✐r♦✱ ❧♦❣♦ k é ♠ú❧t✐♣❧♦

  7

  7 ❞❡ 7✳ ❙❡❣✉❡ q✉❡ 2k ❡ 3k t❛♠❜é♠ sã♦ ♠ú❧t✐♣❧♦s ❞❡ 7 ❡ ♣♦rt❛♥t♦ ♦s três ❢❛t♦r❡s k 7m, 2k 7m 3 ❡ 3k 7m sã♦ ♠ú❧t✐♣❧♦s ❞❡ 7✳ ❈♦♥❝❧✉í♠♦s q✉❡ N é ♠ú❧t✐♣❧♦

  ❞❡ 7 ✳ 2k 7m 2k

  7m m

  ✭✐✐✮ ❙❡ 2k é ♠ú❧t✐♣❧♦ ❞❡ 7✱ ❡♥tã♦ ✏ é ✐♥t❡✐r♦✱ ❧♦❣♦ 2k é

  7

  7 ♠ú❧t✐♣❧♦ ❞❡ 7✳ ❈♦♠♦ 2 ❡ 7 sã♦ ♣r✐♠♦s ❡♥tr❡ s✐✱ s❡❣✉❡ q✉❡ k é ♠ú❧t✐♣❧♦ ❞❡ 7✱ ♦ q✉❡ ❧❡✈❛ ❛♦ ❝❛s♦ ❛♥t❡r✐♦r✳

  ✭✐✐✐✮ ❙❡ 3k 7m é ♠ú❧t✐♣❧♦ ❞❡ 7✱ ❛♥❛❧♦❣❛♠❡♥t❡ ❝♦♥❝❧✉í♠♦s q✉❡ k é ♠ú❧t✐♣❧♦ ❞❡ 7✳ ✷✺✳ ❆s ❝♦♥❞✐çõ❡s ❞♦ ♣r♦❜❧❡♠❛ ❡q✉✐✈❛❧❡♠ ❛ ❞✐③❡r q✉❡✿

  2a ✁ 5 ✏ 2 ☎ ♣a 1q ✁ 7 ✏ 2 ☎ ♣a 2q ✁ 9 ✏ 2 ☎ ♣a 3q ✁ 11,

  é ♠ú❧t✐♣❧♦ ❞❡ ✺✱ ✼✱ ✾ ❡ ✶✶✱ ❞♦♥❞❡ é ♠ú❧t✐♣❧♦ ❞❡ 5 ☎ 7 ☎ 9 ☎ 11 ✏ 3.465✳ ❆ss✐♠✱ ♦ ♠❡♥♦r ✈❛❧♦r ❞❡ a é t❛❧ q✉❡ 2a ✁ 5 ✏ 3.465✱ ♦✉ s❡❥❛✱ a ✏ 1.735✳ 8 11 m 2 m

  2 2 , m, n ✷✻✳ ❉❡ 2 ✏ n P N✱ s❡❣✉❡ 2 ✏ ♣n ✁ 48q ☎ ♣n 48q✳ q p

  ▲♦❣♦✱ n 48 ✏ 2 ❡ n ✁ 48 ✏ 2 ✱ ❝♦♠ p q ✏ m ❡ p ➔ q✳ q p p q 5 q

  ✁p ✁p

  ❆ss✐♠✱ 2 ✁ 2 ✏ 2 ☎ ♣2 ✁ 1q ✏ 96 ✏ 2 ☎ 3 ❡✱ ❝♦♠♦ 2 ✁ 1 é í♠♣❛r✱ s❡❣✉❡ q✉❡ p q 5

  ✁p

  2 ✏ 2 ❡ 2 ✁ 1 ✏ 3✳

  ❉❛í✱ p ✏ 5, q ✁ p ✏ 2, q ✏ 7 ❡✱ ♣♦rt❛♥t♦✱ m ✏ 12 ❡ n ✏ 80✳ ✷✼✳ ◆♦t❡ q✉❡ 2

  3 2 222 ✑ ✁1 ♣mod 5q ♣3 q ✑ 1 ♣mod 5q,

  ✺✳✷ ❆❧❣✉♥s ❆s♣❡❝t♦s ❞❛ ❉✐✈✐s✐❜✐❧✐❞❛❞❡ ♥❛ ➪❧❣❡❜r❛ ✶✸✻ ❡

  4 333 ✑ ✁1 ♣mod 5q

  4 444 333 ✑ ✁1 ♣mod 5q. 444 333

  4

  4 P♦rt❛♥t♦✱ 3 ❞❡✐①❛ r❡st♦ ✁1 1 ✏ 0 q✉❛♥❞♦ ❞✐✈✐❞✐❞♦ ♣♦r ✺✳ ▲♦❣♦✱ 3 é ❞✐✈✐sí✈❡❧ ♣♦r ✺✳ 3

  ✷✽✳ ❯♠ ❝✉❜♦ ♣❡r❢❡✐t♦ é ✉♠ ♥ú♠❡r♦ ❞❛ ❢♦r♠❛ a ✱ ♦♥❞❡ a é ✉♠ ♥ú♠❡r♦ ♥❛t✉r❛❧✳ ❈♦♠♦ 4 2 4 8 8 3 2 3 6

  9 ✏ ♣3 q ✏ 3 ✱ ♦s ❝✉❜♦s ♣❡r❢❡✐t♦s q✉❡ ❞✐✈✐❞❡♠ 3 ✱ sã♦ 3 ❡ ♣3 q ✏ 3 ✳

  ✷✾✳ ❖❜s❡r✈❛♠♦s ♣r✐♠❡✐r❛♠❡♥t❡ q✉❡✱ s❡ ✉♠ ♥ú♠❡r♦ n ♥ã♦ é ❞✐✈✐sí✈❡❧ ♣♦r ✸ ❡♥t❪❛♦ ❡❧❡ é 3 ❞❛ ❢♦r♠❛ 3k 1 ♦✉ 3k 2✱ ❧♦❣♦ n é ❞❛ ❢♦r♠❛ 9k 1 ♦✉ 9k 8✳ ❙✉♣♦♥❤❛ q✉❡ ♥❡♥❤✉♠ ❞♦s ✐♥t❡✐r♦s a✱ b ❡ c s❡❥❛ ❞✐✈✐sí✈❡❧ ♣♦r ✸✳ ❙❡❣✉❡ q✉❡ ♦s ❝✉❜♦s ❞❡ss❡s ♥ú♠❡r♦s sã♦ ❞❛ ❢♦r♠❛ 9k 1 ♦✉ 9k 8✳ 3 3 3 b c

  ❈♦♥s✐❞❡r❛♥❞♦ t♦❞❛s ❛s ♣♦ss✐❜✐❧✐❞❛❡s ♣❛r❛ ❛ s♦♠❛ ❞❡ três ❝✉❜♦s✱ ❛ s♦♠❛ a s❡rá ❞❛ ❢♦r♠❛ 9k 1✱ 9k 3✱ 9k 6 ♦✉ 9k 8✿ 3 3 1 2 3 a b3 c 9k 1 9k 1 9k 1 9k

  3 9k 1 1 9k 2 1 9k 3 8 9k

  1 9k 1 1 1 9k 2 2 8 9k 3 3 8 9k

  8 3 9k 3 8 9k 3 8 9k 8 9k

  6 b c P♦rt❛♥t♦✱ ♦❜t❡♠♦s a ♥ã♦ ❞✐✈✐sí✈❡❧ ♣♦r ✾✳ ❖✉tr❛ s♦❧✉çã♦✿ ❖❜s❡r✈❛♠♦s ♣r✐♠❡✐r❛♠❡♥t❡ q✉❡✱ s❡ ✉♠ ♥ú♠❡r♦ n ♥ã♦ é ❞✐✈✐sí✈❡❧ 3

  ♣♦r ✸ ❡♥tã♦ ❡❧❡ é ❞❛ ❢♦r♠❛ 3k 1 ♦✉ 3k 2✱ ❧♦❣♦ n é ❞❛ ❢♦r♠❛ 9k 1 ♦✉ 9k 8✳ 3 3 P♦rt❛♥t♦✱ s❡ n ♥ã♦ é ❞✐✈✐sí✈❡❧ ♣♦r ✸ ❡♥tã♦ n ✑ 1 ♣mod 9q ♦✉ n ✑ 8 ♣mod 9q✳ ❙✉♣♦♥❤❛ q✉❡ ♥❡♥❤✉♠ ❞♦s ✐♥t❡✐r♦s a✱ b ❡ c s❡❥❛ ❞✐✈✐sí✈❡❧ ♣♦r ✸✳ ❙❡❣✉❡ q✉❡ ♦s ❝✉❜♦s ❞❡ss❡s ♥ú♠❡r♦s ❝♦♥❣r✉❡♥t❡s ❛ ✶ ♦✉ ❛ ✽ ♠ó❞✉❧♦ ✾✳ ❈♦♥s✐❞❡r❛♥❞♦ t♦❞❛s ❛s ♣♦ss✐❜✐❧✐❞❛❞❡s ♣❛r❛ ❛ s♦♠❛ ❞❡ três ❝✉❜♦s t❡r❡♠♦s✿ 3 3 3 a b c 3 3 3 ✑ 1 1 1 ✑ 3 ♣mod 9q a b c 3 3 3 ✑ 1 1 8 ✑ 1 ♣mod 9q a b c 3 3 3 ✑ 1 8 8 ✑ 8 ♣mod 9q a b c 3 3 3 ✑ 8 8 8 ✑ 6 ♣mod 9q b c

  ✺✳✷ ❆❧❣✉♥s ❆s♣❡❝t♦s ❞❛ ❉✐✈✐s✐❜✐❧✐❞❛❞❡ ♥❛ ➪❧❣❡❜r❛ ✶✸✼ ✸✵✳ ❯t✐❧✐③❛♥❞♦ ♦ ♣r♦❝❡ss♦ ❞❛s ❞✐✈✐sõ❡s s✉❝❡ss✐✈❛s✱ ♣❛r❛ ♦s ✐♥t❡✐r♦s ♣♦s✐t✐✈♦s a✱ b✱ ♦❜té♠✲ s❡✿ a

  ✏ b ☎ 1 r; ➔ r ➔ b b 1 ; 1 ✏ r ☎ 5 r ➔ r ➔ r r 1 2 ; 0 2 1

  ✏ r ☎ 3 r ➔ r ➔ r r 1 2 3 ; 0 3 2 ✏ r ☎ 3 r ➔ r ➔ r r 2 3 4 ; 0 4 3

  ✏ r ☎ 1 r ➔ r ➔ r r 3 r 4 4 ✏ ☎ 3 3 P♦rt❛♥t♦✱ r ✏ ♣a, bq ✏ 20 ❡ r ✏ 60✳ ❙✉❜st✐t✉✐♥❞♦ ❡ss❡s ✈❛❧♦r❡s ♥❛s ❡q✉❛çõ❡s ❛♥t❡r✐♦r❡s ❡♥❝♦♥tr❛✲s❡ a ✏ 6180 ❡ b ✏ 5200✳

  ✺✳✸ ❆❧❣✉♥s ❚ó♣✐❝♦s ❞❡ ●❡♦♠❡tr✐❛ ❊✉❝❧✐❞✐❛♥❛ ✶✸✽

  ✺✳✸ ❆❧❣✉♥s ❚ó♣✐❝♦s ❞❡ ●❡♦♠❡tr✐❛ ❊✉❝❧✐❞✐❛♥❛

  ✶✳ ❙❡❥❛ ABC ✉♠ tr✐â♥❣✉❧♦ ✐sós❝❡❧❡s ❞❡ ❜❛s❡ BC✱ ❡ AI✱ AM ❡ AH✱ ❛s s✉♣♦st❛s ❜✐ss❡tr✐③✱ ♠❡❞✐❛♥❛ ❡ ❛❧t✉r❛ r❡❧❛t✐✈❛ à ❜❛s❡ BC✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳

  A B H M I C ❱❛♠♦s ♠♦str❛r q✉❡ AI, AM ❡ AH sã♦ ❝♦✐♥❝✐❞❡♥t❡s✳ ◆♦t❡ q✉❡ ♦s tr✐â♥❣✉❧♦s ABM ❡ ACM sã♦ ❝♦♥❣r✉❡♥t❡s ♣❡❧♦ ❝❛s♦ LLL✱ ♣♦✐s AB ✏ AC, BM

  ✏ CM ❡ AM é ❝♦✐♥❝✐❞❡♥t❡ ❛♦s ❞♦✐s tr✐â♥❣✉❧♦s✳ BAI CAI

  ❖s tr✐â♥❣✉❧♦s ABI ❡ ACI sã♦ ❝♦♥❣r✉❡♥t❡s ♣❡❧♦ ❝❛s♦ ALA✱ ♣♦✐s ③ ✑ ③ ✱ AB ABI ACI

  ✏ AC ❡ ③ ✑ ③ ✱ ♦♥❞❡ BI ✏ CI✱ ❧♦❣♦ I ✑ M✳ O ❖s tr✐â♥❣✉❧♦s ABH ❡ ACH sã♦ ❝♦♥❣r✉❡♥t❡s ♣❡❧♦ ❝❛s♦ LAA ✱ ♣♦✐s AB ✏ AC✱

  ✆

  ④ ABH ACH AHB AHC ✑ ④ ❡ ④ ✏ ④ ✏ 90 ✱ ♦♥❞❡ BH ✏ CH✱ ❧♦❣♦ H ✑ I ✑ M✳

  P♦rt❛♥t♦✱ AI, AM ❡ AH sã♦ ❝♦✐♥❝✐❞❡♥t❡s✳ ✷✳ ❈♦♥s✐❞❡r❡ ♦ tr✐â♥❣✉❧♦ ABC✱ ♦♥❞❡ P ❝♦✐♥❝✐❞❡ ❝♦♠ H✳

  A B P C ✑ H

  BAP CAP AP B AP C

  ❈♦♠♦ ③ ✏ ③ ✱ AP é ❝♦♠✉♠ ❛♦s tr✐â♥❣✉❧♦s BAP ❡ CAP ❡ ③ ✏ ③ ✱ ❡♥tã♦ ♦s tr✐â♥❣✉❧♦s BAP ❡ CAP sã♦ ❝♦♥❣r✉❡♥t❡s ♣❡❧♦ ❝❛s♦ ALA✱ ❡♥tã♦ AB ✏ AC

  ✺✳✸ ❆❧❣✉♥s ❚ó♣✐❝♦s ❞❡ ●❡♦♠❡tr✐❛ ❊✉❝❧✐❞✐❛♥❛ ✶✸✾

  ❉❛ ♠❡s♠❛ ❢♦r♠❛✱ ♣♦❞❡✲s❡ ♦❜s❡r✈❛r q✉❡ ♦s tr✐â♥❣✉❧♦s AMB ❡ AMC sã♦ ❝♦♥❣r✉❡♥t❡s

  ✆

  AM B AM C ♣❡❧♦ ❝❛s♦ LAL✱ ❝♦♠ BM ✏ CM✱ ④ ✏ ④ ✏ 90 ❡ AM ✏ AH✳

  A B M C ✑ H

  ✸✳ ❈♦♥s✐❞❡r❡ ♦ tr✐â♥❣✉❧♦ r❡tâ♥❣✉❧♦ ABC✱ ♦♥❞❡ P H ✏ 2 cm é ❛ ❞✐stâ♥❝✐❛ ❞♦ ♣é ❞❛ ❜✐ss❡tr✐③ à ❤✐♣♦t❡♥✉s❛ BC✳

  A P B O ❖s tr✐â♥❣✉❧♦s ABP ❡ HBP sã♦ ❝♦♥❣r✉❡♥t❡s ♣❡❧♦ ❝❛s♦ LAA ✱ ♣♦✐s BP é ❧❛❞♦ ❝♦♠✉♠

  ✆

  ABP HBP BAP BHP ❛♦s ❞♦✐s tr✐â♥❣✉❧♦s✱ ③ ✏ ④ ❡ ③ ✏ ④ ✏ 90 ✱ ♦♥❞❡ AP ✏ HP ✏ 2 cm✳

  ✹✳ ❖❜s❡r✈❡ ♦ tr✐â♥❣✉❧♦ ❛❜❛✐①♦ A D E F B C

  ◆❡❧❡ ♦ q✉❛❞r✐❧át❡r♦ BCED é ✉♠ tr❛♣é③✐♦ ✐sós❝❡❧❡s✱ ♦♥❞❡ BC k DE ❡ BD ✏ CE✳

  ✺✳✸ ❆❧❣✉♥s ❚ó♣✐❝♦s ❞❡ ●❡♦♠❡tr✐❛ ❊✉❝❧✐❞✐❛♥❛ ✶✹✵

  CEB BCD ✐♠♣❧✐❝❛ ❞✐③❡r q✉❡ ④ ✏ ④ ✱ ♣♦rt❛♥t♦ ♦ tr✐â♥❣✉❧♦ BCF é ✐sós❝❡❧❡s ❞❡ ❜❛s❡ BC ❡ BF ✏ CF ✳

  DB ✺✳ ◆❛ ✜❣✉r❛ ❛❜❛✐①♦✱ ♣r♦❧♦♥❣✉❡ ❛ s❡♠✐rr❡t❛ ÝÝÑ ❛té t♦❝❛r ❡♠ C ♥❛ r❡t❛ s✳ ❖ â♥❣✉❧♦

  BCA ③ é ✐❣✉❛❧ ❛ α✱ ♣♦✐s sã♦ ❛❧t❡r♥♦s ✐♥t❡r♥♦s✳

  C A s

  β γ

  B α r

  D ◆♦ tr✐â♥❣✉❧♦ ABC✱ ♣❡❧♦ t❡♦r❡♠❛ ❞♦ â♥❣✉❧♦ ❡①t❡r♥♦✱ t❡♠♦s q✉❡

  ③ BCA β ✏ γ ñ α β ✏ γ.

  ✻✳ ❖s tr✐â♥❣✉❧♦s ABM ❡ ACM sã♦ ✐sós❝❡❧❡s ❞❡ ❜❛s❡s AB ❡ AC✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳ A B M C ABM BAM CAM ACM ABM

  ❈♦♠ ✐ss♦✱ ④ ✏ ④ ❡ ④ ✏ ④ ✳ ◆♦t❡ q✉❡ ❛ s♦♠❛ ❞♦s â♥❣✉❧♦s ④ ❡ ④

  ACM BAC é ✐❣✉❛❧ ❛♦ ③ ✱ ✐ss♦ só ❛❝♦♥t❡❝❡ ♥♦ tr✐â♥❣✉❧♦ r❡tâ♥❣✉❧♦✳

  ✆ ✆ ✆

  ④ ④ ③ ABM ACM BAC ABM 2 ACM BAC .

  ✏ 180 ñ 2 ☎ ④ ☎ ④ ✏ 180 ñ ③ ✏ 90 ✼✳ ❈♦♥s✐❞❡r❡ ♦ tr✐â♥❣✉❧♦ ABC ❡ I ♦ ♣♦♥t♦ ❞❡ ✐t❡rs❡❝çã♦ ❞❛s ❜✐ss❡tr✐③❡s ✐♥t❡r♥❛s ❞♦s

  ✈ért✐❝❡s B ❡ C✳

  ✺✳✸ ❆❧❣✉♥s ❚ó♣✐❝♦s ❞❡ ●❡♦♠❡tr✐❛ ❊✉❝❧✐❞✐❛♥❛ ✶✹✶

  ✆

  IBI a ④ BI a C

  ③

  I a CI ✏ 360

  ✆

  ñ 90

  ✆

  1

  2 ③

  BAC

  90

  ♣

  ③ CIB

  I a

  90

  ✆

  ✏ 360

  ✆

  ñ ♣

  I a ✏ 90

  ✆

  ✁

  1

  ③

  I I a ❚❡♠♦s q✉❡

  A B C

  1

  I ❏á ✈✐♠♦s ♥♦ ♣r♦❜❧❡♠❛ 7 ❞❡ss❛ s❡çã♦ q✉❡

  ③ BIC

  ✏ ③ BAC

  ③ ABI

  ③ ACI

  ❧♦♦♦♦♦♦♠♦♦♦♦♦♦♥ ñ ③ BIC

  ✏ ③ BAC

  1

  2 ♣

  B

  2 ♣

  ✽✳ ◆❛ ✜❣✉r❛ ❛❜❛✐①♦✱ ♦❜s❡r✈❡ ♦ q✉❛❞r✐❧át❡r♦ IBI a C ✳ A B C

  C ❧♦♦♦♦♠♦♦♦♦♥

  ñ ③ BIC

  ✏ ③ BAC 180

  ✆

  ✁ ③ BIC

  ñ ③ BIC

  ✏ 90

  ✆

  1

  2 ③ BAC.

  2 ③ BAC.

  ✺✳✸ ❆❧❣✉♥s ❚ó♣✐❝♦s ❞❡ ●❡♦♠❡tr✐❛ ❊✉❝❧✐❞✐❛♥❛ ✶✹✷

  ✾✳ ❈♦♥str♦✐ ❛ ✜❣✉r❛ ❡ ♠♦♥t❛ ✉♠ s✐st❡♠❛✱ ✉s❛♥❞♦ ♦ t❡♦r❡♠❛ ❞♦ â♥❣✉❧♦ ❡①t❡r♥♦✳ A B C D

  ✩

  1 ✬

  ♣ ♣ A B ADC

  ✬ ✏ ④

  ✬ ✫

  2 ADC ADB B C ñ ④ ✁ ④ ✏ ♣ ✁ ♣

  ✬ ✬

  1 ✬ ✪

  A ♣ C ♣ ADB ✏ ④

  2 ✶✵✳ ❯s❡ ♣♦r ❞✉❛s ✈❡③❡s ♦ t❡♦r❡♠❛ ❞♦ â♥❣✉❧♦ ❡①t❡r♥♦ ♥❛ ✜❣✉r❛✳

  A D G C B E ♣ ♣

  F A B G D E ❚❡♠♦s q✉❡ ♣ ✏ ♣ ❡ ♣ ✏ ♣ ✱ ♣♦rt❛♥t♦

  ✆ ✆

  ♣ ♣ ♣ ♣ ♣ ♣ ♣ F G C A B C D E .

  ✏ 180 ñ ♣ ✏ 180

  ✶ ✶

  ✶✶✳ ❯♠❛ s♦❧✉çã♦✿ ❚r❛❝❡ ❛s ♣❛r❛❧❡❧❛s DB ❡ DC ❡♠ r❡❧❛çã♦ à AB ❡ AC✱ r❡s♣❡❝t✐✲

  ✶ ✶ ✶

  BAC B DC ABD BDB ✈❛♠❡♥t❡✳ ❚❡♠♦s q✉❡ ③ ✏ ④ ✱ ④ ✏ ④ ✭❖♣♦st♦s P❡❧♦ ❱ért✐❝❡✮ ❡

  ✶

  ④ ④ ④

  ACD CDC BAC ABD ACD ✏ ④ ✱ ♦ q✉❡ ❝♦♥❝❧✉✐ q✉❡ α ✏ ③ ✳

  ❖✉tr❛ s♦❧✉çã♦✿ Pr♦❧♦♥❣✉❡ BD✱ ✐♥t❡rs❡❝t❛♥❞♦ AC ♥✉♠ ♣♦♥t♦ P ✱ ♦♥❞❡ ④ ④ DP C BAC ABD.

  ✏ ③

  ✺✳✸ ❆❧❣✉♥s ❚ó♣✐❝♦s ❞❡ ●❡♦♠❡tr✐❛ ❊✉❝❧✐❞✐❛♥❛ ✶✹✸

  ✁ 45

  ✆

  ✁ 65

  ✆

  ✁ 60

  ✆

  ✏ 55

  ✆ .

  P♦rt❛♥t♦✱ ④ AGD

  ✏ 180

  ✆

  ✆

  ❡ ④ GDA

  ✁ 55

  ✆

  ✏ 80

  ✆

  ✳ ▲♦❣♦✱ ♥♦ tr✐â♥❣✉❧♦ CGH t❡♠♦s x

  80

  ✆

  60

  ✆

  ✏ 180

  ✆

  ✏ 180

  ✆

  ❖ â♥❣✉❧♦ α é â♥❣✉❧♦ ❡①t❡r♥♦ ❞♦ tr✐â♥❣✉❧♦ CDP ✱ ♣♦rt❛♥t♦ α

  ❧♦♦♦♦♠♦♦♦♦♥ ñ P 2 B

  ✏ ④ DP C ④ ACD

  ñ α ✏ ③ BAC ④ ABD ④ ACD. ✶✷✳ P❡❧♦ ❝❛s♦ ❞❡ ❝♦♥❣r✉ê♥❝✐❛ ALA✱ ♦ tr✐â♥❣✉❧♦ ABC é ❝♦♥❣r✉❡♥t❡ ❛♦ tr✐â♥❣✉❧♦ DEC✳ ✶✸✳ ◆♦t❡ q✉❡

  α ✏ β, γ ✏ δ ñ ∆ABC ✏ ∆CDA,

  ♣❡❧♦ ❝❛s♦ ❞❡ ❝♦♥❣r✉ê♥❝✐❛ ALA✱ ♣♦✐s AC é ❝♦♠✉♠ ❛♦s ❞♦✐s tr✐â♥❣✉❧♦s✳ ✶✹✳ ❈♦♥s✐❞❡r❡ ♦ s❡❣♠❡♥t♦ AB ❡ ♦s ♣♦♥t♦s ✐♥t❡r♥♦s P 1 ❡ P 2

  A B P 1 P 2 ❚❡♠♦s q✉❡✿

  AP 1 P 1 B

  AP 2 P 2 B ñ

  AB ✁ P 1 B P 1 B

  ✏ AB

  ✁ P 2 B P 2 B ñ AB ☎ P 2 B ✁ P 1 B ☎ P 2 B

  ❧♦♦♦♦♠♦♦♦♦♥ ✏ AB ☎ P 1 B ✁ P 1 B ☎ P 2 B

  AB ☎ P 1 B AB

  ✏ 45

  ñ P 2 B ✏ P 1 B ñ P 1 ✑ P 2 .

  ✶✺✳ ❈♦♥str✉✐r ❛ ✜❣✉r❛✳ ✶✻✳ ❈♦♠♦ ABC ❡ DEF sã♦ tr✐â♥❣✉❧♦s ❡q✉✐❧át❡r♦s✱ ❝❛❞❛ ✉♠ ❞♦s s❡✉s â♥❣✉❧♦s ✐♥t❡r♥♦s

  ♠❡❞❡ 60

  ✆

  ✳ ◆♦ tr✐â♥❣✉❧♦ AGD✱ t❡♠♦s ④ GAD

  ✏ 180

  ✆

  ✁ 75

  ✆

  ✁ 60

  ✆

  ,

  ✺✳✸ ❆❧❣✉♥s ❚ó♣✐❝♦s ❞❡ ●❡♦♠❡tr✐❛ ❊✉❝❧✐❞✐❛♥❛ ✶✹✹

  ❞♦♥❞❡

  ✆ x .

  ✏ 40 F E B H

  ✆

  C x G

  ✆ ✆

  75

  65 A D ✶✼✳ ❙❡❥❛ O ♦ ♣♦♥t♦ ❞❡ ✐♥t❡rs❡çã♦ ❞❛s ❞✉❛s ❞✐❛❣♦♥❛✐s ❞♦ r❡tâ♥❣✉❧♦✳ ❊♥tã♦ AO ✏ BO✱

  OAB OBA ♣♦rt❛♥t♦ ♦ tr✐â♥❣✉❧♦ AOB é ✐sós❝❡❧❡s ❡ ❧♦❣♦ ③ ✏ ③ ✏ y✳

  B A y

  ✆

  180 ✁ x x

  O

  ✆

  ❈♦♠♦ ❛ s♦♠❛ ❞♦s â♥❣✉❧♦s ✐♥t❡r♥♦s ❞❡ ✉♠ tr✐â♥❣✉❧♦ é 180 ✱ ♥♦ tr✐â♥❣✉❧♦ AOB t❡♠♦s✿

  ✆ ✆

  2y ♣180 ✁ yq ✏ 180 ñ x ✏ 2y. ✶✽✳ ❙❡❥❛ E ♦ ♣♦♥t♦ s♦❜r❡ ❛ r❡t❛ BD t❛❧ q✉❡ ♦ tr✐â♥❣✉❧♦ AEB s❡❥❛ r❡tâ♥❣✉❧♦ ♥♦ ✈ért✐❝❡

  E ✳

  C

  3

  5

  3 A

  3

  ✆

  30 E

  4 D B ◆♦ tr✐â♥❣✉❧♦ r❡tâ♥❣✉❧♦ ∆AEB t❡♠♦s✿

  ❄ ❄ EB

  3 EB

  3

  3

  ✆

  cos 30 ✏ ñ ✏ ñ EB ✏

  ✺✳✸ ❆❧❣✉♥s ❚ó♣✐❝♦s ❞❡ ●❡♦♠❡tr✐❛ ❊✉❝❧✐❞✐❛♥❛ ✶✹✺

  AE

  1 AE

  3

  ✆ sen 30 .

  ✏ ñ ✏ ñ AE ✏ AB

  2

  3

  2 ❆❣♦r❛✱ ❛♣❧✐❝❛♥❞♦ ♦ ❚❡♦r❡♠❛ ❞❡ P✐tá❣♦r❛s ♥♦ tr✐â♥❣✉❧♦ AED ♦❜t❡♠♦s 2 ❄ 2

  ✂ ✡ ✂ ✡ ❄ 2 2 2 2

  3

  3

  3 2 AD ED

  4 3. ✏ AE ñ AD ✏ ñ AD ✏ 25 12

  2

  2 ❛

  ❄

  25 12 3 cm ❉❛í✱ ❝♦♥❝❧✉í♠♦s q✉❡ AD ✏ ✳

  ✶✾✳ ❊♠ ✉♠ tr✐â♥❣✉❧♦ ✐sós❝❡❧❡s✱ ❛ ❛❧t✉r❛ r❡❧❛t✐✈❛ à ❜❛s❡ ❝♦✐♥❝✐❞❡ ❝♦♠ ❛ ♠❡❞✐❛♥❛✳ ❚r❛✲ ç❛♥❞♦ ❡st❛ ❛❧t✉r❛✱ ♦❜t❡♠♦s ❞♦✐s tr✐â♥❣✉❧♦s r❡tâ♥❣✉❧♦s ❝♦♠ ❝❛t❡t♦s ♠❡❞✐♥❞♦ h ❡ 5 ❡ ❤✐♣♦t❡♥✉s❛ 13✳ P❡❧♦ ❚❡♦r❡♠❛ ❞❡ P✐tá❣♦r❛s✱ t❡♠♦s✿ 2 2 2 2 2 2 ❄ h

  5 144 ✏ 13 ñ h ✏ 13 ✁ 5 ✏ 144 ñ h ✏ ✏ 12. b

  10 2 ☎ h ☎ 12

  ▲♦❣♦✱ ❛ ár❡❛ ❞♦ tr✐â♥❣✉❧♦ é A ✏ ✏ ✏ 60 cm ✳

  2

  2

  13

  5

  24

  13 ✷✵✳ ▲❡♠❜r❡ q✉❡✱ ♥✉♠ tr✐â♥❣✉❧♦✱ q✉❛❧q✉❡r ❧❛❞♦ é ♠❛✐♦r q✉❡ ❛ ❞✐❢❡r❡♥ç❛ ❡ ♠❡♥♦r q✉❡ ❛ s♦♠❛ ❞♦s ♦✉tr♦s ❞♦✐s✳ ❉♦ tr✐â♥❣✉❧♦ ADB t❡♠♦s

  AD ✁ AB ➔ BD ➔ AD AB,

  ✺✳✸ ❆❧❣✉♥s ❚ó♣✐❝♦s ❞❡ ●❡♦♠❡tr✐❛ ❊✉❝❧✐❞✐❛♥❛ ✶✹✻

  ❡ ❞♦ tr✐â♥❣✉❧♦ CBD s❡❣✉❡ q✉❡ BC ✁ CD ➔ BD ➔ BC CD. ❙✉❜st✐t✉✐♥❞♦ ♦s ✈❛❧♦r❡s ❝♦♥❤❡❝✐❞♦s ♦❜t❡♠♦s✿

  9 ✁ 5 ➔ BD ➔ 5 9 e 17 ✁ 5 ➔ BD ➔ 17 5,

  ♦✉ s❡❥❛✱

  4 ➔ BD ➔ 14 e 12 ➔ BD ➔ 22. ❉❛s ❞✉❛s ❞❡s✐❣✉❛❧❞❛❞❡s ❝♦♥❝❧✉í♠♦s q✉❡

  12 ➔ BD ➔ 14. ❈♦♠♦ BD é ✐♥t❡✐r♦✱ só ♣♦❞❡♠♦s t❡r BD ✏ 13✳

  ✷✶✳ ❖ tr✐â♥❣✉❧♦ ❞❡ ❘❡✉❧❡❛✉① é ❢♦r♠❛❞♦ ♣♦r 4 r❡❣✐õ❡s✿ ✉♠ tr✐â♥❣✉❧♦ ❡q✉✐❧át❡r♦ ❡ três ❝❛❧♦t❛s✳ ❝❛❞❛ ❝❛❧♦t❛ é ✉♠ s❡①t♦ ❞❡ ✉♠ ❝ír❝✉❧♦ ❞❡ r❛✐♦ 1 ❞♦ q✉❛❧ ❢♦✐ r❡t✐r❛❞♦ ✉♠ tr✐â♥❣✉❧♦ ❡q✉✐❧át❡r♦ ❞❡ ❧❛❞♦ 1✳ P❡❧♦ ❚❡♦r❡♠❛ ❞❡ P✐tá❣♦r❛s✱ ❛ ❛❧t✉r❛ ❞♦ tr✐â♥❣✉❧♦ ❡q✉✐❧át❡r♦ é✿

  ❞ 2 ❄ ✂ ✡

  1

  3 h 1 ,

  ✏ ✁ ✏

  2

  2 ❧♦❣♦ ❛ ár❡❛ ❞♦ tr✐â♥❣✉❧♦ ✈❛❧❡✿

  ❄

  3 ❄

  1 ☎

  3 2

  2 cm .

  ✏

  2

  4 π

  ❆ ár❡❛ ❞❡ ✉♠ s❡t♦r ❝✐r❝✉❧❛r é ✉♠ s❡①t♦ ❞❛ ár❡❛ ❞♦ ❝ír❝✉❧♦✱ ♦✉ s❡❥❛✱ ✐❣✉❛❧ ❛ ✳ ▲♦❣♦✱

  6 ❛ ár❡❛ ❞❛ ❝❛❧♦t❛ é ❛ ❞✐❢❡r❡♥ç❛✿

  ❄ π

  3 . ✁

  6

  4 P♦rt❛♥t♦✱ ❛ ár❡❛ ❞♦ tr✐â♥❣✉❧♦ ❞❡ ❘❡✉❧❡❛✉① é ❄ ❄ ❄

  ✂ ✡ π

  3 3 π

  3 2 3 cm .

  ☎ ✁ ✏ ✁

  6

  4

  4

  2

  2

  ✺✳✸ ❆❧❣✉♥s ❚ó♣✐❝♦s ❞❡ ●❡♦♠❡tr✐❛ ❊✉❝❧✐❞✐❛♥❛ ✶✹✼

  ✷✷✳ ❖s s❡❣♠❡♥t♦s AD ❡ BC sã♦ ♣❛r❛❧❡❧♦s✳ ❊♥tã♦✱ ✈❛❧❡♠ ❛s s❡❣✉✐♥t❡s ✐❣✉❛❧❞❛❞❡s ❡♥tr❡ OAM OCB OBC OM A

  ♦s â♥❣✉❧♦s✿ ④ ✏ ④ ❡ ④ ✏ ④ B C O M A D

  ▲♦❣♦✱ ♣❡❧♦ ❝❛s♦ ➶♥❣✉❧♦✲➶♥❣✉❧♦✱ ♦s tr✐â♥❣✉❧♦s AOM ❡ COB sã♦ s❡♠❡❧❤❛♥t❡s✳ ❈♦♠♦ ♦ ♣♦♥t♦ M é ♦ ♣♦♥t♦ ♠é❞✐♦ ❞♦ s❡❣♠❡♥t♦ AD✱ t❡♠♦s q✉❡

  BC ✏ AD ✏ 2 ☎ AM. P♦rt❛♥t♦✱ ❛ ♣r♦♣♦rçã♦ ❡♥tr❡ ❛s ♠❡❞✐❞❛s ❞♦s tr✐â♥❣✉❧♦s AOM ❡ COB é ❞❡ 1 ♣❛r❛ 2✱ ✐st♦ é✱ ♦ ❝♦♠♣r✐♠❡♥t♦ ❞❡ ❝❛❞❛ ❧❛❞♦✱ ❡ t❛♠❜é♠ ❞❛ ❛❧t✉r❛✱ ❞♦ tr✐â♥❣✉❧♦ COB ♠❡❞❡ ♦ ❞♦❜r♦ ❞♦ ❝♦♠♣r✐♠❡♥t♦ ❞♦ s❡✉ ❧❛❞♦ ❝♦rr❡s♣♦♥❞❡♥t❡ ♥♦ tr✐â♥❣✉❧♦ AOM✳ ■ss♦ ♠♦str❛ q✉❡

  BO ✏ 2 ☎ OM. ❈♦♠♦ ❛ ❛❧t✉r❛ ❞♦ tr✐â♥❣✉❧♦ AOM é ♦ ❞♦❜r♦ ❞❛ ❛❧t✉r❛ ❞♦ tr✐â♥❣✉❧♦ COB✱ ❡♥tã♦ ❛ ♣r♦♣♦rçã♦ ❡♥tr❡s s✉❛s ár❡❛s é ❞❡ 1 ♣❛r❛ 4✱ ♦✉ s❡❥❛

  S AOM

  1 . ✏

  S COB

  4 ◆♦t❡ q✉❡ ♦ s❡❣♠❡♥t♦ AH ❢✉♥❝✐♦♥❛ ❝♦♠♦ ❛❧t✉r❛ ❝♦♠✉♠ ♣❛r❛ ♦s tr✐â♥❣✉❧♦s BAO ❡ OAM

  ❝♦♠♦ ♠♦str❛ ❛ ✜❣✉r❛ ❛❜❛✐①♦✿ B C O H M A D

  ✺✳✸ ❆❧❣✉♥s ❚ó♣✐❝♦s ❞❡ ●❡♦♠❡tr✐❛ ❊✉❝❧✐❞✐❛♥❛ ✶✹✽

  ❈♦♥❝❧✉í♠♦s ❛ss✐♠ q✉❡ BO OM

  ☎ AH ☎ AH S BAO OAM .

  ✏ ✏ 2 ☎ ✏ 2 ☎ S

  2

  2 ❉❡ ♠❛♥❡✐r❛ ❛♥á❧♦❣❛✱ ♦s tr✐â♥❣✉❧♦s BAO ❡ OBC ❝♦♠♣❛rt✐❧❤❛♠ ❛ ♠❡s♠❛ ❛❧t✉r❛✳ ❈♦♠♦ OC ✏ 2 ☎ AO✱ ❡♥tã♦ S OBC AOB .

  ✏ 2 ☎ S ❈❤❛♠❡♠♦s ❞❡ a ❛ ♠❡❞✐❞❛ ❞❛ ár❡❛ ❞♦ tr✐â♥❣✉❧♦ AOM✳ ❆❝✐♠❛✱ ❛❝❛❜❛♠♦s ❞❡ ♠♦str❛r q✉❡ ❛ ár❡❛ ❞♦ tr✐â♥❣✉❧♦ AOB é ✐❣✉❛❧ ❛ 2a✱ ❡ q✉❡ ❛ ár❡❛ ❞♦ tr✐â♥❣✉❧♦ OBC é ✐❣✉❛❧ ❛ 4a✱ ❝♦♠♦ ♠♦str❛❞♦ ♥❛ ✜❣✉r❛ ❛ s❡❣✉✐r✿

  B C 4a

  O 2a a

  M A D ❆❣♦r❛✱ ♦❜s❡r✈❡ q✉❡ ♦s tr✐â♥❣✉❧♦s ABC ❡ ADC sã♦ ❝♦♥❣r✉❡♥t❡s✱ ♣❡❧♦ ❝❛s♦ ▲▲▲✳ ▲♦❣♦ ❡ss❡s ❞♦✐s tr✐â♥❣✉❧♦s tê♠ ❛ ♠❡s♠❛ ár❡❛✳ ❈♦♠♦ ❛ ár❡❛ ❞♦ tr✐â♥❣✉❧♦ ABC é ✐❣✉❛❧ ❛ 6a✱ ♣❛r❛ q✉❡ ♦ tr✐â♥❣✉❧♦ ACD t❡♥❤❛ ❛ ♠❡s♠❛ ár❡❛ é ♥❡❝❡ssár✐♦ q✉❡ ❛ ár❡❛ ❞♦ 2 q✉❛❞r✐❧át❡r♦ MOCD s❡❥❛ ✐❣✉❛❧ ❛ 5a✳ P❡❧♦ ❞❛❞♦ ❞♦ ♣r♦❜❧❡♠❛ t❡♠♦s q✉❡ 5a ✏ 5 cm ✳ 2

  ❉❛í ❞❡❝♦rr❡ q✉❡ a ✏ 1 cm ✳ ❙❡❣✉❡ q✉❡ ❛ ár❡❛ ❞♦ tr✐â♥❣✉❧♦ AOM é ✐❣✉❛❧ ❛ ✶✳ ✷✸✳ ✭❛✮ ❙❡❥❛♠ X ❡ Y ♦s ♣♦♥t♦s ♥♦s q✉❛✐s ❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ❞❡ ❝❡♥tr♦ O t❛♥❣❡♥❝✐❛ ♦s s❡❣✲

  ♠❡♥t♦s BC ❡ BE✱ ❝♦♠♦ ✐❧✉str❛❞♦ ♥❛ ✜❣✉r❛ ❛ s❡❣✉✐r✳ ❖❜s❡r✈❡♠♦s ♣r✐♠❡✐r♦ q✉❡ s❡ ❞❡s❡♥❤❛♠♦s ♦s s❡❣♠❡♥t♦s q✉❡ ♣❛rt❡♠ ❞♦ ❝❡♥tr♦ O ❞❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ❡ t❡r♠✐♥❛♠ ❡♠

  X ❡ Y ✱ ♦❜t❡♠♦s ❝♦♠♦ r❡s✉❧t❛❞♦ ✉♠ q✉❛❞r❛❞♦ BY OX✳

  ✺✳✸ ❆❧❣✉♥s ❚ó♣✐❝♦s ❞❡ ●❡♦♠❡tr✐❛ ❊✉❝❧✐❞✐❛♥❛ ✶✹✾

  C D O

  X A B Y E ❉❡ ❢❛t♦✱ ❝♦♠♦ X ❡ Y sã♦ ♣♦♥t♦s ❞❡ t❛♥❣ê♥❝✐❛✱ OX ❞❡✈❡ s❡r ♦rt♦❣♦♥❛❧ ❛ CB ❡ OY ❞❡✈❡ s❡r ♦rt♦❣♦♥❛❧ ❛ BE✳ ■ss♦ ♠♦str❛ q✉❡ BY OX é ✉♠ r❡tâ♥❣✉❧♦✳ ▼❛s ♦s ❧❛❞♦s OX

  ❡ OY sã♦ r❛✐♦s ♣❛r❛ ❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ❡✱ ♣♦rt❛♥t♦✱ tê♠ ♦ ♠❡s♠♦ ❝♦♠♣r✐♠❡♥t♦✳ ❈♦♥❝❧✉í♠♦s ❛ss✐♠ q✉❡ BY OX é ❞❡ ❢❛t♦ ✉♠ q✉❛❞r❛❞♦✳ ▲♦❣♦✱ BO é ❛ ❞✐❛❣♦♥❛❧ ❞♦

  ✆

  OBE q✉❛❞r❛❞♦ BY OX✱ ❡ ❡♥tã♦ ④ ✏ 45 ✳ L 1 C D L 2 X O A B Y E

  ✆

  CAB 1 2 ❈♦♠♦ ③ ✏ 45 ✱ ❡♥tã♦ ❛s r❡t❛s L ❡ L sã♦ ♣❛r❛❧❡❧❛s✳

  ✭❜✮ ❖❜s❡r✈❡ q✉❡ ♦s tr✐â♥❣✉❧♦s ABC ❡ AOC ❝♦♠♣❛rt✐❧❤❛♠ ❛ ♠❡s♠❛ ❜❛s❡ AC✳ P♦r ♦✉tr♦ ❧❛❞♦✱ ❛ ❞✐stâ♥❝✐❛ ❞♦ ✈ért✐❝❡ B ❛♦ s❡❣♠❡♥t♦ AC ❡ ❛ ❞✐stâ♥❝✐❛ ❞♦ ✈ért✐❝❡ O ❛♦ 1 s❡❣♠❡♥t♦ AC ❝♦✐♥❝✐❞❡♠✱ ❛♠❜❛s✱ ❝♦♠ ❛ ❞✐stâ♥❝✐❛ ❡♥tr❡ ❛s ❞✉❛s ❧✐♥❤❛s ♣❛r❛❧❡❧❛s L 2

  ❡ L ✱ ❝♦♠♦ ♣♦❞❡♠♦s ✈❡r ♥❛ ✜❣✉r❛ ❛ s❡❣✉✐r✳ L 1 C D d L 2 d

  O A B E

  ✺✳✸ ❆❧❣✉♥s ❚ó♣✐❝♦s ❞❡ ●❡♦♠❡tr✐❛ ❊✉❝❧✐❞✐❛♥❛ ✶✺✵

  ❈❤❛♠❡♠♦s ❞❡ d t❛❧ ❞✐stâ♥❝✐❛✳ ❊♥tã♦✱ ♣❡❧❛ ❢ór♠✉❧❛ ❞❛ ár❡❛ ❞❡ ✉♠ tr✐â♥❣✉❧♦✱ ❝♦♥✲ ❝❧✉í♠♦s q✉❡

  AC ☎ d S . ABC AOC

  ✏ ✏ S 2

  2 ABC 2 ❙❡ ❛ ár❡❛ ❞♦ q✉❛❞r❛❞♦ ABCD é 36 cm ✱ ❡♥tã♦ S ✏ 18 cm ✳ P♦rt❛♥t♦✱ ❛ ár❡❛ 2

  ❞♦ tr✐â♥❣✉❧♦ AOC é ✐❣✉❛❧ ❛ 18 cm ✳ ✷✹✳ P♦❞❡♠♦s ❝♦♠♣❧❡t❛r ♦ ❞❡s❡♥❤♦ ❞❡ ♠♦❞♦ q✉❡✱ ♥❛ ✜❣✉r❛ s❡❣✉✐♥t❡✱ HAQE s❡❥❛ ✉♠ r❡tâ♥❣✉❧♦✳

  Q A P R H S E

  ✆ ✆

  HP S P QA ❙❡ ❝❤❛♠❛♠♦s ③ ✏ a ✱ ❡♥tã♦ ③ ✏ a ✳ ❖❜s❡r✈❡✱ q✉❡ ❞❡st❡ ♠♦❞♦✱ ♦s tr✐â♥❣✉❧♦s P HS

  ❡ QAP ♣♦ss✉❡♠ ♦s ♠❡s♠♦s â♥❣✉❧♦s ✐♥t❡r♥♦s✱ ❝♦♠♦ ✐❧✉str❛❞♦ ♥❛ ✜❣✉r❛ ❛ s❡❣✉✐r✳ Q A a

  P R a

  H S E ❈♦♠♦ SP QR é ✉♠ q✉❛❞r❛❞♦✱ ✈❡♠♦s q✉❡ ♦ ❝♦♠♣r✐♠❡♥t♦ ❞❛ ❤✐♦t❡♥✉s❛ ❞♦ tr✐â♥❣✉❧♦ P HS

  ❝♦✐♥❝✐❞❡ ❝♦♠ ♦ ❝♦♠♣r✐♠❡♥t♦ ❞❛ ❤✐♣♦t❡♥✉s❛ ❞♦ tr✐â♥❣✉❧♦ QAP ✳ ❈♦♥❝❧✉í♠♦s q✉❡ P HS ❡ QAP sã♦ tr✐â♥❣✉❧♦s ❝♦♥❣r✉❡♥t❡s✳ ▲♦❣♦✱ AQ ✏ P H ✏ 12. ❈♦♠♦ HAQE é ✉♠ r❡tâ♥❣✉❧♦✱ t❡♠♦s q✉❡ AQ ✏ HE✳ ▲♦❣♦✱ t❡♠♦s q✉❡

  HE ✏ 12. ✭✺✳✷✮

  ✺✳✸ ❆❧❣✉♥s ❚ó♣✐❝♦s ❞❡ ●❡♦♠❡tr✐❛ ❊✉❝❧✐❞✐❛♥❛ ✶✺✶

  P❡❧♦ ♠❡s♠♦ ♠♦t✐✈♦✱ AH ✏ QE ✏ 17 ❡ ❡♥tã♦ AP ✏ AH ✁ P H ✏ 17 ✁ 12 ✏ 5✱ ❝♦♠♦ ♠♦str❛ ❛ ✜❣✉r❛ ❛❜❛✐①♦✳

  12 A Q a

  5 P

  17 a R

  12 H S E ▲❡♠❜r❛♥❞♦ q✉❡ ♦s tr✐â♥❣✉❧♦s P HS ❡ QAP sã♦ ❝♦♥❣r✉❡♥t❡s✱ ❝♦♥❝❧✉í♠♦s q✉❡

  HS ✏ AP ✏ 5 ✭✺✳✸✮

  ❉❡ ✺✳✷ ❡ ✺✳✸ s❡❣✉❡✲s❡ q✉❡ SE ✏ 12 ✁ 5 ✏ 7✳ ✷✺✳ ❖❜s❡r✈❡♠♦s ♣r✐♠❡✐r♦ q✉❡ s❡ ♣r♦❧♦♥❣❛r♠♦s ♦s s❡❣♠❡♥t♦s AB ❡ CD✱ ❝♦♥s❡❣✉✐r❡♠♦s ♦ tr✐â♥❣✉❧♦ AP D✱ ❝♦♥❣r✉❡♥t❡ ❛♦ tr✐â♥❣✉❧♦ ADC✱ ❝♦♠♦ ♠♦str❛❞♦ ♥❛ ✜❣✉r❛ s❡❣✉✐♥t❡✳

  P B D O A M C ❊❧❡s sã♦ ❝♦♥❣r✉❡♥t❡s ♣♦rq✉❡ tê♠ ♦s ♠❡s♠♦s â♥❣✉❧♦s ✐♥t❡r♥♦s ❡ ❝♦♠♣❛rt✐❧❤❛♠ ♦ ♠❡s♠♦ ❝❛t❡t♦ AD✳ ▲♦❣♦✱ P D ✏ DC✳ ❆❣♦r❛✱ tr❛ç❛♠♦s ♦ s❡❣♠❡♥t♦ QD ♦rt♦❣♦♥❛❧ ❛♦ s❡❣♠❡♥t♦ P B✳ ❖s tr✐â♥❣✉❧♦s AQD ❡ AMD ♣♦ss✉❡♠ ♦s ♠❡s♠♦s â♥❣✉❧♦s ✐♥t❡r♥♦s ❡ ❝♦♠♣❛rt✐❧❤❛♠ ❛ ♠❡s♠❛ ❤✐♣♦t❡♥✉s❛ AD✳ ▲♦❣♦✱ ❡ss❡s tr✐â♥❣✉❧♦s sã♦ ❝♦♥❣r✉❡♥t❡s ❡ QD ✏ DM ✏ 10✳

  ✺✳✸ ❆❧❣✉♥s ❚ó♣✐❝♦s ❞❡ ●❡♦♠❡tr✐❛ ❊✉❝❧✐❞✐❛♥❛ ✶✺✷

  P Q B D O A M C P♦r ♦✉tr♦ ❧❛❞♦✱ ❝♦♠♦ QD ❡ BC sã♦ ♣❛r❛❧❡❧♦s✱ ❡♥tã♦ ♦s tr✐â♥❣✉❧♦s P QD ❡ P BC ♣♦ss✉❡♠ ♦s ♠❡s♠♦s â♥❣✉❧♦s ✐♥t❡r♥♦s✳ ❊❧❡s sã♦ s❡♠❡❧❤❛♥t❡s✳ ❈♦♠♦ P C ✏ 2 ☎ P D✱ ❡♥tã♦ ❛ ♣r♦♣♦rçã♦ ❞♦s tr✐â♥❣✉❧♦s é ❞❡ ✶ ♣❛r❛ ✷✳ ■ss♦ ✐♠♣❧✐❝❛ q✉❡

  BC ✏ 2 ☎ QD ✏ 2 ☎ 10 ✏ 20. ▲♦❣♦✱ BO ✏ BC ✁ OC ✏ 20 ✁ 12 ✏ 8✳

  ✷✻✳ ❈❤❛♠❡♠♦s ❞❡ P ♦ ♣♦♥t♦ ❞❡ ✐♥t❡rs❡çã♦ ❞♦s s❡❣♠❡♥t♦s AC ❡ DM✳ ❈♦♠♦ ❛ s♦♠❛ ❞♦s

  ✆

  â♥❣✉❧♦s ✐♥t❡r♥♦s ❞♦ tr✐â♥❣✉❧♦ AP M é 180 ✱ ❡♥tã♦

  ✆

  ④ M AP AM P .

  ✏ 90 ✁ ④

  ✆

  ④ M AP P AD P AD AM P

  ▼❛s ④ ✏ 90 ✳ ❊♥tã♦✱ ❝♦♥❝❧✉í♠♦s q✉❡ ④ ✏ ④ ✳ ❆❝❛❜❛♠♦s ❞❡ ♠♦str❛r ❛ss✐♠ q✉❡ ♦s tr✐â♥❣✉❧♦s r❡tâ♥❣✉❧♦s MAD ❡ ADC ♣♦ss✉❡♠ ♦s ♠❡s♠♦s â♥❣✉❧♦s ✐♥t❡r♥♦s✳ ❊❧❡s sã♦ ♣♦rt❛♥t♦ s❡♠❡❧❤❛♥t❡s✳

  A M B D C ❆♣❧✐❝❛♥❞♦ ❛ s❡♠❡❧❤❛♥ç❛ ❡♥tr❡ ❡ss❡s tr✐â♥❣✉❧♦s t❡♠♦s✿

  AD M A .

  ✏ ✭✺✳✹✮

  ✺✳✸ ❆❧❣✉♥s ❚ó♣✐❝♦s ❞❡ ●❡♦♠❡tr✐❛ ❊✉❝❧✐❞✐❛♥❛ ✶✺✸

  ❙❛❜❡♠♦s q✉❡ AD ✏ 4 ❡ q✉❡ DC ✏ 2 ☎ MA✳ ❯s❛♥❞♦ ❡ss❡s ❢❛t♦s ❡♠ ✺✳✹ ♦❜t❡♠♦s ❄ ❄

  2

  2 M A ✏ 2 ☎ ❡ ♣♦rt❛♥t♦ DC ✏ 4 ☎ ✳ ❋✐♥❛❧♠❡♥t❡✱ ❛ ár❡❛ ❞♦ r❡tâ♥❣✉❧♦ ABCD

  ❄

  2 r❡s✉❧t❛ AD ☎ DC ✏ 16 ☎ ✳ ✷✼✳ ❈❛❞❛ â♥❣✉❧♦ ✐♥t❡r♥♦ ❞♦ ♣❡♥tá❣♦♥♦ r❡❣✉❧❛r t❡♠ ♠❡❞✐❞❛

  360

  ✆ a . i

  ✏ 180 ✁ ✏ 180 ✁ 72 ✏ 108

  5

  ✆

  CDE ❈♦♠♦ DC ✏ DE✱ ♦ tr✐â♥❣✉❧♦ CDE é ✐sós❝❧❡s ❞❡ ✈ért✐❝❡ D✱ ❡✱ ❝♦♠♦ ④ ✏ 180 ✱ t❡♠♦s

  180 ✁ 108

  ✆

  ④ DCE DEC .

  ✏ ④ ✏ ✏ 36

  2 D

  ✆

  108

  ✆

  36 E C

  ✆

  P

  72 A B ❈♦♠♦ ♦s tr✐â♥❣✉❧♦s CDE ❡ DEA sã♦ ❝♦♥❣r✉❡♥t❡s ✭▲❆▲✮✱ t❡♠♦s t❛♠❜é♠ q✉❡

  ✆ ✆ ✆

  ④ DAE ADE EAP P EA

  ✏ ④ ✏ 36 ✳ ❈♦♠♦ ③ ✏ 36 ❡ ③ ✏ 108 ✁ 36 ✏ 72 ✱ t❡♠♦s q✉❡

  ✆

  ③ EP A

  ✏ 180 ✁ 36 ✁ 72 ✏ 72 ✱ ❧♦❣♦ ♦ tr✐â♥❣✉❧♦ EAP é ✐sós❝❡❧❡s ❞❡ ✈ért✐❝❡ A✳ ❈♦♠ ✐ss♦✱ AP ✏ EA ✏ a✳

  ✆

  ❖s tr✐â♥❣✉❧♦s DP E ❡ DEA ♣♦ss✉❡♠✱ ❝❛❞❛ ✉♠✱ ❞♦✐s â♥❣✉❧♦s ❞❡ ♠❡❞✐❞❛ 36 ✱ ❢❛③❡♥❞♦ ❝♦♠ q✉❡ s❡✉s t❡r❝❡✐r♦s â♥❣✉❧♦s t❡♥❤❛♠ t❛♠❜é♠ ❛ ♠❡s♠❛ ♠❡❞✐❞❛✳ ❆ss✐♠✱ ❡ss❡s tr✐â♥❣✉❧♦s sã♦ s❡♠❡❧❤❛♥t❡s✱ ❝♦♠

  P D DE .

  ✏ EA AD P D

  ❈♦♠♦ EA ✏ DE ✏ a ❡ AD ✏ AP ✏ a P D✱ t❡♠♦s P D a

  , ✏ a a P D

  ❧♦❣♦✱ 2 2 a , ☎ P D P D ✏ a

  ✺✳✸ ❆❧❣✉♥s ❚ó♣✐❝♦s ❞❡ ●❡♦♠❡tr✐❛ ❊✉❝❧✐❞✐❛♥❛ ✶✺✹

  ❡ ❡♥tã♦✱ 2 2 P D a ☎ P D ✁ a ✏ 0.

  ❘❡s♦❧✈❡♥❞♦ ❛ ❡q✉❛çã♦✱ t❡♠♦s ❄ 2

  5a ✁a ✟ ☎ P D .

  ✏

  2 ❚♦♠❛♥❞♦ ❛ s♦❧✉çã♦ ♣♦s✐t✐✈❛✱ t❡♠♦s

  ❄ a

  5 ♣ ✁ 1q P D .

  ✏ a b

  2 ✷✽✳ ◆♦ tr✐â♥❣✉❧♦ ABC✱ t❡♠♦s q✉❡ h é ❛ ❛❧t✉r❛ r❡❧❛t✐✈❛ ❛♦ ❧❛❞♦ a✱ h é ❛ ❛❧t✉r❛ r❡❧❛t✐✈❛ c

  ❛♦ ❧❛❞♦ b ❡ h é ❛ ❛❧t✉r❛ r❡❧❛t✐✈❛ ❛♦ ❧❛❞♦ c✳ ❙❡♥❞♦ ❛ss✐♠✱ ♣♦❞❡♠♦s ❡s❝r❡✈❡r três r❡❧❛çõ❡s q✉❡ ❢♦r♥❡❝❡♠ ❛ ♠❡s♠❛ ár❡❛✱ ♦✉ s❡❥❛✱ ❛ ár❡❛ ❞♦ tr✐â♥❣✉❧♦ ABC é a b c a b c

  ☎ h ☎ h ☎ h .

  ✏ ✏

  2

  2

  2 a b c a a ❉❡ss❛ ❢♦r♠❛✱ t❡♠♦s q✉❡ a☎h ✏ b☎h ✏ c☎h ✳ ❈♦♠♦ a☎h ✏ h 1 ✱ ♣♦❞❡♠♦s r❡❡s❝r❡✈❡r a ❛ ❡①♣r❡ssã♦ ❛♥t❡r✐♦r ♥❛ ❢♦r♠❛ a b c h a h h c 1 ✏ 1 ✏ b 1 ,

  1

  1

  1 ♦ q✉❡ ♠♦str❛ q✉❡ ♦s tr✐â♥❣✉❧♦s ❞❡ ❧❛❞♦s a✱ b✱ c ❡ ❧❛❞♦s h h h 1 ✱ a b c 1 ✱ 1 sã♦ s❡♠❡❧❤❛♥t❡s✳

  ✷✾✳ ❚r❛❝❡ ❛ ❞✐❛❣♦♥❛❧ ❞♦ q✉❛❞r❛❞♦ q✉❡ é ♣❡r♣❡♥❞✐❝✉❧❛r ❛♦ ❞✐â♠❡tr♦ ❞♦ s❡♠✐❝ír❝✉❧♦ ♣❛r❛ ❧♦❝❛❧✐③❛r ♦ ❝❡♥tr♦ ❞♦ s❡♠✐❝ír❝✉❧♦✳ ❚r❛❝❡ ❛s r❡t❛s q✉❡ ♣❛ss❛♠ ♣❡❧♦ ❝❡♥tr♦ ❞♦ s❡♠✐❝ír✲ ❝✉❧♦ ❡ ♣❡❧❛ ✐♥t❡rs❡❝çã♦ ❞♦ s❡♠✐❝ír❝✉❧♦ ❝♦♠ ♦s ❧❛❞♦s ❞♦ q✉❛❞r❛❞♦✳ ❈♦♠♦ ♦ ✈ért✐❝❡ ✐♥❢❡r✐♦r ❞♦ q✉❛❞r❛❞♦ é ✉♠ â♥❣✉❧♦ r❡t♦ ❡♥tã♦ ♦ ✈ért✐❝❡ ✐♥❢❡r✐♦r ♣❡rt❡♥❝❡ à ❝✐r❝✉♥❢❡✲ rê♥❝✐❛ q✉❡ ❝♦♥té♠ ♦ s❡♠✐❝ír❝✉❧♦✱ ♣♦rt❛♥t♦ ♦ s❡❣♠❡♥t♦ q✉❡ ✉♥❡ ♦ ❝❡♥tr♦ ❛ ❡st❡ ✈ért✐❝❡ é ✐❣✉❛❧ ❛♦ r❛✐♦ ❞♦ s❡♠✐❝ír❝✉❧♦✳ ❯s❛♥❞♦ ❛s ♣r♦♣r✐❡❞❛❞❡s q✉❡ ❛♣r❡♥❞❡♠♦s s♦❜r❡ ♦s tr✐â♥❣✉❧♦s ✐sós❝❡❧❡s ❛ r❡s♣❡✐t♦ ❞❡ ♠❡❞✐❛♥❛✱ ❜✐ss❡tr✐③ ❡ ❛❧t✉r❛✱ t❡♠♦s✱ ♣♦rt❛♥t♦✱ ❛s s❡❣✉✐♥t❡s ♠❡❞✐❞❛s ♥❛ ✜❣✉r❛ ❛❜❛✐①♦✳

  ❄

  2 ❖ q✉❛❞r❛❞♦ é ❝♦♥st✐t✉í❞♦ ♣♦r ❞♦✐s q✉❛❞r❛❞♦s✱ ✉♠ ❞❡ ❧❛❞♦ ✶ ❡ ♦✉tr♦ ❞❡ ❧❛❞♦ ❡

  2

  ✺✳✹ Pr✐♥❝í♣✐♦ ❋✉♥❞❛♠❡♥t❛❧ ❞❡ ❈♦♥t❛❣❡♠ ✶✺✺

  1 ❄

  2

  2 ❄

  2 ❞♦✐s r❡tâ♥❣✉❧♦s ❞❡ ❧❛❞♦s ✶ ❡ ✳ ▲♦❣♦ ❛ ár❡❛ S ❞♦ q✉❛❞r❛❞♦ é✿

  2 2

  ✂❄ ✡ ❄ 2

  2

  2

  3 S

  2 2. ✏ 1 ☎ 1 ☎ ✏

  2

  2

  2

  ✺✳✹ Pr✐♥❝í♣✐♦ ❋✉♥❞❛♠❡♥t❛❧ ❞❡ ❈♦♥t❛❣❡♠

  ✶✳ ✭❛✮ P❛r❛ ❡s❝♦❧❤❡r ♦ ♣♦rt❛✲✈♦③✱ t❡♠♦s 10 ♣♦ss✐❜✐❧✐❞❛❞❡s✱ ❥á q✉❡ sã♦ ❞❡③ ❛❧✉♥♦s✳ ❊s❝♦❧❤✐❞♦ ♦ ♣♦rt❛✲✈♦③✱ t❡♠♦s ❛❣♦r❛ 9 ♣♦ss✐❜✐❧✐❞❛❞❡s ♣❛r❛ ❡s❝♦❧❤❡r ♦ ❛❧✉♥♦ q✉❡ s❡rá ♦ ❞✐r❡t♦r ❞❡ ❛rt❡s✳ ❋✐♥❛❧♠❡♥t❡✱ ♣❛r❛ ❡s❝♦❧❤❡r ♦ ❛ss❡ss♦r té❝♥✐❝♦✱ r❡st❛♠ 8 ♣♦ss✐❜✐❧✐❞❛❞❡s✳ ▲♦❣♦✱ t❡♠♦s

  10 ☎ 9 ☎ 8 ✏ 720

  ♠❛♥❡✐r❛s ❞✐❢❡r❡♥t❡s ♣❛r❛ ❡s❝♦❧❤❡r ❛ ❝♦♠✐ssã♦ ♣❡❞✐❞❛✳ ✭❜✮ P❛r❛ ❛ ❡s❝♦❧❤❛ ❞♦ ♣♦rt❛✲✈♦③ t❡♠♦s três ♣♦ss✐❜✐❧✐❞❛❞❡s ❞❡♥tr❡ ▼❛r❝❡❧♦✱ ❘❡♥❛t♦

  ❡ ▲❡❛♥❞r♦✳ ❊s❝♦❧❤✐❞♦ ♦ ♣♦rt❛✲✈♦③✱ r❡st❛♠ ❞✉❛s ♣♦ss✐❜✐❧✐❞❛❞❡s ♣❛r❛ ❡s❝♦❧❤❡r ♦ ❞✐r❡t♦r ❞❡ ❛rt❡s✳ ❊s❝♦❧❤✐❞♦ ♦s ❞♦✐s ❝❛r❣♦s ❛♥t❡r✐♦r❡s✱ só r❡st❛ ✉♠❛ ♣♦ss✐❜✐❧✐❞❛❞❡ ♣❛r❛ ❡s❝♦❧❤❡r ♦ ú❧t✐♠♦ ❝❛r❣♦✳ ▲♦❣♦✱ t❡♠♦s

  3 ☎ 2 ☎ 1 ✏ 6

  ♠❛♥❡✐r❛s ❞✐❢❡r❡♥t❡s ♣❛r❛ ❡s❝♦❧❤❡r ✐♠❛ ❝♦♠✐ssã♦ q✉❡ t❡♥❤❛ ♦s ❛❧✉♥♦s ▼❛r❝❡❧♦✱ ▲❡❛♥❞r♦ ❡ ❘❡♥❛t♦✳ ❯♠❛ ♦✉tr❛ ♠❛♥❡✐r❛ ❞❡ ❢❛③❡r ❡ss❡ ✐t❡♠ é ❧✐st❛r t♦❞❛s ❛s ❝♦♠✐ssõ❡s ❝♦♠ t♦❞♦s ♦s ❛❧✉♥♦s✳

  ✭❝✮ ◆ã♦ ❤❛✈❡♥❞♦ ♠❛✐s ❝❛r❣♦s✱ ❛s ❝♦♠✐ssõ❡s ❧✐st❛❞❛s ♥♦ ✐t❡♠ ♣bq sã♦ t♦❞❛s ✐❣✉❛✐s✳ P❛r❛ ❝♦♥t❛r q✉❛♥t❛s sã♦ ❛s ❝♦♠✐ssõ❡s s❡♠ ❝❛r❣♦✱ ✈❛♠♦s ❛❣r✉♣❛r ❛s ❝♦♠✐ssõ❡s

  ✺✳✹ Pr✐♥❝í♣✐♦ ❋✉♥❞❛♠❡♥t❛❧ ❞❡ ❈♦♥t❛❣❡♠ ✶✺✻

  ❝♦♠ ❝❛r❣♦s ✭♣♦rt❛✲✈♦③✱ ❞✐r❡t♦r ❞❡ ❛rt❡s ❡ ❛ss❡ss♦r té❝♥✐❝♦✮ ❡♠ ❣r✉♣♦s ❞❡ s❡✐s q✉❡ t❡♥❤❛♠ ♦s ♠❡s♠♦s três ❛❧✉♥♦s✳ ❈♦♠♦ sã♦ 720 ❝♦♠✐ssõ❡s ❝♦♠ ❝❛r❣♦✱ ❡ sã♦ ❣r✉♣♦s ❞❡ 6 ❝♦♠ ❛s ♠❡s♠❛s ♣❡ss♦❛s✱ ♦❜t❡♠♦s

  720 ✏ 120

  6 ♠❛♥❡✐r❛s ❞✐❢❡r❡♥t❡s ❞❡ ❝♦♠♣♦r ✉♠❛ ❝♦♠✐ssã♦ s❡♠ ❝❛r❣♦s✳

  ✷✳ P♦❞❡♠♦s ✐♥s❝r❡✈❡r ♦ ❞♦❞❡❝á❣♦♥♦ r❡❣✉❧❛r ❡♠ ✉♠ ❝ír❝✉❧♦ ❝♦♠♦ ♠♦str❛ ❛ s❡❣✉✐♥t❡ ✜❣✉r❛✿ ❊♥tã♦ ♦s ❛r❝♦s q✉❡ ❝♦rr❡s♣♦♥❞❡♠ ❛ ❝❛❞❛ ✉♠ ❞♦s ❧❛❞♦s ❞♦ ❞♦❞❡❝á❣♦♥♦ ❞❡✈❡♠ ♠❡❞✐r ✂ ✡

  360

  ✆

  ✏ 30 ✳

  12 ✭❛✮ ❖s tr✐â♥❣✉❧♦s ❡q✉✐❧át❡r♦s q✉❡ ♣♦❞❡♠♦s ❢♦r♠❛r ❡st❛rã♦ t❛♠❜é♠ ✐♥s❝r✐t♦s ♥♦

  ❝ír❝✉❧♦✳ ❙❛❜❡♠♦s q✉❡ ❛ ♠❡❞✐❞❛ ❞♦ ❛r❝♦ q✉❡ ❝♦rr❡s♣♦♥❞❡ ❛ ✉♠ ❧❛❞♦ ❞❡ ✉♠ ✂ ✡

  360

  ✆

  tr✐â♥❣✉❧♦ ❡q✉✐❧át❡r♦ ✐♥s❝r✐t♦ ❡♠ ✉♠❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ❞❡✈❡ s❡r ✏ 120 ✳

  3

  ✆

  120

  ✆

  120

  ✆

  120 ❙❡ ✜①❛r♠♦s ✉♠ ✈ért✐❝❡✱ ❞✐❣❛♠♦s A ♥♦ ❣rá✜❝♦✱ ❡①✐st✐rá ✉♠ ú♥✐❝♦ tr✐â♥❣✉❧♦ ❡q✉✐❧át❡r♦ ✉s❛♥❞♦ ♦ ✈ért✐❝❡ A✳

  ✺✳✹ Pr✐♥❝í♣✐♦ ❋✉♥❞❛♠❡♥t❛❧ ❞❡ ❈♦♥t❛❣❡♠ ✶✺✼

  A P♦❞❡♠♦s ❝♦♥t❛r ❛ss✐♠ 12 tr✐â♥❣✉❧♦s ✭✉♠ ♣❛r❛ ❝❛❞❛ ✈ért✐❝❡✮✳ ◆❡st❡ ❝❛s♦✱ ❝❛❞❛

  12 tr✐â♥❣✉❧♦ s❡r✐❛ ❝♦♥t❛❞♦ três ✈❡③❡s✳ ❈♦♥❝❧✉í♠♦s q✉❡ ❡①✐st❡♠ ✏ 4 tr✐â♥❣✉❧♦s

  3 ❡q✉✐❧át❡r♦s ✐♥s❝r✐t♦s ♥♦ ❞♦❞❡❝á❣♦♥♦✳

  ✭❜✮ P❛r❛ ❝♦♥t❛r ♦ ♥ú♠❡r♦ ❞❡ tr✐â♥❣✉❧♦s ❡s❝❛❧❡♥♦s✱ ♣♦❞❡♠♦s ❝♦♥t❛r ♦ ♥ú♠❡r♦ t♦t❛❧ ❞❡ tr✐â♥❣✉❧♦s ❡ s✉❜tr❛✐r ♦ ♥ú♠❡r♦ ❞❡ tr✐â♥❣✉❧♦s ✐sós❝❡❧❡s ❡ ❡q✉✐❧át❡r♦s✳ Pr✐♠❡✐r♦✱ ❝♦♥t❛r❡♠♦s ♦s tr✐â♥❣✉❧♦s ✐sós❝❡❧❡s q✉❡ ♥ã♦ sã♦ ❡q✉✐❧át❡r♦s✳ ❋✐①❡♠♦s ✉♠ ✈ért✐❝❡✱ ❞✐❣❛♠♦s✱ A✳ ❊①✐st❡♠ q✉❛tr♦ tr✐â♥❣✉❧♦s ✐sós❝❡❧❡s ♥ã♦ ❡q✉✐❧át❡r♦s ✐♥s❝r✐t♦s ♥♦ ❞♦❞❡❝á❣♦♥♦ ❝✉❥♦ ✈ért✐❝❡ A ❝♦rr❡s♣♦♥❞❡ ❛♦ â♥❣✉❧♦ ❞❡s✐❣✉❛❧✳ P♦❞❡♠♦s ❝♦♥t❛r✱ ❛ss✐♠✱ q✉❛tr♦ tr✐â♥❣✉❧♦s ✐sós❝❡❧❡s ♣♦r ❝❛❞❛ ✈ért✐❝❡ ❞♦ ❞♦❞❡✲ ❝á❣♦♥♦✱ ✐st♦ é✱ 4 ☎ 12 ✏ 48 tr✐â♥❣✉❧♦s ✐sós❝❡❧❡s ♥ã♦ ❡q✉✐❧át❡r♦s ♥♦ t♦t❛❧✳ ❈♦♥t❡♠♦s ♦ ♥ú♠❡r♦ t♦t❛❧ ❞❡ tr✐â♥❣✉❧♦s✳ P❛r❛ ❞❡t❡r♠✐♥❛r ✉♠ tr✐â♥❣✉❧♦✱ ❞❡✲ ✈❡♠♦s ❡s❝♦❧❤❡r três ✈ért✐❝❡s ♥♦ ❝♦♥❥✉♥t♦ ❞❡ 12 ✈ért✐❝❡s ❞♦ ❞♦❞❡❝á❣♦♥♦✳ P❛r❛ ❡s❝♦❧❤❡r ♦ ♣r✐♠❡✐r♦ ✈ért✐❝❡ ❞♦ tr✐â♥❣✉❧♦✱ t❡♠♦s 12 ❛❧t❡r♥❛t✐✈❛s✱ ♣❛r❛ ♦ s❡❣✉♥❞♦✱

  11 ❡ ♣❛r❛ ♦ t❡r❝❡✐r♦✱ r❡st❛♠ 10✳ ▼❛s ♥♦ ♣r♦❞✉t♦ 12 ☎ 11 ☎ 10 ❡st❛♠♦s ❝♦♥t❛♥❞♦

  ✈ár✐❛s ✈❡③❡s ✉♠ ♠❡s♠♦ tr✐â♥❣✉❧♦✳ ❖❜s❡r✈❡ q✉❡ ❤á s❡✐s ❢♦r♠❛s ♣♦ssí✈❡✐s ♣❛r❛ ♦r❞❡♥❛r ♦s ✈ért✐❝❡s ❞❡ ✉♠ tr✐â♥❣✉❧♦✳ P♦rt❛♥t♦✱ ♥♦ ♣r♦❞✉t♦ 12 ☎ 11 ☎ 10✱ ❝❛❞❛ tr✐â♥❣✉❧♦ ❡stá s❡♥❞♦ ❝♦♥t❛❞♦ s❡✐s ✈❡✲

  12 ☎ 11 ☎ 10

  ③❡s✳ ❈♦♥❝❧✉í♠♦s q✉❡ ❤á✱ ♥♦ t♦t❛❧✱ ✏ 220 tr✐â♥❣✉❧♦s ✐♥s❝r✐t♦s ♥♦

  6 ❞♦❞❡❝á❣♦♥♦✳ ❉❛í ♦ ♥ú♠❡r♦ ❞❡ tr✐â♥❣✉❧♦s ❡s❝❛❧❡♥♦s s❡rá ✐❣✉❛❧ ❛

  220 ✁ 4 ☎ 12 ✁ 4 ✏ 168. ✸✳ ✭❛✮ ❈♦♠❡ç❛♠♦s ❡s❝r❡✈❡♥❞♦ ♦s ♣r✐♠❡✐r♦s ♥ú♠❡r♦s ❞❛ ❧✐st❛✿ 12345, 12354, 12435, 12453, 12534, 12543, 13245, 13254, 13425, 13452.

  ▲♦❣♦✱ ♦ ❞é❝✐♠♦ ♥ú♠❡r♦ é 13452✳

  ✺✳✹ Pr✐♥❝í♣✐♦ ❋✉♥❞❛♠❡♥t❛❧ ❞❡ ❈♦♥t❛❣❡♠ ✶✺✽

  ✉♠ ♥ú♠❡r♦ ❞❡ ❝✐♥❝♦ ❞í❣✐t♦s ❝♦♠❡ç❛ ❝♦♠ ♦ ❞í❣✐t♦ ✶✱ ❡st❡ ♥ú♠❡r♦ é ♠❡♥♦r ❞♦ q✉❡ ✉♠ q✉❡ ❝♦♠❡ç❛ ❝♦♠ ♦ ♥ú♠❡r♦ ✷✳ P♦r s✉❛ ✈❡③✱ ❡st❡ ♥ú♠❡r♦ é ♠❡♥♦r ❞♦ q✉❡ ✉♠ q✉❡ ❝♦♠❡ç❛ ❝♦♠ ✸✱ ❡ ❛ss✐♠ ♣♦r ❞✐❛♥t❡✳ ❊♥tã♦✱ ❝♦♥t❛r❡♠♦s q✉❛♥t♦s ❞❡st❡s ♥ú♠❡r♦s ❝♦♠❡ç❛♠ ❝♦♠ ❝❛❞❛ ✉♠ ❞❡ss❡s ❛❧❣❛r✐s♠♦s✳ ❙❡ ✜①❛r♠♦s ♦ ♣r✐♠❡✐r♦ ❞í❣✐t♦✱ ♣♦r ❡①❡♠♣❧♦ ✶✱ ✈❛♠♦s t❡r q✉❛tr♦ ❡s❝♦❧❤❛s ♣❛r❛ ♦ s❡❣✉♥❞♦ ❞í❣✐t♦✳ P❛r❛ ♦ t❡r❝❡✐r♦ ❞í❣✐t♦ ✈❛♠♦s t❡r só três ❡s❝♦❧❤❛s✳ ❚❡r❡♠♦s ❞✉❛s ❡s❝♦❧❤❛s ♣❛r❛ ♦ q✉❛rt♦ ❞í❣✐t♦ ❡ ✉♠❛ ú♥✐❝❛ ❡s❝♦❧❤❛ ♣❛r❛ ♦ ú❧t✐♠♦ ❞í❣✐t♦✳ ▲♦❣♦✱ t❡♠♦s 4 ☎ 3 ☎ 2 ☎ 1 ✏ 24 ♥ú♠❡r♦s ❞❡ ❝✐♥❝♦ ❞í❣✐t♦s✱ q✉❡ s❡rã♦ ♦s ♣r✐♠❡✐r♦s ♥ú♠❡r♦s ❞❛ ❧✐st❛✳ ❆♥❛❧♦❣❛♠❡♥t❡✱ t❡r❡♠♦s ✷✹ ♥ú♠❡r♦s q✉❡ ❝♦♠❡ç❛♠ ❝♦♠ ✷✱ q✉❡ s❡rã♦ ♦s ♣ró①✐♠♦s ✷✹ ♥ú♠❡r♦s ❞❛ ❧✐st❛✳ ❉❛ ♠❡s♠❛ ❢♦r♠❛✱ t❡r❡♠♦s ✷✹ ♥ú♠❡r♦s q✉❡ ❝♦♠❡ç❛♠ ❝♦♠ ♦ ❞í❣✐t♦ ✸✱ t♦t❛❧✐③❛♥❞♦ ✼✷ ♥ú♠❡r♦s ❞❛ ❧✐st❛✳ ❖ q✉❡ s✐❣♥✐✜❝❛ ❞✐③❡r q✉❡ ♦ ♣r✐♠❡✐r♦ ♥ú♠❡r♦ q✉❡ ❝♦♠❡ç❛ ❝♦♠ ✹✱ ❡stá ♥❛ ♣♦s✐çã♦ ✼✸ ❡ ♦ ú❧t✐♠♦ ♥❛ ♣♦s✐çã♦ ✾✻✳ ❈♦♥❝❧✉✐✲s❡ q✉❡ ♦ ♥ú♠❡r♦ ♥❛ ♣♦s✐çã♦ ✽✺ ❝♦♠❡ç❛ ❝♦♠ ✹✳ ❘❡♣❡t✐♠♦s ♦ ♣r♦❝❡ss♦ ♣❛r❛ ❞❡t❡r♠✐♥❛r ♦ s❡❣✉♥❞♦ ❞í❣✐t♦✳ ❚❡♠♦s 3 ☎ 2 ☎ 1 ✏ 6 ♥ú♠❡r♦s q✉❡ ❝♦♠❡ç❛♠ ❝♦♠ ✹✶✱ ❡ ❡st❡s sã♦ ❝❧❛r❛♠❡♥t❡ ♠❡♥♦r❡s q✉❡ ♦s ♥ú♠❡r♦s q✉❡ ❝♦♠❡ç❛♠ ❝♦♠ ✹✷ ✭q✉❡ t❛♠❜é♠ sã♦ ✻✮✳ ❈♦♠♦ 72 6 6 ✏ 84✱ t❡♠♦s q✉❡ ♦ ♥ú♠❡r♦ q✉❡ ♦❝✉♣❛ ❛ ♣♦s✐çã♦ ✽✺ é ♦ ♣r✐♠❡✐r♦ ♥ú♠❡r♦ q✉❡ ❝♦♠❡ç❛ ❝♦♠ ✹✸✱ ♦✉ s❡❥❛✱ é ♦ ♥ú♠❡r♦ 43125✳

  ✹✳ ✭❛✮ ❯♠ ❛♣❡rt♦ ❞❡ ♠ã♦ é ❞❛❞♦ ❡♥tr❡ ❞✉❛s ♣❡ss♦❛s✳ ▲♦❣♦✱ q✉❛♥❞♦ s♦♠❛♠♦s ♦s ❛♣❡rt♦s ❞❡ ♠ã♦ ❞❡ t♦❞❛s ❛s ♣❡ss♦❛s✱ ❝❛❞❛ ❛♣❡rt♦ ❞❡ ♠ã♦ é ❝♦♥t❛❞♦ ❞✉❛s ✈❡③❡s✳ ▲♦❣♦✱ ❛ s♦♠❛ ❞❡ q✉❛♥t❛s ✈❡③❡s ❝❛❞❛ ♣❡ss♦❛ ❛♣❡rt♦✉ ❛ ♠ã♦ ❞❡ ❛❧❣✉é♠ é ♣❛r✱ ♣♦✐s é ♦ ❞♦❜r♦ ❞❡ ❛❧❣✉♠ ♥ú♠❡r♦✳

  ✭❜✮ ❈♦♠♦ sã♦ ✾✾ ♣❡ss♦❛s✱ s❡ ❝❛❞❛ ✉♠❛ ❛♣❡rt❛ss❡ ❛ ♠ã♦ ❞❡ ❛❧❣✉é♠ três ✈❡③❡s✱ ✐ss♦ ❞❛r✐❛ ✉♠ t♦t❛❧ ❞❡ 3 ☎ 99 ✏ 297 ❛♣❡rt♦s ❞❡ ♠ã♦✳ P❡❧♦ ✐t❡♠ ❛♥t❡r✐♦r✱ ✈✐♠♦s q✉❡ ❡ss❡ t♦t❛❧ ❞❡ ❛♣❡rt♦s ❞❡✈❡ s❡r ♣❛r✳ ▲♦❣♦ ♥ã♦ é ♣♦ssí✈❡❧ q✉❡ ❝❛❞❛ ♣❡ss♦❛ t❡♥❤❛ ❞❛❞♦ três ❛♣❡rt♦s ❞❡ ♠ã♦✳

  ✺✳ P❛r❛ ❝❛❞❛ ✉♠ ❞♦s ♦✐t♦ ❜r✐♥q✉❡❞♦s r❡st❛♥t❡s✱ ❞♦ ♥ú♠❡r♦ ✸ ❛♦ ♥ú♠❡r♦ ✶✵✱ ❞❡✈❡r❡♠♦s ❞❡❝✐❞✐r s❡ ❡❧❡ ✐rá ♣❡rt❡♥❝❡r ❛ ❱✐❝t♦r✱ ♦✉ ▼❛②❛r❛ ♦✉ ❞❡✈❡rá ♣❡r♠❛♥❡❝❡r ❝♦♠ s❡✉ ♣❛✐✳ ❙❡ ♠✉❧t✐♣❧✐❝❛r♠♦s ❡♥tã♦

  3 ☎ 3 ☎ ☎ ☎ 3

  ❧♦♦♦♠♦♦♦♥ 8 vezes ❝♦♥t❛r❡♠♦s ❛s ❢♦r♠❛s ❞❡ ❞✐✈✐❞✐r ♦s ❜r✐♥q✉❡❞♦s ❡♥tr❡ ❱✐❝t♦r✱ ▼❛②❛r❛ ❡ s❡✉ ♣❛✐✱ ✐♥✲ ❝❧✉✐♥❞♦ ♦s ❝❛s♦s q✉❡ s❡✉ ♣❛✐ ✜❝❛ s❡♠ ♥❡♥❤✉♠ ❜r✐♥q✉❡❞♦✳ ❘❡st❛rá ❡♥tã♦ ❝♦♥t❛r ♦ ♥ú♠❡r♦ ❞❡ ❢♦r♠❛s ❞❡ ❞✐✈✐❞✐r t♦❞♦s ♦s ❜r✐♥q✉❡❞♦s ❡♥tr❡ ❱✐❝t♦r ❡ ▼❛②❛r❛ ✭s❡♠ ❞❡✐①❛r 8

  ♥❛❞❛ ♣❛r❛ s❡✉ ♣❛✐✮✱ ❡ s✉❜tr❛✐r ❡ss❡ ♥ú♠❡r♦ ❞❡ 3 ✳

  ✺✳✹ Pr✐♥❝í♣✐♦ ❋✉♥❞❛♠❡♥t❛❧ ❞❡ ❈♦♥t❛❣❡♠ ✶✺✾

  P❛r❛ ❞✐✈✐❞✐r ♦s ❜r✐♥q✉❡❞♦s ❡♥tr❡ ❱✐❝t♦r ❡ ▼❛②❛r❛ ❞❡✈❡r❡♠♦s ❞❡❝✐❞✐r✱ ♣♦r ❝❛❞❛ ✉♠ ❞♦s ♦✐t♦ ❜r✐♥q✉❡❞♦s✱ ♣❛r❛ q✉❛❧ ❞♦s ❞♦✐s ✐rá✳ ❆ss✐♠✱ t❡r❡♠♦s

  2 ☎ 2 ☎ ☎ ☎ 2

  ❧♦♦♦♠♦♦♦♥ 8 vezes ❢♦r♠❛s ❞❡ ❞✐✈✐❞✐r ♦s ❜r✐♥q✉❡❞♦s ❡♥tr❡ ❡❧❡s✳ ▲♦❣♦✱ t❡r❡♠♦s 8 8

  3 ✁ 2 ✏ 6305

  ❢♦r♠❛s ❞❡ ❞✐✈✐❞✐r ♦s ❜r✐♥q✉❡❞♦s✳ ✻✳ ✭❛✮ ❖ ❛❧❣❛r✐s♠♦ ✶ ♥ã♦ ♣♦❞❡ s❡r r❡♣❡t✐❞♦✱ ♣♦rq✉❡ ♥ã♦ é ♣♦ssí✈❡❧ ❡s❝r❡✈❡r ✶✷ ❝♦♠♦

  ✉♠❛ s♦♠❛ ❞❛ ❢♦r♠❛ 1 1 x✱ ♦♥❞❡ x é ✉♠ ❛❧❣❛r✐s♠♦✳ ❉❡ ❢❛t♦✱ ❝♦♠♦ x é ♥♦ ♠á①✐♠♦ ✾✱ ❡st❛ s♦♠❛ s❡rá ♥♦ ♠á①✐♠♦ ✶✶✳ ❖ ❛❧❣❛r✐s♠♦ ✹ ♥ã♦ ♣♦❞❡ s❡r r❡♣❡t✐❞♦✱ ♣♦✐s✱ ♥❡st❡ ❝❛s♦✱ ♦ ♥ú♠❡r♦ t❡r✐❛ q✉❡ s❡r ✹✹✹✱ q✉❡ t❡♠ três ❛❧❣❛r✐s♠♦s ✐❣✉❛✐s ❡ ♥ã♦ ❡st❛r✐❛ ❞❡ ❛❝♦r❞♦ ❝♦♠ ♦ ❡♥✉♥❝✐❛❞♦✳ ❖s ❛❧❣❛r✐s♠♦s ✼✱ ✽ ❡ ✾ ♥ã♦ ♣♦❞❡♠ s❡r r❡♣❡t✐❞♦s✱ ♣♦✐s✱ ♥❡st❡ ❝❛s♦✱ ❛ s♦♠❛ ❞♦s ❛❧❣❛r✐s♠♦s ✉❧tr❛♣❛ss❛r✐❛ ✶✷✳ ❆ss✐♠✱ ♦ ❛❧❣❛r✐s♠♦ r❡♣❡t✐❞♦ só ♣♦❞❡ s❡r ✷✱ ✸✱ ✺ ♦✉ ✻✳ ❈♦♠ ✷✱ ✸ ❡ ✺ ♣♦❞❡♠♦s ❢♦r♠❛r ♥♦✈❡ ♥ú♠❡r♦s✿ 228, 282, 822, 336, 363, 633, 552, 525, e 255.

  ❈♦♠ ♦ ❛❧❣❛r✐s♠♦ ✻ ♣♦❞❡♠♦s ❢♦r♠❛r ❞♦✐s ♥ú♠❡r♦s✿ 606 e 660.

  P♦rt❛♥t♦✱ ❛ q✉❛♥t✐❞❛❞❡ ❞❡ ♥ú♠❡r♦s ❡s❝r✐t❛ é ✶✶✳ ✭❜✮ ❆ s♦♠❛ ❞❡ três ♥ú♠❡r♦s í♠♣❛r❡s é ✉♠ ♥ú♠❡r♦ í♠♣❛r✳ ▲♦❣♦✱ ♥❡♥❤✉♠ ❞♦s

  ♥ú♠❡r♦s ❡s❝r✐t♦s t❡♠ ♦s três ❛❧❣❛r✐s♠♦s í♠♣❛r❡s✳ ✼✳ ❈❛❞❛ ❥♦❣♦ ❞✐s♣✉t❛❞♦ sã♦ ❞✐str✐❜✉í❞♦s ❞♦✐s ♣♦♥t♦s✱ ♥♦ ❝❛s♦ ❞❡ ❡♠♣❛t❡ ♦✉ três ♣♦♥t♦s✱

  ❝❛s♦ ♥ã♦ ♦❝♦rr❛ ❡♠♣❛t❡✳ ❝♦♠♦ ❝❛❞❛ ✉♠ ❞♦s t✐♠❡s ❥♦❣♦✉ ✉♠❛ ú♥✐❝❛ ✈❡③ ❝♦♠ s❡✉s três ❛❞✈❡rsár✐♦s✱ ❢♦r❛♠ ❞✐s♣✉t❛❞♦s ❛♦ t♦❞♦ s❡✐s ❥♦❣♦s✱ ♥♦s q✉❛✐s ❢♦r❛♠ ❞✐str✐❜✉í❞♦s

  5

  3

  3

  2 ✏ 13 ♣♦♥t♦s✳ ❆ ú♥✐❝❛ ♠❛♥❡✐r❛ ❞❡ ♣❛r❝❡❧❛r ✶✸ ♣♦♥t♦s ❡♠ s❡✐s ♣❛r❝❡❧❛s ❞❡

  ✷ ♦✉ ✸ é 13 ✏ 3 2 2 2 2 2✳ ▲♦❣♦✱ ❝✐♥❝♦ ❞♦s s❡✐s ❥♦❣♦s t❡r♠✐♥❛r❛♠ ❡♠♣❛t❛❞♦s✳ ✽✳ ✭❛✮ P♦❞❡♠♦s ♣✐♥t❛r ♦ ❝ír❝✉❧♦ ✶ ❝♦♠ q✉❛❧q✉❡r ✉♠❛ ❞❛s três ❝♦r❡s✳ ❖ ❝ír❝✉❧♦ ✷ ❞❡✈❡

  ❡♥tã♦ s❡r ♣✐♥t❛❞❛ ❞❡ ✉♠❛ ❝♦r ❞✐❢❡r❡♥t❡ ❞❛ ♣r✐♠❡✐r❛✱ r❡st❛♥❞♦ ❞✉❛s ❝♦r❡s ♣❛r❛

  ✺✳✹ Pr✐♥❝í♣✐♦ ❋✉♥❞❛♠❡♥t❛❧ ❞❡ ❈♦♥t❛❣❡♠ ✶✻✵

  3

  1

  2 ✭❜✮ ❱❛♠♦s ❞✐✈✐❞✐r ❡♠ ❞♦✐s ❝❛s♦s ❛s ♠❛♥❡✐r❛s ❞❡ ♣✐♥t❛r ❛ ✜❣✉r❛ ✷✳

  1

  2

  4

  3 Pr✐♠❡✐r♦ ❝❛s♦✿ ♦s ❝ír❝✉❧♦s ✶ ❡ ✸ sã♦ ♣✐♥t❛❞♦s ❞❡ ♠❡s♠❛ ❝♦r✳ ❊st❛ ❝♦r ♣♦❞❡ s❡r ❡s❝♦❧❤✐❞❛ ❞❡ três ♠❛♥❡✐r❛s ❞✐❢❡r❡♥t❡s✳ ❆♣ós ❡ss❛ ❡s❝♦❧❤❛✱ ❛ ❝♦r ❞♦ ❝ír❝✉❧♦ ✷ ♣♦❞❡ s❡r ❡s❝♦❧❤✐❞❛ ❞❡ ❞✉❛s ♠❛♥❡✐r❛s ❞✐❢❡r❡♥t❡s✱ ❜❡♠ ❝♦♠♦ ♦ ❞♦ ❝ír❝✉❧♦ ✹✳ ❖ ♥ú♠❡r♦ ❞❡ ♠❛♥❡✐r❛s ❞❡ ♣✐♥t❛r ❛ ✜❣✉r❛ ✷ ♥❡ss❡ ❝❛s♦ é 3 ☎ 2 ☎ 2 ✏ 12✳ ❙❡❣✉♥❞♦ ❝❛s♦✿ ♦s ❝ír❝✉❧♦s ✶ ❡ ✸ sã♦ ♣✐♥t❛❞♦s ❞❡ ❝♦r❡s ❞✐❢❡r❡♥t❡s✳ ◆❡ss❡ ❝❛s♦✱ ❛ ❝♦r ❞♦ ❝ír❝✉❧♦ ✶ ♣♦❞❡ s❡r ❡s❝♦❧❤✐❞❛ ❞❡ três ♠❛♥❡✐r❛s ❞✐❢❡r❡♥t❡s ❡ ❛♣ós ✐ss♦✱ r❡st❛♠ ❞✉❛s ♣♦ss✐❜✐❧✐❞❛❞❡s ♣❛r❛ ❛ ❝♦r ❞♦ ❝ír❝✉❧♦ ✸✳ P❛r❛ ♦s ❝ír❝✉❧♦s ✷ ❡ ✹ ❤á ❛♣❡♥❛s ✉♠❛ ♣♦ss✐❜✐❧✐❞❛❞❡ q✉❡ é ❛ ❝♦r q✉❡ ♥ã♦ ❢♦✐ ✉s❛❞❛ ♥♦s ❝ír❝✉❧♦s ✶ ❡ ✸✳ ▲♦❣♦✱ ♦ ♥ú♠❡r♦ ❞❡ ♠❛♥❡✐r❛s ❞❡ ♣✐♥t❛r ❛ ✜❣✉r❛ ✷✱ ♥❡ss❡ ❝❛s♦✱ é 3 ☎ 2 ☎ 1 ✏ 6✳ ◆♦ t♦t❛❧✱ ❛ ✜❣✉r❛ ✷ ♣♦❞❡ s❡r ♣✐♥t❛❞❛ ❞❡ 12 6 ✏ 18 ♠❛♥❡✐r❛s ❞✐❢❡r❡♥t❡s✳

  ✭❝✮ ❖s ❝ír❝✉❧♦s ❞❡ ✶ ❛ ✹ ❢♦r♠❛♠ ❛ ✜❣✉r❛ ❞♦ ✐t❡♠ ❛♥t❡r✐♦r ❡ ♣♦rt❛♥t♦✱ ♣❛r❛ ♣✐♥tá✲❧♦s✱ ❤á ✶✽ ♣♦ss✐❜✐❧✐❞❛❞❡s✳ P❛r❛ ♣✐♥t❛r ♦ ❝ír❝✉❧♦ ✺✱ t❡♠✲s❡ ❞✉❛s ❝♦r❡s ❞✐s♣♦♥í✈❡✐s✱ ♣♦✐s ♦ ❝ír❝✉❧♦ ✹ ❥á ❡stá ♣✐♥t❛❞♦✳ ▲♦❣♦✱ t❡♠♦s 18 ☎ 2 ✏ 36 ♣♦ss✐❜✐❧✐❞❛❞❡s ♣❛r❛ ♣✐♥t❛r ♦s ❝ír❝✉❧♦s ❞❡ ✶ ❛ ✺✳ ❱❛♠♦s ❞✐✈✐❞✐r ♥♦ss❛ ❝♦♥t❛❣❡♠ ❡♠ ❞♦✐s ❝❛s♦s✳ Pr✐♠❡✐r♦ ❝❛s♦✿ ❖s ❝ír❝✉❧♦s ✸ ❡ ✻ sã♦ ♣✐♥t❛❞♦s ❞❡ ♠❡s♠❛ ❝♦r✳ ◆❡ss❡ ❝❛s♦✱ t❡♠♦s ✉♠❛ ú♥✐❝❛ ❡s❝♦❧❤❛ ♣❛r❛ ❛ ❝♦r ❞♦ ❝ír❝✉❧♦ ✻ ✭♣♦✐s ♦ ❝ír❝✉❧♦ ✸ ❥á ❢♦✐ ♣✐♥t❛❞♦✮ ❡ ❞✉❛s ♣❛r❛ ♦ ❝ír❝✉❧♦ ✼✱ ♦✉ s❡❥❛✱ t❡♠♦s 1 ☎ 2 ✏ 2 ♣♦ss✐❜✐❧✐❞❛❞❡s✳ ❙❡❣✉♥❞♦ ❝❛s♦✿ ♦s ❝ír❝✉❧♦s ✸ ❡ ✻ sã♦ ♣✐♥t❛❞♦s ❞❡ ❝♦r❡s ❞✐❢❡r❡♥t❡s✳ ◆❡ss❡ ❝❛s♦ t❛♠❜é♠ t❡♠♦s ✉♠❛ ú♥✐❝❛ ❡s❝♦❧❤❛ ♣❛r❛ ❛ ❝♦r ❞♦ ❝ír❝✉❧♦ ✻ ✭❞✐❢❡r❡♥t❡ ❞❛s ❝♦r❡s ❞♦s ❝ír❝✉❧♦s ✸ ❡ ✹✮ ❡ s♦❜r❛ ❛♣❡♥❛s ✉♠❛ ❝♦r ♣❛r❛ ♦ ❝ír❝✉❧♦ ✼✳ ❆q✉✐✱

  ✺✳✹ Pr✐♥❝í♣✐♦ ❋✉♥❞❛♠❡♥t❛❧ ❞❡ ❈♦♥t❛❣❡♠ ✶✻✶

  1

  2

  5

  4

  3

  6

  7 t❡♠♦s ❛♣❡♥❛s ✉♠❛ ♣♦ss✐❜✐❧✐❞❛❞❡✳ ◆♦ t♦t❛❧ ❤á 36 ☎ 2 36 ☎ 1 ✏ 108 ♠❛♥❡✐r❛s ❞✐❢❡r❡♥t❡s ❞❡ ♣✐♥t❛r ❛ ❋✐❣✉r❛ ✸✳

  ✾✳ ✭❛✮ P❛r❛ ♦❜t❡r ❛ ♣♦s✐çã♦ ❞♦ ♥ú♠❡r♦ ✼✽✺✷✸ ♥❡st❛ ❧✐st❛✱ ❝❛❧❝✉❧❛r❡♠♦s q✉❛♥t♦s ♥ú♠❡r♦s

  ❡stã♦ ❛♥t❡s ❞❡❧❡ ♥❛ ❧✐st❛✳ ❖s ♣r✐♠❡✐r♦s ♥ú♠❡r♦s✱ ♥❡st❛ ❧✐st❛✱ sã♦ ❛q✉❡❧❡s q✉❡ ✐♥✐❝✐❛♠ ❝♦♠ ✷✳ P❡r♠✉t❛♥❞♦ ♦s ♦✉tr♦s ✹ ❛❧❣❛r✐s♠♦s✱ t❡♠♦s ✷✹ ♥ú♠❡r♦s q✉❡ ✐♥✐❝✐❛♠ ❝♦♠ ✷✳ ❖s ♣ró①✐♠♦s ♥ú♠❡r♦s sã♦ ♦s ✐♥✐❝✐❛❞♦s ❝♦♠ ✸✳ ❈♦♠♦ ♥♦ ❝❛s♦ ❛♥t❡r✐♦r✱ tr❛t❛✲s❡ ❞❡ ✷✹ ♥ú♠❡r♦s✳ ❙❡❣✉❡♠✲s❡ ♦s ♥ú♠❡r♦s ✐♥✐❝✐❛❞♦s ❝♦♠ ✺✱ q✉❡ t❛♠❜é♠ sã♦ ✷✹ ♥ú♠❡r♦s✳ ❖s ♣ró①✐♠♦s ♥ú♠❡r♦s sã♦ ♦s ✐♥✐❝✐❛❞♦s ❝♦♠ ✼✷✱ ✼✸ ❡ ✼✺✳ P❛r❛ ❝❛❞❛ ✉♠ ❞❡st❡s ❝❛s♦s✱ ♣❡r♠✉t❛♥❞♦s ♦s ♦✉tr♦s ✸ ❛❧❣❛r✐s♠♦s✱ t❡♠♦s ♠❛✐s ✻ ♥ú♠❡r♦s✳ ❋✐♥❛❧♠❡♥t❡✱ t❡♠♦s ♦s ♥ú♠❡r♦s ✐♥✐❝✐❛❞♦s ❝♦♠ ✼✽✷ ❡ ✼✽✸✱ ❝♦♠ ✷ ♥ú♠❡r♦s ❡♠ ❝❛❞❛ ❝❛s♦✳ ❙❡♥❞♦ ❛ss✐♠✱ ❛ ♣♦s✐çã♦ ❞♦ ♥ú♠❡r♦ ✼✽✺✷✸ é

  3 ☎ 24 3 ☎ 6 2 ☎ 2 1 ✏ 95. ✭❜✮ P❡r♠✉t❛♥❞♦ ♦s ✺ ❛❧❣❛r✐s♠♦s✱ ✈❡♠♦s q✉❡ ❛ ❧✐st❛ t❡♠ ✉♠ t♦t❛❧ ❞❡ ✶✷✵ ♥ú♠❡r♦s✳ P❛r❛

  ❝❛❧❝✉❧❛r ❛ s♦♠❛ ❞❡st❡s ♥ú♠❡r♦s✱ ❞❡✈❡♠♦s ♣❡r❝❡❜❡r q✉❡ ❝❛❞❛ ❛❧❣❛r✐s♠♦ ❞♦ ♥ú♠❡r♦ ✼✽✺✷✸ ❛♣❛r❡❝❡ ♥❛ ♣♦s✐çã♦ ❞❛s ✉♥✐❞❛❞❡s ❡♠ ✷✹ ♥ú♠❡r♦s✳ ❆ss✐♠ t❡♠♦s ♣7 8 5 2

  3 q ☎ 24 ✏ 600 ✉♥✐❞❛❞❡s✳ ❖ ♠❡s♠♦ ♦❝♦rr❡ ♥❛ ♣♦s✐çã♦ ❞❛s ❞❡③❡♥❛s✱ ❝❡♥t❡♥❛s✱ ♠✐❧❤❛r❡s ❡ ❞❡③❡♥❛s ❞❡ ♠✐❧❤❛r❡s✳ P♦rt❛♥t♦✱ ❛ s♦♠❛ é ✐❣✉❛❧ ❛

  600 ☎ 10000 600 ☎ 1000 600 ☎ 100 600 ☎ 10 600 ✏ 6.666.600. ✶✵✳ ❊①✐st❡♠ q✉❛tr♦ t✐♣♦s ♣♦ssí✈❡✐s ❞❡ ❝♦❧✉♥❛s✳ ❆ s❛❜❡r✿ ♣0, 0q; ♣0, 1q; ♣1, 0qe♣1, 1q✳

  ❆s r❡❣r❛s s❡ r❡s✉♠❡♠ ❛ ♥ã♦ ♣r❡❡♥❝❤❡r♠♦s ✉♠❛ ❞❡t❡r♠✐♥❛❞❛ ❝♦❧✉♥❛ ❝♦♠ ❛ ♠❡s♠❛ ❝♦♥✜❣✉r❛çã♦ ❞❛ ❝♦❧✉♥❛ ✐♠❡❞✐❛t❛♠❡♥t❡ ❛♥t❡r✐♦r✳ ❆ss✐♠✱ ✉♠❛ ✈❡③ ❡s❝♦❧❤✐❞♦ ♦s ♥ú♠❡✲

  ✺✳✹ Pr✐♥❝í♣✐♦ ❋✉♥❞❛♠❡♥t❛❧ ❞❡ ❈♦♥t❛❣❡♠ ✶✻✷

  ◆♦ ✐♥í❝✐♦✱ ♣♦❞❡♠♦s ♣r❡❡♥❝❤❡r ❧✐✈r❡♠❡♥t❡ ❝♦♠♦ ♣r❡❡♥❝❤❡r ❛ ❝♦❧✉♥❛ ♠❛✐s à ❡sq✉❡r❞❛ ❡ ✐ss♦ ♣♦❞❡ s❡r ❢❡✐t♦ ❞❡ q✉❛tr♦ ❢♦r♠❛s✳ ❊♠ s❡q✉✐❞❛✱ ❛♦ ♣r❡❡♥❝❤❡r♠♦s ❛s ♣ró①✐♠❛s ❝♦❧✉♥❛s à ❞✐r❡✐t❛✱ t❡r❡♠♦s três ♦♣çõ❡s✳ P♦rt❛♥t♦✱ ♦ t♦t❛❧ ❞❡ ♣r❡❡♥❝❤✐♠❡♥t♦ é✿ 2014 4 .

  ☎ 3 ☎ 3 ☎ 3 ☎ ☎ ☎ 3 ✏ 4 ☎ 3 1 , a 2 1 , b 2 1 , c 2 ✶✶✳ ❖ ❝✉❜♦ t❡♠ três ♣❛r❡s ❞❡ ❢❛❝❡s ♦♣♦st❛s✿ ♣a q, ♣b q, ♣c q ❡ s❛❜❡♠♦s q✉❡ a 1 a 2 b 1 b 2 c 1 c 2

  2

  3

  4

  5

  6 ✏ 1 ✏ 21✳ ◆❛ ♣r✐♠❡✐r❛ ❥♦❣❛❞❛✱ 1 2 1 b 2 c 1 c 2

  ❞✐❣❛♠♦s q✉❡ ❛s ❢❛❝❡s ❞❡ ❜❛✐①♦ ❡ ❞❡ ❝✐♠❛ s❡❥❛♠ a ❡ a ✳ ❈♦♠♦ b ✏ 15✱ 1 2 a ❝♦♥❝❧✉í♠♦s q✉❡ a ✏ 6✳ 1 2

  ◆❛ s❡❣✉♥❞❛ ❥♦❣❛❞❛✱ ❞✐❣❛♠♦s q✉❡ ❛s ❢❛❝❡s ❞❡ ❜❛✐①♦ ❡ ❞❡ ❝✐♠❛ s❡❥❛♠ b ❡ b ✳ ❈♦♠♦ a 1 a 2 c 1 c 2 1 b 2 ✏ 12✱ ❝♦♥❝❧✉í♠♦s q✉❡ b ✏ 9✳ ▲♦❣♦✱

  21 1 a 2 b 1 b 2 c 1 c 2 1 c 2 1 c 2 ✏ a ✏ 6 9 c ✏ 21 ñ c ✏ 6.

  ❚❡♠♦s três ♣❛r❡s ❞❡ ❢❛❝❡s ♦♣♦st❛s ❡ q✉❛♥❞♦ s♦♠❛♠♦s ❢❛❝❡s ♦♣♦st❛s✱ ♦❜t❡♠♦s ♦s s❡❣✉✐♥t❡s r❡s✉❧t❛❞♦s✿ ✻✱ ✾ ♦✉ ✻✳ ❙♦♠❛♥❞♦ ✻ ❛♦ ❛❧❣❛r✐s♠♦ ❣r❛✈❛❞♦ ♥❛ s✉❛ ❢❛❝❡ ♦♣♦st❛✱ s❡❥❛ ❡❧❡ q✉❛❧ ❢♦r✱ r❡s✉❧t❛ ❡♠ ✉♠ ♥ú♠❡r♦ ♠❛✐♦r q✉❡ ✻✳ ▲♦❣♦✱ ❛ ú♥✐❝❛ ♣♦ss✐❜✐❧✐❞❛❞❡ é q✉❡ ❡st❛ s♦♠❛ s❡❥❛ ✾✱ ♣♦rt❛♥t♦✱ ♥❛ ❢❛❝❡ ♦♣♦st❛ ❛♦ ✻ ❞❡✈❡ ❛♣❛r❡❝❡r ♦ ✸✳

  ✶✷✳ ❙❛❜❡✲s❡ q✉❡ ❛ s❡♥❤❛ é ❢♦r♠❛❞❛ ♣♦r q✉❛tr♦ ❛❧❣❛r✐s♠♦s ❞✐st✐♥t♦s✱ ❝✉❥♦ ♣r✐♠❡✐r♦ ❞í❣✐t♦ é ✺ ❡ ♦ ú❧t✐♠♦ é ✸✳ ❉♦ t♦t❛❧ ❞❡ ✶✵ ❛❧❣❛r✐s♠♦s r❡st❛♠ ♦✐t♦ ♣♦ss✐❜✐❧✐❞❛❞❡s ❞❡ ❡s❝♦❧❤❛ ♣❛r❛ ♦ s❡❣✉♥❞♦ ❞í❣✐t♦ ❡ s❡t❡ ♣♦ss✐❜✐❧✐❞❛❞❡s ❞❡ ❡s❝♦❧❤❛ ♣❛r❛ ♦ t❡r❝❡✐r♦ ❞í❣✐t♦✳ ❆ss✐♠✱ t❡♠♦s

  8 ☎7 ✏ 56 ❝♦♠❜✐♥❛çõ❡s ♣♦ssí✈❡✐s✳ ❈♦♠♦ ♥ã♦ é ♥❡❝❡ssár✐♦ ❛❣✉❛r❞❛r ❞♦✐s ♠✐♥✉t♦s ♣❛r❛

  ❛ ♣r✐♠❡✐r❛ t❡♥t❛t✐✈❛✱ ♠❛s t❡♠♦s q✉❡ ❛❣✉❛r❞❛r ✉♠ ✐♥t❡r✈❛❧♦ ❞❡ t❡♠♣♦ ❞❡ ✷ ♠✐♥✉t♦s ♣❛r❛ ❛s ❞❡♠❛✐s t❡♥t❛t✐✈❛s ♦ t❡♠♣♦ ♥❡❝❡ssár✐♦ q✉❡ ❧❡✈❛rá ♣❛r❛ ❛t✐✈❛r ♦ ❛❧❛r♠❡ s❡rá ❞❡ 2 ☎ ♣56 ✁ 1q ✏ 110 ♠✐♥✉t♦s✱ ♦✉ s❡❥❛✱ ✶ ❤♦r❛ ❡ ✺✵ ♠✐♥✉t♦s✳

  ✶✸✳ ❆s ❧❡tr❛s ❛✱ ❜✱ ❝ ❢♦r♠❛♠ 3! ✏ 6 ♣❛❧❛✈r❛s✱ ❞❛s q✉❛✐s ❛♣❡♥❛s ✉♠❛ ❡stá ♥❛ ♦r❞❡♠ ❞❡s❡❥❛❞❛✳ ❆s s❡✐s ❧❡tr❛s ✭❛✱ ❜✱ ❝✱ ❞✱ ❡✱ ❢✮ ❢♦r♠❛♠ 6! ✏ 720 ♣❛❧❛✈r❛s✱ ❞❛s q✉❛✐s ✐♥t❡r❡ss❛ ❛ s❡①t❛ ♣❛rt❡✱ q✉❡ ❛ss✐♠ ♣❡r❢❛③❡♠ 720 ✄ 6 ✏ 120✳

  ✶✹✳ ❈♦♠♣r❡❡♥❞❡♥❞♦ q✉❡ ❛ ❝♦♠✐ssã♦ s❡rá ❢♦r♠❛❞❛ ❝♦♠ ❛ ❡s❝♦❧❤❛ ❞❡ ❝✐♥❝♦ ♣r♦❢❡ss♦r❡s ❞❡♥tr❡ ♦s ✶✷✱ ❡①❝❡t✉❛♥❞♦✲s❡ ❛s ♣♦ss✐❜✐❧✐❞❛❞❡s ❞❡ ❝♦♠✐ssõ❡s ❝♦♠♣♦st❛s ❡①❝❧✉s✐✈❛✲ ♠❡♥t❡ ♣♦r ♣r♦❢❡ss♦r❡s ❞♦ ❊♥s✐♥♦ ❋✉♥❞❛♠❡♥t❛❧✱ t❡♠♦s q✉❡ ❤á ❝✐♥❝♦ ✈❛❣❛s ♣❛r❛ ♦s ✶✷ ♣r♦❢❡ss♦r❡s ❞❛ ❡❞✉❝❛çã♦ ❜ás✐❝❛✱ ♣♦rt❛♥t♦ 12 ☎ 11 ☎ 10 ☎ 9 ☎ 8 ✏ 95040 ♠♦❞♦s ❞❡ ❡s❝♦❧❤❡r ♦s ❝✐♥❝♦ ♣r♦❢❡ss♦r❡s✳ ❈♦♠♦ ❛ ♦r❞❡♠ ❞❡ ❡s❝♦❧❤❛ ♥ã♦ ♦❝❛s✐♦♥❛ ❝♦♠✐ssõ❡s ❞✐✲

  ✺✳✹ Pr✐♥❝í♣✐♦ ❋✉♥❞❛♠❡♥t❛❧ ❞❡ ❈♦♥t❛❣❡♠ ✶✻✸

  ❤á 95040 ✄ ♣5 ☎ 4 ☎ 3 ☎ 2 ☎ 1q ✏ 792 ♠❛♥❡✐r❛s ❞❡ ❡s❝♦❧❤❡r ♦s ❝✐♥❝♦ ♣r♦❢❡ss♦r❡s ❞❛ ❡❞✉✲ ❝❛çã♦ ❜ás✐❝❛✳ ❈♦♠♦ t❡r❡♠♦s q✉❡ t❡r ♣❡❧♦ ♠❡♥♦s ✉♠ ♣r♦❢❡ss♦r ❞♦ ❊♥s✐♥♦ ▼é❞✐♦✱ ❞❡✈❡♠♦s ❞❡s❝❛rt❛r ❛ ♣♦ss✐❜✐❧✐❞❛❞❡ ❞❡ss❛ ❝♦♠✐ssã♦ t❡r ❛♣❡♥❛s ♣r♦❢❡ss♦r❡s ❞♦ ❊♥s✐♥♦ ❋✉♥❞❛♠❡♥t❛❧✳ ■ss♦ ♦❝♦rr❡ ❞❡ 8 ☎ 7 ☎ 6 ☎ 5 ☎ 4 ✄ ♣5 ☎ 4 ☎ 3 ☎ 2 ☎ 1q ✏ 56 ♠❛♥❡✐r❛s✳ ❉❡ss❡ ♠♦❞♦✱ ❤á 792 ✁ 56 ✏ 736 ♠❛♥❡✐r❛s ❞❡ ❢♦r♠❛r ❛ ❝♦♠✐ssã♦✳ 8

  ✶✺✳ ❖ t♦t❛❧ ❞❡ ♥ú♠❡r♦s ❞❡ ♥♦✈❡ ❛❧❣❛r✐s♠♦s t❡♥❞♦ ♦ ❞í❣✐t♦ ✾ ❝♦♠♦ ♣r✐♠❡✐r♦ é 10 ✳ ❖s 7 ♥ú♠❡r♦s ❛♥t✐❣♦s ❞❡ ♦✐t♦ ❛❧❣❛r✐s♠♦s ❝♦♥st❛♠ ♥♦ t♦t❛❧ ❞❡ 3 ☎ 10 ❡✱ ✐♥tr♦❞✉③✐♥❞♦ ♦ ❞í❣✐t♦ ✾ ❝♦♠♦ ♣r✐♠❡✐r♦ ♥ã♦ ❛❧t❡r❛ ♦ t♦t❛❧ ❞❡ss❡s ♥ú♠❡r♦s✱ ❛♣❡♥❛s ♣❛ss❛♠ ❛ t❡r ♥♦✈❡ ❛❧❣❛r✐s♠♦s✳ P♦rt❛♥t♦✱ ❛ q✉❛♥t✐❞❛❞❡ ❞❡ ♥ú♠❡r♦s ❝r✐❛❞♦s é ✐❣✉❛❧ ❛ 8 7 7 7 7 10 .

  ✁ 3 ☎ 10 ✏ 10 ☎ 10 ✁ 3 ☎ 10 ✏ 7 ☎ 10 ✶✻✳ ❊①✐st❡ ✉♠❛ r❡str✐çã♦ ♥♦s ❞í❣✐t♦s q✉❡ ♠❛r❝❛♠ ❛s ❤♦r❛s ❡ ♥♦ ♣r✐♠❡✐r♦ ❞í❣✐t♦ q✉❡ ♠❛r✲

  ❝❛♠ ♦s ♠✐♥✉t♦s✳ ❉❡ss❛ ❢♦r♠❛✱ ❡♠ ✈❡③ ❞❡ ♣❡♥s❛r♠♦s ❡♠ ❝❛❞❛ ❞í❣✐t♦ s❡♣❛r❛❞❛♠❡♥t❡✱ ✈❛♠♦s ♣❡♥s❛r ❡♠ três ❜❧♦❝♦s ❞❡ ❛❧❣❛r✐s♠♦s✳ ❖ ♣r✐♠❡✐r♦✱ q✉❡ é ❢♦r♠❛❞♦ ♣❡❧♦s ❞♦✐s ♣r✐♠❡✐r♦s ❛❧❣❛r✐s♠♦s✱ ♣♦❞❡ ❛ss✉♠✐r s❡t❡ ✈❛❧♦r❡s ❞✐❢❡r❡♥t❡s ✭✵✵✱ ✵✷✱ ✵✹✱ ✵✻✱ ✵✽✱ ✷✵ ♦✉ ✷✷✮❀ ♦ s❡❣✉♥❞♦ é ❢♦r♠❛❞♦ ❛♣❡♥❛s ♣❡❧♦ t❡r❝❡✐r♦ ❞í❣✐t♦ ❡ ♣♦❞❡ ❛ss✉♠✐r três ✈❛❧♦r❡s ✭✵✱ ✷ ♦✉ ✹✮❀ ❡ ♦ ú❧t✐♠♦ ❞í❣✐t♦ ♣♦❞❡ ❛ss✉♠✐r ❝✐♥❝♦ ✈❛❧♦r❡s ✭✵✱ ✷✱ ✹✱ ✻ ♦✉ ✽✮✳ ▲♦❣♦✱ ♦ t♦t❛❧ ❞❡ ✈❡③❡s ❡♠ q✉❡ t♦❞♦s ❛♣❛r❡❝❡♠ ♣❛r❡s é 7 ☎ 3 ☎ 5 ✏ 105✳

  ❈❛♣ít✉❧♦

  6

❈♦♥s✐❞❡r❛çõ❡s ✜♥❛✐s

  P❛r❛ s❡ ❢❛③❡r ✉♠ ❡♥t❡♥❞✐♠❡♥t♦ ♠❡❧❤♦r ❞❛ ✐❞❡❛❧✐③❛çã♦ ❞❡ss❡ ♠❛t❡r✐❛❧ é ♥❡❝❡ssár✐♦ q✉❡ s❡❥❛ ❡①♣♦st♦ ❛ ♠✐♥❤❛ ❢♦r♠❛çã♦ ❛❝❛❞ê♠✐❝❛✱ ❜❡♠ ❝♦♠♦ ❛ ♠✐♥❤❛ ✈✐❞❛ ♣r♦✜ss✐♦♥❛❧ ❛♦ ❧♦♥❣♦ ❞❡ss❡s ú❧t✐♠♦s ✷✵ ❛♥♦s✳

  ❙♦✉ ❊♥❣❡♥❤❡✐r♦ ❆❣rô♥♦♠♦✱ ❢♦r♠❛❞♦ ♣❡❧❛ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ❆❧❛❣♦❛s ✭❯❋❆▲✮✱ ♦♥❞❡ ❝✉rs❡✐ ♣♦r ❧♦♥❣♦s ♦✐t♦ ❛♥♦s ✭✶✾✾✹ ❛ ✷✵✵✷✮✳ ◆❡ss❡ ♣❡rí♦❞♦✱ ♣❛ss❡✐ ♣♦r ✈ár✐♦s ❧♦❝❛✐s ❞❡ tr❛❜❛❧❤♦ ❡ ❢♦✐ ❛í t❛♠❜é♠ q✉❡ t✐✈❡ ❛ ♦♣♦rt✉♥✐❞❛❞❡ ❞❡ ❡♥s✐♥❛r ❝♦♠♦ ❜♦❧s✐st❛ ❛tr❛✈és ❞❡ ✉♠ ❝♦♥✈ê♥✐♦ q✉❡ ❡①✐st✐❛ ❡♥tr❡ ❛ ❯❋❆▲ ❡ ❛ ❙❡❝r❡t❛r✐❛ ❊st❛❞✉❛❧ ❞❡ ❊❞✉❝❛çã♦✳ ❆ ♦♣♦rt✉♥✐❞❛❞❡ ♣❛r❛ ❡♥s✐♥❛r ❛ sér✐❡ q✉❡ ♠❡ ❢♦✐ ❞❛❞❛ ❡r❛ ♠✉✐t♦ ♣❡q✉❡♥♦✱ ♣♦ré♠ ❛❝❡✐t❡✐ ♦ ❞❡s❛✜♦✱ ❛✜♥❛❧ ❡r❛ ❛ r❡❛❧✐③❛çã♦ ❞❡ ✉♠ s♦♥❤♦✳

  ▲♦❣♦ ❛♣ós ♦ tér♠✐♥♦ ❞♦ ❝✉rs♦ ❞❡ ❆❣r♦♥♦♠✐❛✱ ♦ ❈❊❙▼❆❈ ❞❛✈❛ ♦♣♦rt✉♥✐❞❛❞❡ ❞❡ ❢❛③❡r ✉♠ ❝✉rs♦ ❞❡ ❢♦r♠❛çã♦ ♣❡❞❛❣ó❣✐❝❛ ♥❛s ár❡❛s ❞❡ ▼❛t❡♠át✐❝❛✱ ❋ís✐❝❛✱ ❇✐♦❧♦❣✐❛ ❡ ◗✉í♠✐❝❛ ♣❛r❛ ♣r♦❢❡ss♦r❡s q✉❡ ❛t✉❛✈❛♠ ❡♠ s❛❧❛ ❞❡ ❛✉❧❛ ♥❛ ár❡❛ ❞❛ ❡❞✉❝❛çã♦✱ ♣♦ré♠ ♥ã♦ t✐♥❤❛♠ ❛ ❧✐❝❡♥❝✐❛t✉r❛✳

  ◆❡ss❡ ♣❡rí♦❞♦ q✉❡ ❡st❛✈❛ ❢❛③❡♥❞♦ ♦ ❝✉rs♦ ❞❡ ❢♦r♠❛çã♦ ♣❡❞❛❣ó❣✐❝❛✱ ❛ ❙❡❝r❡t❛r✐❛ ❊st❛✲ ❞✉❛❧ ❞❛ ❊❞✉❝❛çã♦ ❞❡ ❆❧❛❣♦❛s ❛❜r✐✉ ✐♥s❝r✐çõ❡s ♣❛r❛ ♠♦♥✐t♦r✐❛✱ ♦♥❞❡ ✜③ ✐♥s❝r✐çã♦ ❡ ❛♣ós s❡r s❡❧❡❝✐♦♥❛❞♦✱ ♣❛ss❡✐ ❞♦✐s ❛♥♦s ❝♦♠♦ ♠♦♥✐t♦r ❡♠ s❛❧❛ ❞❡ ❛✉❧❛✳ ❆♣ós ♦ tér♠✐♥♦ ❞❡ss❡ ❝✉rs♦✱ ❛ ❙❡❝r❡t❛r✐❛ ❊st❛❞✉❛❧ ❛❜r✐✉ ❡❞✐t❛❧ ❞❡ ❝♦♥❝✉rs♦ ♣❛r❛ ♣r♦❢❡ss♦r ❡❢❡t✐✈♦✱ ♥♦ q✉❛❧ ❢✉✐ ❛♣r♦✈❛❞♦✱ ❡ ❛ ❙❡❝r❡t❛r✐❛ ▼✉♥✐❝✐♣❛❧ ❞❛ ❊❞✉❝❛çã♦ ✭❙❊▼❊❉✮ ❛❜r✐✉ ♦ ❡❞✐t❛❧ ❞♦ ❝♦♥❝✉rs♦ ♥♦ ❛♥♦ ♣♦st❡r✐♦r✱ ♦♥❞❡ ❢✉✐ ❛♣r♦✈❛❞♦ t❛♠❜é♠✱ ♦♥❞❡ s♦✉ ♣r♦❢❡ss♦r ❡❢❡t✐✈♦ ❞❛s r❡❞❡s ❡st❛❞✉❛❧ ❡ ♠✉♥✐❝✐♣❛❧ ❞❛ ❡❞✉❝❛çã♦✱ ❛té ♦s ❞✐❛s ❞❡ ❤♦❥❡✳

  ❋✐q✉❡✐ q✉❛s❡ ✶✵ ❛♥♦s ❛♣❡♥❛s tr❛❜❛❧❤❛♥❞♦ s❡♠ ♥❡♥❤✉♠ t✐♣♦ ❞❡ ❝✉rs♦ ♦✉ ❢♦r♠❛çã♦ q✉❡ ♣✉❞❡ss❡ s❡r ❞❛❞♦ ♣❛r❛ ♦s ♣r♦❢❡ss♦r❡s ❝♦♠♦ ❢♦r♠❛ ❞❡ ❛♣❡r❢❡✐ç♦❛r ❛ ❞✐❞át✐❝❛ ❡ s❡✉ ❝♦♥❤❡❝✐✲

  ✶✻✺ ♠❡♥t♦ ❡♠ s❛❧❛ ❞❡ ❛✉❧❛✳ ❋♦✐ q✉❛♥❞♦ ❛♣❛r❡❝❡✉ ❛ ♦♣♦rt✉♥✐❞❛❞❡ ❞❡ ✐♥❣r❡ss❛r ♥♦ ♠❡str❛❞♦✱ ❛tr❛✈és ❞❡ ✉♠❛ ♣r♦✈❛ ♥❛❝✐♦♥❛❧ ❞❡ ❛❝❡ss♦ ❛♦ ❝✉rs♦ ❞♦ P❘❖❋▼❆❚✳ ❋✉✐ ❝❧❛ss✐✜❝❛❞♦ ❡♥tr❡ ♦s ✸✵ ❛❧✉♥♦s ♣❛r❛ ❢❛③❡r ♦ ♠❡str❛❞♦ ♣r♦✜ss✐♦♥❛❧ ❡♠ ▼❛t❡♠át✐❝❛ ❝♦♠ ♣♦❧♦ ♥❛ ❯❋❆▲✳

  ◆♦ ♣r✐♠❡✐r♦ s❡♠❡str❡ ❞♦ ❝✉rs♦ s❡♥t✐ ♠✉✐t❛s ❞✐✜❝✉❧❞❛❞❡s✱ ♣♦✐s ❛ ❧✐♥❣✉❛❣❡♠ ✉s❛❞❛ ❡ ❛té ♦s ❛ss✉♥t♦s ❛❜♦r❞❛❞♦s ♣♦r ♠✉✐t❛s ✈❡③❡s ♣❛r❛ ♠✐♠ ❡r❛ ✐♥❝♦♠♣r❡❡♥sí✈❡❧ ♦✉ ♥✉♥❝❛ t✐♥❤❛ ♦✉✈✐❞♦ ❢❛❧❛r✱ ❞❡✈✐❞♦ ❛♦s ❝✉rs♦s q✉❡ ❡✉ t✐♥❤❛ ❢❡✐t♦ q✉❡ ♥ã♦ ❡r❛ ✈♦❧t❛❞♦ ♣❛r❛ ❛ ár❡❛✳ ❉❡ss❛ ❢♦r♠❛✱ ♣❡r❝❡❜✐✱ ❞❡♣♦✐s ❞❡ ❞❡③ ❛♥♦s ❡♠ s❛❧❛ ❞❡ ❛✉❧❛✱ q✉❡ ❡✉ ❡♥s✐♥❛✈❛ ♠❛t❡♠át✐❝❛ ❞❡ ✉♠❛ ❢♦r♠❛ ❡q✉✐✈♦❝❛❞❛✱ ♦♥❞❡ ❡✉ t✐♥❤❛ ❛ ♣r❡♦❝✉♣❛çã♣ ❛♣❡♥❛s ❞❡ r❡s♦❧✈❡r ❡①❡r❝í❝✐♦s ❡♠ s❛❧❛✱ s❡♠ t❡r ♦ ❝✉✐❞❛❞♦ ❡♠ tr❛♥s♠✐t✐r ❛s ❞❡✜♥✐çõ❡s ❡ ❛s ♣r♦♣r✐❡❞❛❞❡s q✉❡ ❛❧✐ ❡①✐st❡♠✳ ■ss♦ ♠❡ ❞❡s♣❡rt♦✉ ♣❛r❛ q✉❡ ❡✉ r❡❛❧✐③❛ss❡ ✉♠ tr❛❜❛❧❤♦ ❥✉♥t♦ ❛♦s ♠❡✉s ❝♦❧❡❣❛s ❞❡ ♣r♦✜ssã♦✱ ♦♥❞❡ ❡ss❛ tr❛♥s♠✐ssã♦ ❢❛❧❤❛ ❞❡ ❝♦♥❤❡❝✐♠❡♥t♦s q✉❡ ❡✉ ♣❛ss❛✈❛ ♣❛r❛ ♠❡✉s ❛❧✉♥♦s ❞✉r❛♥t❡ ❡ss❡ ♣❡rí♦❞♦ q✉❡ ❧❡❝✐♦♥❡✐ ♥ã♦ ❛❝♦♥t❡❝❡ss❡ ❝♦♠ ❡ss❡s ♠❡✉s ❝♦❧❡❣❛s✱ ♥♦ q✉❛❧ t✐♥❤❛ ✉♠❛ ❢♦r♠❛çã♦ ❛❝❛❞ê♠✐❝❛ ♠✉✐t♦ ♣❛r❡❝✐❞❛ ❝♦♠ ❛ ♠✐♥❤❛✳ ❚❡♠♦s ♥❛ ❡s❝♦❧❛✱ ❡♥❣❡♥❤❡✐r♦s✱ ❝♦♥t❛❞♦r✱ ❛❣rô♥♦♠♦s q✉❡ ❡♥s✐♥❛♠ ▼❛t❡♠át✐❝❛ ❡ t❛♠❜é♠ t❡♠ ♦ ❝✉rs♦ ❞❡ ❢♦r♠❛çã♦ ♣❡❞❛❣ó❣✐❝❛✳

  ❖✉tr♦ ❢❛t♦ q✉❡ ♠❡ ❝❤❛♠♦✉ ❛t❡♥çã♦✱ ♥❛ é♣♦❝❛ q✉❡ ✐♥✐❝✐❡✐ ♦ ♠❡str❛❞♦✱ ❢♦✐ q✉❡ ♥❡ss❛ ❡s❝♦❧❛ ♥ã♦ ❤❛✈✐❛ ♥❡♥❤✉♠ t✐♣♦ ❞❡ ♣r❡♣❛r❛çã♦ ♣❛r❛ ❛ ❖❧✐♠♣í❛❞❛ ❇r❛s✐❧❡✐r❛ ❞❡ ▼❛t❡♠át✐❝❛ ❞❛s ❊s❝♦❧❛s Pú❜❧✐❝❛s ✭❖❇▼❊P✮ ❡✱ ♠❡s♠♦ ❛ss✐♠✱ ✉♠ ❛❧✉♥♦ ❝♦♥s❡❣✉✐✉ ❝♦♥q✉✐st❛r ♠❡♥çã♦ ❤♦♥r♦s❛ ♥❛ ❖❇▼❊P ✷✵✶✷✳ ❉❡ss❛ ♠❛♥❡✐r❛✱ r❡s♦❧✈✐ ❥✉♥t❛r ♦ út✐❧ ❛♦ ❛❣r❛❞á✈❡❧✱ ♦♥❞❡ ❛ ✐♥s♣✐r❛çã♦ s❡r✐❛ ♣❡♥s❛r q✉❡ s❡ é ♣♦ssí✈❡❧ ✉♠ ❛❧✉♥♦ ❝♦♥s❡❣✉✐r s❡ ❞❡st❛❝❛r ♥❛❝✐♦♥❛❧♠❡♥t❡ s❡♠ q✉❛❧q✉❡r t✐♣♦ ❞❡ tr❡✐♥❛♠❡♥t♦✱ ❡♥tã♦ ♦s ♦✉tr♦s ♣♦❞❡r✐❛♠ ❝♦♥s❡❣✉✐r✱ ❝❛s♦ ❤♦✉✈❡ss❡ ✉♠ ✐♥❝❡♥t✐✈♦ ♣❛r❛ ❡❧❡s✳ ❆ ✐❞❡✐❛ ❡r❛ ♠♦t✐✈❛r ♦s ♣r♦❢❡ss♦r❡s ❛ ♣❡sq✉✐s❛r q✉❡stõ❡s ✈♦❧t❛❞❛s ♣❛r❛ ❛ ♦❧✐♠♣í❛❞❛ ❡ r❡s♦❧✈ê✲❧❛s ✉s❛♥❞♦ ✉♠❛ ❧✐♥❣✉❛❣❡♠ ❛❞❡q✉❛❞❛✱ ♣r❡♦❝✉♣❛♥❞♦✲s❡ ❝♦♠ ♦ ❢♦r♠❛❧✐s♠♦ ♠❛t❡♠át✐❝♦ ❡ ❛♣❧✐❝❛♥❞♦ ❞❡✜♥✐çõ❡s✱ ♣r♦♣r✐❡❞❛❞❡s ❡ t❡♦r❡♠❛s ❞❡ ♠❛♥❡✐r❛ ❝♦♠♣r❡❡♥sí✈❡❧ ♣❛r❛ ♦ ❛❧✉♥♦✳ ❙❡♥❞♦ ❛ss✐♠✱ t❛♠❜é♠ ❤♦✉✈❡ ❛ ♥❡❝❡ss✐❞❛❞❡ ❞❡ ♣r♦♠♦✈❡r ❡♥❝♦♥tr♦s ❡♥tr❡ ♦s ♣r♦❢❡ss♦r❡s ❞❡ ▼❛t❡♠át✐❝❛✱ ❝♦♦r❞❡♥❛çã♦ ❡ ❞✐r❡çã♦ ♣❛r❛ s❡r❡♠ ❞❡❜❛t✐❞♦s ♦s ♣❧❛♥❡❥❛♠❡♥t♦s ♣❛r❛ ❛ r❡❛❧✐③❛çã♦ ❞❡ss❡s ❡♥❝♦♥tr♦s ❡ ❞✐s❝✉t✐r ♦s ❝♦♥t❡ú❞♦s ❡♥✈♦❧✈✐❞♦s ♥❡ss❛s q✉❡stõ❡s ♣❛r❛ s❡r ❛♣r❡s❡♥t❛❞♦ ❞❡ ❢♦r♠❛ ❛❞❡q✉❛❞❛ ♣❛r❛ ♦ ❛❧✉♥♦✳

  ◆♦ ❛♥♦ ♣♦st❡r✐♦r✱ ❝♦♠❡ç❛♠♦s ❛ ♣r❡♣❛r❛çã♦ ❝♦♠ ♦s ❛❧✉♥♦s ❡ ❝♦♥s❡❣✉✐♠♦s ✉♠ r❡s✉❧t❛❞♦✱ ♥❛ ❖❇▼❊P ✷✵✶✸✱ ♠❡❧❤♦r ❞♦ q✉❡ ♥♦ ❛♥♦ ❛♥t❡r✐♦r✱ ❝♦♠ ✉♠❛ ♠❡❞❛❧❤❛ ❞❡ ❜r♦♥③❡ ❡ ❞✉❛s ♠❡♥çõ❡s ❤♦♥r♦s❛s✳ ❖ q✉❡ ❛♦ ♥♦ss♦ ✈❡r ❢♦✐ ✉♠ ❣❛♥❤♦ s✐❣♥✐✜❝❛t✐✈♦ ♣❛r❛ ❛ ❡s❝♦❧❛ ❡ q✉❡ ♦ tr❛❜❛❧❤♦ ❡st❛✈❛ ♥♦ ❝❛♠✐♥❤♦ ❝❡rt♦✳ ❊♠ ✷✵✶✹✱ ❛ ♣r❡♣❛r❛çã♦ s❡ ❡st❡♥❞❡✉ ♥ã♦ só ♣❛r❛ ♦s ❛❧✉♥♦s✱ ♠❛s t❛♠❜é♠ ♣❛r❛ ♦s ♣r♦❢❡ss♦r❡s✳ ■❞❡❛❧✐③❛♠♦s ❡♥❝♦♥tr♦s ♣❛r❛ ❞✐s❝✉t✐r♠♦s ♣r♦❜❧❡✲ ♠❛s ❡♥✈♦❧✈✐❞♦s ♥❛s ♦❧✐♠♣í❛❞❛s✱ ❜❡♠ ❝♦♠♦ ❛ ❢♦r♠❛ ❞❡ ❛❜♦r❞❛r ❡ss❡s t✐♣♦s ❞❡ q✉❡stõ❡s ♥❛ ❡①♣❧❛♥❛çã♦ ♣❛r❛ ♦ ❛❧✉♥❛❞♦✳ ❊ss❡ tr❛❜❛❧❤♦ ✈❡♠ ❞❛❞♦ r❡s✉❧t❛❞♦ ❡ ♦s ♣r♦❢❡ss♦r❡s ✈ê♠ s❡ ♠♦str❛♥❞♦ ✐♥t❡r❡ss❛❞♦s ❡ ♠♦t✐✈❛❞♦s ♣❛r❛ ❡ss❡ t✐♣♦ ❞❡ s✐t✉❛çã♦✱ ♣♦✐s ❡st❛♠♦s ❞✐s❝✉t✐♥❞♦ ❢♦r♠❛s ❞❡ ❝♦♠♦ ❛❜♦r❞❛r ♦s ❛ss✉♥t♦s ❡♥✈♦❧✈✐❞♦s ❝♦♠ ♦s ❝♦♥t❡ú❞♦s ❞❡ ▼❛t❡♠át✐❝❛✱ ♦♥❞❡

  ✶✻✻ ❝❛❞❛ ✉♠ ♣♦♥t✉❛ s✉❛s ❞✐✜❝✉❧❞❛❞❡s ❡ ♠♦str❛ s✉❛s ❤❛❜✐❧✐❞❛❞❡s ♣❛r❛ r❡s♦❧✉çã♦ ❞❛s q✉❡stõ❡s ♣r♦♣♦st❛s ♥❡ss❡s ❡♥❝♦♥tr♦s✱ ♦♥❞❡ ❛♥t❡s ♣❛r❡❝✐❛ ♥ã♦ ❤❛✈❡r ❝❛♠✐♥❤♦s ♣❛r❛ r❡s♦❧✈ê✲❧❛s✳

  ❋♦✐ ✐♥s♣✐r❛❞♦ ♥❡ss❛ ❡①♣❡r✐ê♥❝✐❛ ❡ ♥❛s ❞✐✜❝✉❧❞❛❞❡s q✉❡ ♣❛ss❡✐ ❞✉r❛♥t❡ ❡ss❡ ♣❡rí♦❞♦ ❞❡ ❞♦❝ê♥❝✐❛ q✉❡ ❡ss❡ ♠❛t❡r✐❛❧ ❢♦✐ ♣r♦❞✉③✐❞♦✱ ♣❛r❛ ♠♦t✐✈❛r ♦ ♣r♦❢❡ss♦r q✉❡ s❡♥t❡ ❞✐✜❝✉❧❞❛❞❡s ❡♠ r❡s♦❧✈❡r ♣r♦❜❧❡♠❛s q✉❡ ❡♥✈♦❧✈❡♠ ♦ r✐❣♦r ♠❛t❡♠át✐❝♦✱ ✉s❛♥❞♦ ❛s ❞❡✜♥✐çõ❡s✱ ♣r♦♣r✐❡❞❛✲ ❞❡s ❡ t❡♦r❡♠❛s ❞♦s ❛ss✉♥t♦s q✉❡ sã♦ ❛❜♦r❞❛❞♦s ❡♠ t♦❞♦ ♦ ❛ss✉♥t♦ ❞♦ ❡♥s✐♥♦ ♠é❞✐♦✳

  ❊ss❡ ♠❛t❡r✐❛❧ ❞❡✈❡ s❡r ✉s❛❞♦ ❝♦♥t✐♥✉❛♠❡♥t❡ ♣❡❧♦ ♣r♦✜ss✐♦♥❛❧ ❞❛ ❡❞✉❝❛çã♦ q✉❡ ❧❡❝✐♦♥❛ ▼❛t❡♠át✐❝❛ ❞♦ ❊♥s✐♥♦ ▼é❞✐♦ ❡ ❋✉♥❞❛♠❡♥t❛❧ ♣❛r❛ ❛❜r❛♥❣❡r ♦ ❝♦♥❤❡❝✐♠❡♥t♦ ♥❛ ár❡❛ ❡ t❡r ✐❞❡✐❛s ♣❛r❛ ❧❛♥ç❛r ❥✉♥t♦ ❛ s❡✉s ❝♦❧❡❣❛s ❞❡ tr❛❜❛❧❤♦ ❡ ❛❧✉♥♦s✱ ❞❡ ♠♦❞♦ ❛ t♦r♥❛r ✉♠❛ ❞✐s❝✐♣❧✐♥❛ ♠❛✐s ❛❣r❛❞á✈❡❧ ♥♦ s❡✉ ❡st✉❞♦✳

  ❉❡s❡❥♦ ❛ t♦❞♦s ✉♠ ❜♦♠ ❡st✉❞♦ ❡ ❡s♣❡r♦ q✉❡ ♦ ♠❛t❡r✐❛❧ t❡♥❤❛ s✐❞♦ ❞❡ ❜♦♠ ❛❣r❛❞♦ ❡ q✉❡ ❡❧❡ s❡❥❛ ❛♣r♦✈❡✐t❛❞♦ ♣❛r❛ ♠♦t✐✈❛r ✈♦❝ê ♣r♦❢❡ss♦r q✉❡ ❜✉s❝❛ s❡♠♣r❡ ✉♠ ❛♣❡r❢❡✐ç♦❛♠❡♥t♦ ♥❛ s✉❛ ❝❛rr❡✐r❛✳

  

❘❡❢❡rê♥❝✐❛s ❇✐❜❧✐♦❣rá✜❝❛s

  ❬✻❪ ▼❯◆■❩ ◆❊❚❖✱ ❆✳ ❈✳ ●❡♦♠❡tr✐❛✳ ❘✐♦ ❞❡ ❏❛♥❡✐r♦✿ ❙❇▼✱ ✷✵✶✸✳ ❬✼❪ ▼❯◆■❩ ◆❊❚❖✱ ❆✳ ❈✳ ❚ó♣✐❝♦s ❞❡ ▼❛t❡♠át✐❝❛ ❊❧❡♠❡♥t❛r✿ ❣❡♦♠❡tr✐❛ ❡✉❝❧✐❞✐❛♥❛ ♣❧❛♥❛✳

  

  ✶✳ ❬✶✶❪ ❘❡✈✐st❛ ❈á❧❝✉❧♦✱ ▼❛t❡♠át✐❝❛ ❇♦♠ P❛r❛ ❚♦❞♦s✱ ❡❞✐çã♦ ✶✷✳ ❬✶✷❪ ❘❡✈✐st❛ ❞♦ Pr♦❢❡ss♦r ❞❡ ▼❛t❡♠át✐❝❛✱ ♥

  ♦

  ❆❧❡❣r❡ ✲ ❘❙✿ ❆❘❚▼❊❉ ❡❞✐t♦r❛✱ ✶✾✾✻✳ ❬✶✵❪ ❘❡✈✐st❛ ❊✉r❡❦❛✱ ♥

  ❡❞✐çã♦ ✲ ❘✐♦ ❞❡ ❏❛♥❡✐r♦✿ ❙❇▼✱ ✷✵✶✵✳ ❬✾❪ P❆❘❘❆✱ ❈✳ ❙❆■❩✱ ✳✳✳ ❬❡t ❛❧❪✳ ❉✐❞át✐❝❛ ❞❛ ▼❛t❡♠át✐❝❛✿ r❡✢❡①õ❡s ♣❡❞❛❣ó❣✐❝❛s✳ P♦rt♦

  ❛

  ♣r♦❜❧❡♠❛s ❡ s♦❧✉çõ❡s✳ ✷

  ❡❞✐çã♦✳ ❘✐♦ ❞❡ ❏❛♥❡✐r♦✿ ❙❇▼✱ ✷✵✶✷✳ ❬✽❪ ❖▲■❱❊■❘❆✱ ❑✳ ■✳ ▼✳ ❋❊❘◆➪◆❉❊❙✱ ❆✳ ❏✳ ❈✳ ■♥✐❝✐❛çã♦ à ▼❛t❡♠át✐❝❛✿ ✉♠ ❝✉rs♦ ❝♦♠

  ❛

  ✶

  ❡❞✐çã♦✳ ❘✐♦ ❞❡ ❏❛♥❡✐r♦✿ ❙❇▼✱ ✷✵✶✸✳

  ❬✶❪ ❇❊❘▲■◆●❍❖❋❋✱ ❲✳ P✳✱ ●❖❯❱✃❆✱ ❋✳ ◗✳ ❆ ▼❛t❡♠át✐❝❛ ❛tr❛✈és ❞♦s t❡♠♣♦s✿ ✉♠ ❣✉✐❛ ❢á❝✐❧ ❡ ♣rát✐❝♦ ♣❛r❛ ♣r♦❢❡ss♦r❡s ❡ ❡♥t✉s✐❛st❛s✳ ✷

  ❛

  ❬✺❪ ▲■▼❆✱ ❊✳ ▲✳ ☎ ☎ ☎ ❬❡t ❛❧❪✳ ❚❡♠❛s ❡ Pr♦❜❧❡♠❛s ❊❧❡♠❡♥t❛r❡s✳ ✺

  ❡❞✐çã♦✳ ❘✐♦ ❞❡ ❏❛♥❡✐r♦✿ ❙❇▼✱ ✷✵✵✻✳

  ❛

  ❬✹❪ ▲■▼❆✱ ❊✳ ▲✳ ☎ ☎ ☎ ❬❡t ❛❧❪✳ ❆ ▼❛t❡♠át✐❝❛ ❞♦ ❡♥s✐♥♦ ♠é❞✐♦✳ ❱♦❧✉♠❡ ✷✱ ✻

  ❡❞✐çã♦✳ ❙ã♦ P❛✉❧♦✿ ❙❛r❛✐✈❛✱ ✷✵✶✸✳

  ❛

  ❬✸❪ ■❊❩❩■✱ ●✳☎ ☎ ☎ ❬❡t ❛❧❪✳ ▼❛t❡♠át✐❝❛✿ ❝✐ê♥❝✐❛ ❡ ❛♣❧✐❝❛çõ❡s✳ ❱♦❧✉♠❡ ✶✱ ✼

  ❬✷❪ ❉▼■❚❘■✱ ❋✳ ☎ ☎ ☎ ❬❡t ❛❧❪✳ ❈í❝✉❧♦s ▼❛t❡♠át✐❝❛✿ ❆ ❊①♣❡r✐ê♥❝✐❛ ❘✉ss❛✳ ❘✐♦ ❞❡ ❏❛♥❡✐r♦✿ ■▼P❆✱ ✷✵✶✷✳

  ❡❞✐çã♦✳ ❙ã♦ P❛✉❧♦✿ ❇❧✉❝❤❡r✱ ✷✵✶✵✳

  ❛

  ✽✹✱ ❛♥♦ ✸✷✱ ✷✵✶✹✳

  

❮♥❞✐❝❡ ❘❡♠✐ss✐✈♦

  ❆❧❣♦r✐t♠♦ ❞❛ ❉✐✈✐sã♦✱ ✶✽ Pr✐♥❝í♣✐♦ ❞❛ ■♥❞✉çã♦ ❋✐♥✐t❛✱ ✺✱ ✷✾ ❆♣♦❧ô♥✐♦ ❞❡ P❡r❣❛✱ ✼✸ Pr✐♥❝í♣✐♦ ❋✉♥❞❛♠❡♥t❛❧ ❞❛ ❈♦♥t❛❣❡♠✱ ✶✵✶ ❆r✐t♠ét✐❝❛ ❞♦s ❘❡st♦s✱ ✸✵ Pr✐♥❝í♣✐♦ ▼✉❧t✐♣❧✐❝❛t✐✈♦✱ ✶✵✶ ❆①✐♦♠❛ ❞❡ P❡❛♥♦✱ ✺

  ❘❛③ã♦ ❞❡ ❙❡♠❡❧❤❛♥ç❛✱ ✻✻✱ ✽✾ ❇❛s❡ ♠é❞✐❛✱ ✺✻ ❘❡t❛ ❞❡ ❊✉❧❡r✱ ✼✼ ❇✐ss❡tr✐③ ❊①t❡r♥❛✱ ✼✷

  ❙❡♠❡❧❤❛♥ç❛ ❞❡ ❚r✐â♥❣✉❧♦s✱ ✻✻ ❇✐ss❡tr✐③ ■♥t❡r♥❛✱ ✼✷

  ❚❡♦r❡♠❛ ❞❛ ❇✐ss❡tr✐③✱ ✼✷✱ ✼✹ ❈ír❝✉❧♦ ❞❡ ❆♣♦❧ô♥✐♦✱ ✼✹

  ❚❡♦r❡♠❛ ❞❡ P✐tá❣♦r❛s✱ ✼✶ ❈ír❝✉❧♦ ❞❡ ❊✉❧❡r✱ ✼✾

  ❚❡♦r❡♠❛ ❞♦ â♥❣✉❧♦ ❡①t❡r♥♦✱ ✹✻✱ ✹✽ ❈❧á✉❞✐♦ Pt♦❧♦♠❡✉✱ ✼✹

  ❚❡♦r❡♠❛ ❋✉♥❞❛♠❡♥t❛❧ ❞❛ ❆r✐t♠ét✐❝❛✱ ✷✽ ❈♦♥❣r✉ê♥❝✐❛ ▼♦❞✉❧❛r✱ ✸✵

  ❚r✐â♥❣✉❧♦s ❝♦♥❣r✉❡♥t❡s✱ ✹✶ ❉❡s✐❣✉❛❧❞❛❞❡ ❚r✐❛♥❣✉❧❛r✱ ✹✽ ❉✐✈✐s✐❜✐❧✐❞❛❞❡✱ ✶✽ ●❡♦♠❡tr✐❛ ❊✉❝❧✐❞✐❛♥❛✱ ✻✷ ▲❡✐ ❞❡ ●❛✉ss✱ ✷✸ ▲❡♠❛ ❞❡ ❊✉❝❧✐❞❡s✱ ✷✼ ▼á①✐♠♦ ❉✐✈✐s♦r ❝♦♠✉♠✱ ✷✶ ▼í♥✐♠♦ ▼ú❧t✐♣❧♦ ❈♦♠✉♠✱ ✷✹ ▼❡❞✐❛♥❛ ❞❡ ❊✉❧❡r✱ ✺✻✱ ✼✼ ◆ú♠❡r♦ ❈♦♠♣♦st♦✱ ✷✼ ◆ú♠❡r♦ Pr✐♠♦✱ ✷✼ P❡q✉❡♥♦ ❚❡♦r❡♠❛ ❞❡ ❋❡r♠❛t✱ ✸✷ Pr✐♥❝í♣✐♦ ❞❛ ❈❛s❛ ❞♦s P♦♠❜♦s✱ ✶✸

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