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

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Mestrado Profissional em Matemática em Rede Nacional PROFMAT

Mestrado Profissional em Matemática em Rede Nacional PROFMATDISSERTAđấO DE MESTRADO Uma Matemática Voltada para o Professor do Ensino MédioGeraldo 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✷✵✶✺ 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 MestradoProfissional 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❡❛ ❞♦ 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✳ ❖ ♠❛t❡r✐❛❧ ❛ s❡r ❛♣r❡s❡♥t❛❞♦ ❞❡✈❡ s❡r ✉s❛❞♦ ♣❡❧♦ ♣r♦❢❡ss♦r ❝♦♥t✐♥✉❛♠❡♥t❡✱ ♥♦ ✐♥t✉✐t♦❞❡ ❡♥❝♦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♠ ❢❛❧❛ré ✉♠ ♣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♦sq✉❡ 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❛ ◆♦ ✐♥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 ❖❜s❡r✈❡ q✉❡♣1 2 ☎ ☎ ☎ nq ♣n 1q ✏ n 2✳ ✏ 3 ☎ ♣3 ☎ n2 33✁ n3 3☎ n2 ☎ 3 3 ☎ n ☎ 32 ♣n 1q3 ♣n 2q3 ♣n 3q3 ✁ rn3 ♣n 1q3 ♣n 2q3 s ✏ ♣n 3q3 ✁ n3✏ n3 P❡❧♦ Pr✐♥❝í♣✐♦ ❞❛ ■♥❞✉çã♦ ❋✐♥✐t❛✱ ❞❡✈❡♠♦s ♠♦str❛r q✉❡ P ♣n 1q é ✈❡r❞❛❞❡✐r❛✱ ♦✉ s❡❥❛✱ q✉❡♣n 1q3 ♣n 1 1q3 ♣n 2 1q3 ✏ 3t, t P N.➱ ✈❡r❞❛❞❡✱ ♣n 1q3 ♣n 2q3 ✏ 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❡❥❛ n3 13♣1 1q3 ♣1 2q3 ✏ 36 ✏ 3 ☎ 12. ❊①❡♠♣❧♦ ✶✳✶✳✷✳ Pr♦✈❡ q✉❡ n3 ♣n 1q3 ♣n 2q3 é ♠ú❧t✐♣❧♦ ❞❡ 3✳▼♦str❛r❡♠♦s✱ ♣♦r ✐♥❞✉çã♦✱ q✉❡ ❛ ❛✜r♠❛çã♦ ❛❝✐♠❛ é ✈❡r❞❛❞❡✐r❛✳ P❡r❝❡❜❛ q✉❡ ☎ ♣n 1q ☎ ♣n 1q ❖ 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 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 32☎ n 32 q. ▲♦❣♦✱333222 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✱ ♣❡❧♦333 Pr✐♥❝í♣✐♦ ❞❛ ■♥❞✉çã♦ ■♥✜♥✐t❛✳ ❖✉ s❡❥❛✱ n ♣n 1q ♣n 2q é ❞❛ ❢♦r♠❛ 3k, k P N ❡333❝♦♥❝❧✉✐✲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 M11 ✏ 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♦✱22 M n 1 n✁ M ✏ ♣n 1q ☎ r♣n 1q ✁ 1s ☎ r3 ☎ ♣n 1q 2s ✁ n ☎ ♣n ✁ 1q ☎ ♣3n 2q22 2n✏ ♣n 1q ☎ rn s ☎ r3n 5s ✁ n ☎ ♣n ✁ 1q ☎ ♣3n 2q2 ✏ ♣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 2q22 11n 10 n 2✏ n ☎ ♣n 1q ☎ r3n ✁ 3n s✏ n ☎ ♣n 1q ☎ ♣12n 12q2 . ✏ 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 ❉❛í✱M n 1 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 ✏ ✁ M n ✏ 24s, s P N.❉♦♥❞❡✱ s❡❣✉❡ q✉❡ ■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. 1, ❡ ✐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.✏ ❆ ♣r♦♣r✐❡❞❛❞❡ P ♣n 1q é ✈❡r❞❛❞❡✐r❛✱ ❡✱ ♣♦rt❛♥t♦ P ♣nq t❛♠❜é♠ é ✈❡r❞❛❞❡✐r❛ ♣❛r❛ t♦❞♦ nP N✳ ❈♦♥❝❧✉✐✲s❡ q✉❡ ♦ ♠❛✐♦r ♥ú♠❡r♦ ❞❡ ♣❛rt❡s ♣♦ssí✈❡❧ q✉❡ ♣♦❞❡♠♦s ❞✐✈✐❞✐r ♦ ♣❧❛♥♦ n ☎ ♣n 1q 1❝♦♠ n r❡t❛s é ❞❛❞♦ ♣♦r ✳ 2 ✶✳ ❉❡♠♦♥str❛r ♣♦r ✐♥❞✉çã♦ q✉❡ ♣❛r❛ n ➙ 1 ♥❛t✉r❛❧22 2 ♣n 1q♣2n 1q n 2 .✭❛✮ 1 ☎ ☎ ☎ n ✏333 62 2✭❜✮ 1 ☎ ☎ ☎ n ✏ ♣1 2 ☎ ☎ ☎ nq2 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 n1 ✏ a ♣n ✁ 1q ☎ r ✽✳ ❯s❛♥❞♦ ♦ Pr✐❝í♣✐♦ ❞❛ ■♥❞✉çã♦ ❋✐♥✐t❛✱ ♠♦str❡ q✉❡ ❛ s♦♠❛ ❞♦s n t❡r♠♦s ❞❡ ✉♠❛ P✳❆✳é ❞❛❞❛ ♣♦r 1 a n ♣a q ☎ nS 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 n1✏ a ☎ q ✶✵✳ ▼♦str❡✱ ♣❡❧♦ Pr✐♥❝í♣✐♦ ❞❛ ■♥❞✉çã♦ ❋✐♥✐t❛✱ q✉❡ ❛ s♦♠❛ ❞♦s n t❡r♠♦s ❞❡ ✉♠❛ P✳●✳ é❞❛❞❛ ♣♦r n a1 ♣q ✁ 1qS , 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 ◆❡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✉❡ SS ♥ã♦ é ♠❛✐♦r q✉❡ ✱ ❛ss✐♠ ❝♦♠♦ ❡①✐st❡ ♣❡❧♦ ♠❡♥♦s ✉♠ ❞❡❧❡s q✉❡ ♥ã♦ é ♠❡♥♦r q✉❡ ✳ nn ❖❜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✳ ◆♦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♦✳ ❉❡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♥✐♦✳ ✶✳ ◗✉❛❧ 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♦✳ ✶✷✳ ✭■▼❖✶✾✽✺✮✳ ❉❛❞♦ ✉♠ ❝♦♥❥✉♥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✳ ❉❡♠♦♥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♦ ❛❝✐♠❛✱ ba q ↕ ➔ q 1✱ ♦✉ s❡❥❛✱ b ☎ q ↕ a ➔ b ☎ ♣q 1q✳ ❙♦♠❛♥❞♦ ✁b ☎ q ❛ ❡st❛ ❞❡s✐❣✉❛❧❞❛❞❡✱ ♦❜t❡♠♦s br q✉❡ 0 ↕ a ✁ b ☎ q ➔ b✳ P♦rt❛♥t♦✱ ❞❡✜♥✐♥❞♦ r ✏ a ✁ b ☎ q✱ s❡❣✉❡ q✉❡ a ✏ b ☎ q ✱ ❝♦♠ ↕ r ➔ ⑤b⑤✱ ♣♦✐s b → 0✳ ◆♦ ❝❛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 aq ✏ ❡ 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✐♣❧♦ ❋❛❧❛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 ❡ ❉❡♠♦♥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✉. ❈♦r♦❧ár✐♦ ✷✳✸✳✷✳ ❉❛❞♦s a, b P Z✱ ♥ã♦ ❛♠❜♦s ♥✉❧♦s✱ t❡♠✲s❡ q✉❡✂ ✡ a bmdc , ✏ 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 ✶✳ ❆ s❡❣✉✐r ❞❡♠♦♥str❛r❡♠♦s ✉♠❛ ❝❛r❛❝t❡r✐③❛çã♦ ❞❡ ♥ú♠❡r♦s ♣r✐♠♦s ♠✉✐t♦ út✐❧ ❞♦ ♣♦♥t♦❞❡ ✈✐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✉❡ ▼✉❧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✉❡ bcb 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 ✂ ✡ b c, P♦r ♦✉tr♦ ❧❛❞♦✱ ♣❡❧♦ ❝♦r♦❧ár✐♦ ✷✳✸✳✷✱ t❡♠♦s q✉❡ mdc ✏ 1✱ ❡ mdc mdc♣b, cq ♣b, cq bb ☎ c❝❧❛r❛♠❡♥t❡ ⑤ m ♥❛ ❡q✉❛çã♦ ✷✳✸✱ ♦ q✉❡ ✐♠♣❧✐❝❛ q✉❡ ⑤ m ☎ c✳ ❈♦♠♦ mdcmdc ♣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✳ ❉❡✜♥✐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 aa ☎ 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 ✶ 1 , a 2 , n ✶ ♣❛r❛ t♦❞♦ ♠ú❧t✐♣❧♦ ❝♦♠✉♠ m ❞♦s ♥ú♠❡r♦s a ☎ ☎ ☎ , a ✱ t❡♠✲s❡ q✉❡ m ⑤ m ✳ 1 , nPr♦♣♦s✐çã♦ ✷✳✸✳✸✳ ❙❡❥❛♠ a ☎ ☎ ☎ , a ♥ú♠❡r♦s ✐♥t❡✐r♦s ♥ã♦ ♥✉❧♦s✳ ❊♥tã♦ ❡①✐st❡ ♦ ♥ú♠❡r♦ mmc1 , n ♣a ☎ ☎ ☎ , a q ❡ mmc1 , , 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, 12qé ♦ ♥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✐♠♦✳ ✹ ✻ ✶✷ ✷✷ ✸ ✻ ✷✶ ✸ ✸ ✸ ♣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✱ ❉❡♠♦♥str❛çã♦✿ ❉✐✈✐❞✐r❡♠♦s ❛ ❞❡♠♦♥str❛çã♦ ❡♠ ❞✉❛s s✐t✉❛çõ❡s✿ ♣p ⑤ a ☎ bq ♦✉ ♣p ∤ a ❡ p ∤ bq✳ ❙❡ 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♦ et❛❧ q✉❡ a ☎ b ✏ p ☎ e✳ ▼❛s p ∤ b✱ ❧♦❣♦ mdc♣p, bq ✏ 1 ❡ ♣♦rt❛♥t♦ p ⑤ a✳ 1 , n n12 n ❈♦r♦❧ár✐♦ ✷✳✹✳✶✳ ❙❡❥❛♠ p, p ☎ ☎ ☎ , p ♥ú♠❡r♦s ♣r✐♠♦s✳ ❙❡ p ⑤ p ☎ p ☎ ☎ ☎ p ✱ ❡♥tã♦ pi✏ 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 i2 ♦✉ n ✏ p ☎ p ☎ ☎ ☎ p ✱ ♦♥❞❡ p é ♣r✐♠♦ ❡ i P t1, 2, ☎ ☎ ☎ , r✉✳❙❡ N é ♣r✐♠♦✱ ♥❛❞❛ t❡♠♦s ❛ ❞❡♠♦♥str❛r✳ ❙❡ N é ✉♠ ♥ú♠❡r♦ ❝♦♠♣♦st♦✱ ❡♥tã♦ ❡①✐st❡ 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✉❡ p1 2 r1 2 s ✏ p ☎ p ☎ ☎ ☎ p ❡ q ✏ q ☎ q ☎ ☎ ☎ q ✳ P♦rt❛♥t♦✱1212 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✳ n12 r s12✏ p ☎ p ☎ ☎ ☎ p ✏ q ☎ q ☎ ☎ ☎ q ✭✷✳✹✮ i i , q , 1112 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❛❧✐❞❛❞❡✱11 r❡♦r❞❡♥❛♥❞♦ s❡ ♥❡❝❡ssár✐♦✱ t❡♠✲s❡ q✉❡ p ✏ q ✳ P♦rt❛♥t♦ ❈♦♠♦2323 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 α12 r n .12 r 1 , ✏ ✟p ☎ p ☎ ☎ ☎ p r1 α α122 α rPr♦♣♦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 ✶12 n ,12 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 pi ✱ ♣❛r❛ ❛❧❣✉♠ i P t1, ☎ ☎ ☎ , r✉✳ ❈♦♥s❡q✉❡♥t❡♠❡♥t❡✱ 0 ↕ β ↕ α ✳ P♦r ♦✉tr♦ ❧❛❞♦✱ ♣♦r s❡r α j i, ji ♣r✐♠♦ ❝♦♠ ♦s ❞❡♠❛✐s p j ✘ i, j P t1, 2, ☎ ☎ ☎ , r✉✱ s❡❣✉❡✲s❡ q✉❡ p ✏ p ✳1 α α12 2 α n1 β β1 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 δ12 γ n δ n12 mdc e mmc .1212 ♣a, bq ✏ p ☎ p ☎ ☎ ☎ p n ♣a, bq ✏ p ☎ p ☎ ☎ ☎ p n 1 γ γ1 2 n 2 n γ ❉❡♠♦♥str❛çã♦✿ ◆♦t❡ q✉❡ ♦ ♥ú♠❡r♦ p ☎ p ☎ ☎ ☎ p é ✉♠ ❞✐✈✐s♦r ❝♦♠✉♠ ❞❡ a ❡ b❀ ❧♦❣♦✱ ε ε1 2 r εγ γ1 2 n γ c i i , β i1 2 r1 2 n✏ ✟p ☎ p ☎ ☎ ☎ p ✱ ♦♥❞❡ ε ↕ mintα ✉ ❡✱ ♣♦rt❛♥t♦✱ c ⑤ p ☎ p ☎ ☎ ☎ p ✳1 δ δ1 2 n2 δ n ◆♦t❡ t❛♠❜é♠ q✉❡ ♦ ♥ú♠❡r♦ p ☎ p ☎ ☎ ☎ p é ✉♠ ♠ú❧t✐♣❧♦ ❝♦♠✉♠ ❞❡ a ❡ b✳ ❙❡❥❛ m ̟ ̟12 ̟ r1 2 i i i r , β✉♠ ♠ú❧t✐♣❧♦ ❝♦♠✉♠ ❞❡ a ❡ b❀ ❧♦❣♦✱ m ✏ ✟p ☎ p ☎ ☎ ☎ p ✱ ♦♥❞❡ ̟ ➙ maxtα ✉ ❡✱ δ δ1122 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♦✿ ✭✐✐✮ ❙❡ 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✉❡ m⑤ ♣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✉❡ m⑤ 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 ❉❡♠♦♥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✱ ♣♦✐sa 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 an 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✳ 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✉❡ ✂ ✡ ma 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✉❡ m⑤ ♣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❉♦♥❞❡✱ ✂ ✡ ma mod . ✑ b mdc♣c, mq ❘❡❝✐♣r♦❝❛♠❡♥t❡✱ s❡✂ ✡ ma 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✮✳ n1 ❉❡♠♦♥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 10s❡✱ ❡ 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 ré 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✮✳ ❉❡♠♦♥str❛çã♦✿ ❚❡♠♦s q✉❡ a✁ ♣r n ☎ ☎ ☎ r 1 r q ✏ r n ☎ 10 n ☎ ☎ ☎ r1 ☎ 10 r ✁ ♣r n ☎ ☎ ☎ r 1 r q ✁1 ☎ 102 k ✁1 ✁1 ✁ a1 ☎ ☎ ☎ ✁ a2 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 r2 ✁ r3 ☎ ☎ ☎ ✁ r 2 k ♣mod 11q❙♦♠❛♥❞♦ ♠❡♠❜r♦ ❛ ♠❡♠❜r♦✱ ❛s ❝♦♥❣r✉ê♥❝✐❛s ❛❝✐♠❛✱ t❡♠♦s q✉❡ a r2 k☎ 102 k ✑ r2 k ✁1 ♣mod 11q ✑ ✁r2 k ♣mod 11q✳✳✳ ✳✳✳ r2 k ✏ r n ☎ ♣10 n ✁ 1q ☎ ☎ ☎ r1 ☎ ♣10 ✁ 1q. ☎ 10 ✑ ✁r1 ♣mod 11q r2☎ 102 ✑ r2 ✑ r♣mod 11q r1 ✑ ✁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 ✁ r1 r2 ✁ r3 ☎ ☎ ☎ 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✉❡ 102 k ✑ 1 ♣mod 11q ❡ 102 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, a2 ké ❞✐✈✐sí✈❡❧ ♣♦r ✶✶✳ ✼✳ ✭❍✉♥❣r✐❛✮ Pr♦✈❡ q✉❡ ✺ ❞✐✈✐❞❡ 199 ✭❜✮ 2 100 é ❞✐✈✐❞✐❞♦ ♣♦r 3❀ ✾✳ ❊♥❝♦♥tr❡ ♦ r❡st♦ q✉❡ é ❞❡✐①❛❞♦ q✉❛♥❞♦✭❛✮ 2001 ☎ 2002 ☎ 2003 ☎ 2004 20052 é ❞✐✈✐❞✐❞♦ ♣♦r 7❀ 410☎ ☎ ☎ 10010 é ❞✐✈✐sí✈❡❧ ♣♦r 101✳ 310 210 ✽✳ ▼♦str❡ q✉❡ 110 599✳ 499 399 299 ✶✳ 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 a1 ✱ a2 ❡ b✱ s❡♥❞♦ b ♥ã♦ ♥✉❧♦✱ ♠♦str❡ q✉❡ b ⑤ ♣a1 ✁ a2 q s❡✱ ❡ só s❡✱ a1 ❡ a2❞❡✐①❛♠ r❡st♦s ✐❣✉❛✐s ♥❛ ❞✐✈✐sã♦ ♣♦r b✳ ✭❜✮ ❞❡✐①❛ r❡st♦ 0 ♦✉ 1 q✉❛♥❞♦ ❞✐✈✐❞✐❞♦ ♣♦ 4❀✭❝✮ ❞❡✐①❛ r❡st♦ 0✱ 1 ♦✉ 4 q✉❛♥❞♦ ❞✐✈✐❞✐❞♦ ♣♦r 8✳ ✸✳ ▼♦str❡ q✉❡ t♦❞♦ q✉❛❞r❛❞♦ ♣❡r❢❡✐t♦✿✭❛✮ ❞❡✐①❛ r❡st♦ 0 ♦✉ 1 q✉❛♥❞♦ ❞✐✈✐❞✐❞♦ ♣♦r 3❀ ✭❝✮ mdc ♣2n 1, 9n 4q ✏ 1❀✭❞✮ mdc ♣n! 1, ♣n 1q! 1q ✏ 1✳ ✭❜✮ mdc ♣n 1, n2 n1 q ✏ 1❀ ✷✳ ❙❡❥❛ n P N✳ ▼♦str❡ q✉❡✭❛✮ mdc ♣n, 2n 1q ✏ 1❀ ✭❝✮ ✺✺✶ ❡ ✽✼✹✭❞✮ ✼ ✸✷✺ ❡ ✽ ✹✽✺✳ 1 né ✉♠ ♠ú❧t✐♣❧♦ ❞❡ ✶✵✳ 2468 ✶✵✳ Pr♦✈❡ q✉❡ ♣999994q ✁ 1 é ❞✐✈✐sí✈❡❧ ♣♦r 333331✳✶✶✳ ❊♥❝♦♥tr❡ ♦ ú❧t✐♠♦ ❞í❣✐t♦ ❞♦s ♥ú♠❡r♦s 2015 ✭❛✮ 1989 77750 2✭❜✮ 777222 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❀222015 ✭❝✮ 2014 ♣♦r 7✳54n ✶✹✳ 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 22225555 ✶✼✳ Pr♦✈❡ q✉❡ 2222 é ❞✐✈✐sí✈❡❧ ♣♦r 7✳✶✽✳ ❉❡t❡r♠✐♥❡ ♦ ✈❛❧♦r ❞❡✿22 ✭❛✮ 25 ✁ 24 ❀22✭❜✮ 1975 ✁ 1974 ❀22 ✭❝✮ 555555 ✁ 444444 ❀22✭❞✮ ♣n 1q ✁ n ✳32 ✶✾✳ ❙❡❥❛ n ✏ 9867✳ ❙❡ ✈♦❝ê ❝❛❧❝✉❧❛ss❡ n ✁ n ✱ q✉❛❧ ♥ú♠❡r♦ ❡♥❝♦♥tr❛r✐❛ ♥♦ ❛❧❣❛r✐s♠♦❞❛s ✉♥✐❞❛❞❡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 2é ♠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✉❡811 m2 2 2 2 . 444 333 ✏ n 4✷✼✳ ❖ ♥ú♠❡r♦ 3 é ❞✐✈✐sí✈❡❧ ♣♦r ✺❄4 ✷✽✳ ◗✉❛✐s ♦s ❝✉❜♦s ♣❡r❢❡✐t♦s q✉❡ ❞✐✈✐❞❡♠ 9 ❄333 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✐❝❡ OOA 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 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❡ss❡❥❛♠ ❝♦♥❣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♦ AØ D BØ E CØ F ❞❡✜♥❡ ❛ ❝♦♥❣r✉ê♥❝✐❛ ❡♥tr❡ ♦s tr✐â♥❣✉❧♦s ABC ❡ DEF ✳ ❊♥tã♦ ✈❛❧❡♠✱ s✐♠✉❧tâ♥❡❛♠❡♥t❡✱ ❛s s❡❣✉✐♥t❡s r❡❧❛çõ❡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 A B C D E F✭✐✮ AC ✏ DF ❀ ❉♦♥❞❡ ❝♦♥❝❧✉í♠♦s q✉❡ ④F ✶ ✳ P♦r ♦✉tr♦ ❧❛❞♦✱♣ E✏ ④ DEF ✶ ④F ✶ EF . ✶ ❡ ♣B EF✏ 0 ❡✱ ♣♦rt❛♥t♦✱ ✉♠❛ ❝♦♥tr❛❞✐çã♦✱ ♣♦✐s F ✶ EF→ 0✳ ▲♦❣♦✱ ♥ã♦ ♣♦❞❡ ♦❝♦rr❡r ✭✐✐✮✳❙❡ ♦❝♦rr❡r ✭✐✐✐✮✱ t♦♠❡ F ✶ ♥❛ s❡♠✐rr❡t❛ ÝÝÑDF t❛❧ q✉❡ DF ✶ ✏ AC✳A B C D E F F ✏ ④DEF ✏ ♣C ✭✐✐✮ 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❛ ÝÝÑDF t❛❧ q✉❡ DF ✶ ✏ AC ❡ ❝♦♥s✐❞❡r❡ ♦ tr✐â♥❣✉❧♦ DEF ✶ ✳ ❱❡❥❛ ✜❣✉r❛ ❛❜❛✐①♦✳A B C D E F F P♦rt❛♥t♦✱ ♣❡❧♦ ❛①✐♦♠❛ ✸✳✶✳✶✱ ♦❜t❡♠♦s q✉❡✿∆ABC ✶ ✏ ∆DEF ✶ ,✉♠❛ ✈❡③ q✉❡ AB ✏ DE✱ ♣ A✏ ♣ D❡ AC ✏ DF ✶ ✳❉❡ss❛ ❢♦r♠❛✱ ① F ✶ ✶ 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 EFF 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♦sAB ✏ 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 E C B F G M ◆❛ s❡♠✐rr❡t❛ ÝÝÑBM ✏ ④BDA ✸✳✷ ❆❧❣✉♠❛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❡❥❛✱ ✏ ♣ ④ADC BAD④DAC ♠❛rq✉❡ ✉♠ ♣♦♥t♦ D t❛❧ q✉❡ BD ✏ EF ✳A B C D E F G M A✏ ④ ✳ ❉♦♥❞❡✱ ❝♦♥❝❧✉í♠♦s q✉❡♣ ✏ ④DAC ❡ ④ADC ✏ ④BDA ❡ ACD sã♦ ✐sós❝❡❧❡s✳ ▲♦❣♦✱ ④BAD ❖❜s❡r✈❡ q✉❡ ❞❡ ✸✳✶ t❡♠✲s❡ ♥❡❝❡ss❛r✐❛♠❡♥t❡ q✉❡ DC ✏ EG✳ P♦rt❛♥t♦✱ ♦s tr✐â♥❣✉❧♦sABD F✱ BD ✏ EF ❡ BC ✏ F G✳ CBD✏ ♣ ✏ ∆EF G ✭✸✳✶✮✉♠❛ ✈❡③ q✉❡ ④ ❉❡ss❛ ❢♦r♠❛✱ t❡♠♦s q✉❡∆DBC ◆❡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✳ 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✳ 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 XAYP♦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 X❋✐❣✉r❛ ✸✳✸✿ ➶♥❣✉❧♦ ❡①t❡r♥♦ ❞❡ ✉♠ tr✐â♥❣✉❧♦✳ ❉❡♠♦♥str❛çã♦✿ ❇❛st❛ ♦❜s❡r✈❛r ❛ ✜❣✉r❛ ✸✳✸✱ ❡ q✉❡ ❛ ♣r♦♣♦s✐çã♦ ✸✳✷✳✷ ♥♦s ❣❛r❛♥t❡ q✉❡ ✆ ˆ④ 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 C B D EP❡❧❛ ♣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 ◆❡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 CPr♦♣♦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 1③ ♣ ♣ ♣ ♣ ♣ AP B CBP C B C C B C . ✏ ④ ✏ ✁ ♣ ✏ 2 2▼❛s✱ ❝♦♠♦ ✁ ✠ ✁ ✠ 1 1③ ♣ ♣ ♣ 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 ❧❛❞♦✳ ✆ 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 cb 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 Ac ba 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♦ ④ABD ADB CBD ✏ ④ ➔ ④ ✳❱❡❥❛♠♦s ✉♠ ❡①❡♠♣❧♦ ♣❛r❛ ✐❧✉str❛r ❛ ❞❡s✐❣✉❛❧❞❛❞❡ tr✐❛♥❣✉❧❛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 BPPr♦❧♦♥❣✉❡ ❛ 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. ❜✮ ❱✐♠♦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♦ B B A AF igura2 F igura11 , 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♦s1 A 2 , A2 A 3 , A n1 ❝♦♥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 ❉❡♠♦♥str❛çã♦✿ ❙✉♣♦♥❤❛✱ ♣r✐♠❡✐r♦✱ q✉❡ ♦ q✉❛❞r✐❧át❡r♦ ❝♦♥✈❡①♦ ABCD ♥❛ ✜❣✉r❛ ✸✳✻ é✉♠ ♣❛r❛❧❡❧♦❣r❛♠♦✳ D C A B AD BC A Bk 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♦❝❛♠❡♥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 ❡BD ✳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✳ D C M A BP♦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✳ 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❛❧❡❧♦sAD ❡ 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 CDé ❜❛s❡ ♠é❞✐❛ ❞♦ tr✐â♥❣✉❧♦ ACD✳ P♦rt❛♥t♦ ÐÝÑ ✳ P♦r ♦✉tr♦ ❧❛❞♦✱ MQ é ❜❛s❡ ÐÑM 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❝✉❧♦ ❳❱■■■✳ 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❡♠♦sAB 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♠❛ ♠❡❞✐❞❛✳ 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 ✆ ③DAB 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 ❡ BDCACD ❡ 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 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✳ ❈♦♠♦ ✆ ✆ ✆ ✆ ③BAC 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❛♥❣♦ ❞❡ ❞✐❛❣♦♥❛✐sAC ❡ BD✳D A C M B ❋✐❣✉r❛ ✸✳✶✶✿ ▲♦s❛♥❣♦ ABCD✳ ❈♦♠♦✱ ♣♦r ❞❡✜♥✐çã♦✱ AB ✏ AD✱ CB ✏ CD ❡ AC é ✉♠ ❧❛❞♦ ❝♦♠✉♠ ❛♦s tr✐â♥❣✉❧♦sABC ❡ 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 ❉❡♠♦♥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♥❣✉❧♦ ABPé ✐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 MP 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 ÝÑP, 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 ✆ ④M OP N OP OM P ON P ✏ ④ ✱ ④ ✏ ④ ✏ 90 ❡ OP é ❧❛❞♦ ❝♦♠✉♠ ❛♦s ♠❡s♠♦s✳ ❉❛í✱ P M ✏ P N✱✁ ✠ ✁ ✠ ÝÑ ÝÑP, 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 ✶ rA A ✶ BB 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❡sBC 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 Cé ✈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 ✜❣✉r❛ ✸✳✶✸✱ t❡♠♦s q✉❡AB 3 .✏ BC 2 ✶ u u ✶ A Ar ✶ X X ✶ Y Y ✶ B B s ✶ Z Z ✶ C Ct ❋✐❣✉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✳ ✷ ❊♥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 BÐÝÑ ÐÑ ✶ ✶AB A B ❳ ✏ tC✉✳ ❙❡ ✏ ✱ ❡♥tã♦ ❛ r❡t❛ r é ♣❛r❛❧❡❧❛ à r❡t❛ s✳ ✶ BC B C tC ✶ BB s ✶ ✷ sB ✶ AA 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✳ ▲♦❣♦✱ ✭✐✮ ❙❡ 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 ✁ c⑤A 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 CÐÝÑ ✶ QA BB AQ BAX k❈♦♠♦ ÐÑ ❡ ÐÑ é ❜✐ss❡tr✐③ ❞❡ ④ ✱ ♦❜t❡♠♦s ✶ ✶ ④ABB 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 . ✏ ✏ P C AC❡ ♦ ❝❛s♦ ❡♠ q✉❡ AB → AC ♣♦❞❡ s❡r tr❛t❛❞♦ ❞❡ ♠♦❞♦ ❛♥á❧♦❣♦✳ ◗✉❛♥t♦ ❛♦ ✐t❡♠ ✭✐✐✮✱ s❡❣✉❡ ✐♠❡❞✐❛t❛♠❡♥t❡ ❞♦ ✐t❡♠ ✭✐✮✳ ❙❡♥❞♦ BP ✏ x ❡ CP ✏ y✱ x ct❡♠♦s x y ✏ a ❡✱ ♣❡❧♦ ✐t❡♠ ✭✐✮✱ ✏ ✳ y b A cb x y B C P❘❡s♦❧✈❡♥❞♦ ♦ s✐st❡♠❛ ❞❡ ❞✉❛s ❡q✉❛çõ❡s ✩ x y✬ ✏ a✬✫ x c✬✬✪ ✏ y bac 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 . A ✶ ✶ A ✶ kb kc ✶ ✶ c b ✶ ✶ ✶ B ✶C C Ba 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 CPr♦♣♦s✐çã♦ ✸✳✼✳✶✳ ❙❡❥❛♠ ABC ❡ A tr✐â♥❣✉❧♦s ♥♦ ♣❧❛♥♦ t❛✐s q✉❡ AB BC AC . ✏ ✏ ✶ ✶ ✶ ✶ ✶ ✶ A B B C A C ✶ ✶ ✶✶ ✶ B 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 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 CP♦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❡❣✉✐ré ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ ♦ ❝❛s♦ ▲❆▲ ❞❡ s❡♠❡❧❤❛♥ç❛✳ ✶ ✶ ✶ B CPr♦♣♦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 ✳ AC ✶ ✶ A A , ♣ C C❊♠ ♣❛rt✐❝✉❧❛r✱ ♣ ✏ ♣ ✏ ① ❡ ✏ k✳ ✶ ✶ A C A ✶ A ✶ kc ✶ c ✶ ✶ ✶✶ B C ka B Ca ❊ss❡ ♣ró①✐♠♦ ❝r✐tér✐♦ é ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ ❝❛s♦ ❆❆ ❞❡ s❡♠❡❧❤❛♥ç❛ ❞❡ tr✐â♥❣✉❧♦s✳ ✶ ✶ ✶ B CPr♦♣♦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 , CØ 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.22 222 c . ✭✐✐✐✮ a ✏ b2 ✭✐✈✮ h ✏ m ☎ n. P❛r❛ t♦❞❛s ❛s ❞❡♠♦♥str❛çõ❡s ✐r❡♠♦s ♥♦s ♦r✐❡♥t❛r ♣❡❧❛ ✜❣✉r❛ ✸✳✶✺✳A cb h B C n mH 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 a2 ☎ 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♣m22222 n cc 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❡♠♦s2 a☎ m ☎ n ✏ ♣b ☎ cq ♦✉ ❛✐♥❞❛✱2 ✂ ✡ b2☎ 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♦✳ ✸ ❖ ✐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 Da a A B❋✐❣✉r❛ ✸✳✶✻✿ ❞✐❛❣♦♥❛❧ ❞❡ ✉♠ q✉❛❞r❛❞♦ ❡♠ ❢✉♥çã♦ ❞❡ s❡✉ ❧❛❞♦✳ ❉❛í✱ ♦ t❡♦r❡♠❛ ❞❡ P✐tá❣♦r❛s ❢♦r♥❡❝❡❄22 AC AB BC ✏❄22 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✳ ✸ A aa 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❡♠♦s222 AM✏ AC ✁ CM2 ✁ ✠2 a✏ a ✁2 23a .✏ 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❛❞♦✳ BCPr♦♣♦s✐çã♦ ✸✳✽✳✶✳ ❙❡❥❛ ABC ✉♠ tr✐â♥❣✉❧♦ ❡ P ❡ Q ♣♦♥t♦s s♦❜r❡ ❛ r❡t❛ ÐÑ ✱ ❝♦♠ P P BC❡ Q ❘ BC✳ ❙❡ BP BQ ✆ ③P 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♦sAB AC ❞❡ ✐♥t❡rs❡çã♦ ❝♦♠ ❛s r❡t❛s ÐÑ ❡ ÐÑ ✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳ 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 APk 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✳ 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♦✳ ④P CD DP✏ AB ☎ CD AD ☎ BC ✏ BD. AC .❈♦♥❝❧✉í✲s❡ q✉❡ AC AD☎ BC AB☎ CD BD✏ AC .❊♥tã♦✱ AD☎ BC AC❡ BP ✏ AB☎ CD BP✱ t❡♠♦s ❞❛s ❡q✉❛çõ❡s ✸✳✻ ❡ ✸✳✼ q✉❡ ✏ ③ACB ✭✸✳✼✮❈♦♠♦ BD ✏ DP ✏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❝♦ ②BC ✏ ④BDC ❈♦♠♦ ③BAC ✳A B C D O P ❆ s❡❣✉✐r✱ ✉♠❛ ❝♦♥s❡q✉ê♥❝✐❛ ✐♥t❡r❡ss❛♥t❡ ❞♦ t❡♦r❡♠❛ ❞❡ Pt♦❧♦♠❡✉✳ ❈♦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 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 ❡ ♦sON 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 AOM 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♥❣✉❧♦✳ ✶ Mé ♦ ♣♦♥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 Mé ✉♠ ♣❛r❛❧❡❧♦❣r❛♠♦ ❡✱ ♣♦rt❛♥t♦✱ A ✏ AO ✏ R✳ ✶✶ HH 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❞✐♦ ❞❡ ✶ ✶ HOM ✱ ❝♦♥❝❧✉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♦222 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 cb h dk B C H D a m na ❋✐❣✉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❛❧ ❞❛ ♣r♦❥❡çã♦ ❞❛ ❝❡✈✐❛♥❛ AD s♦❜r❡ ♦ ❧❛❞♦ BC✳ ◆♦ tr✐â♥❣✉❧♦ ABH a ✱ ✈❛❧❡♠ ❛s r❡❧❛çõ❡s✱♣r♦✈❡♥✐❡♥t❡s ❞♦ t❡♦r❡♠❛ ❞❡ P✐tá❣♦r❛s✿ c2 ✏ ♣m ✁ kq2 h2❡ h2 ✏ d2 ✁ k2 . ▲♦❣♦✱ c2✏ m2 ✁ 2 ☎ m ☎ k k2 d2 ✁ k2, ❞♦♥❞❡ c2✏ m2 d2 ✁ 2 ☎ m ☎ k ✭✸✳✽✮Pr♦❝❡❞❡♥❞♦ ❞❛ ♠❡s♠❛ ❢♦r♠❛ ♥♦ tr✐â♥❣✉❧♦ ACH a ✱ t❡♠♦s b2 ✏ ♣n kq2 h2❡ h2 ✏ d2 ✁ k2 . ▲♦❣♦✱ b2✏ n2 2☎ n ☎ k k2 d2 ✁ k2, ❞❛í b2✏ n2 d2 2☎ n ☎ k ✭✸✳✾✮ ▼✉❧t✐♣❧✐❝❛♥❞♦ ✸✳✽ ♣♦r n ❡ ✸✳✾ ♣♦r m✱ ♦❜t❡♠♦s✿ c2☎ n ✏ m2 ☎ n d2 ☎ n ✁ 2 ☎ m ☎ n ☎ k ❡ b2☎ m ✏ n2 ☎ m d2 ☎ m 2 ☎ m ☎ n ☎ k. ❙♦♠❛♥❞♦✲s❡ ♦s ❞♦✐s ♠❡♠❜r♦s ❞❛s ✐❣✉❛❧❞❛❞❡s✱ ✈❡♠✿ c2☎ n b2 ☎ m ✏ m ☎ n ☎ ♣m nq d2 ☎ ♣m nq.❈♦♠♦ m n ✏ a✱ t❡♠♦s✿ c2 ☎ n b2 ☎ m ✁ d2 ☎ a ✏ a ☎ m ☎ n. ❱❡❥❛♠♦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 a22 b c ☎ ☎ ✁ m a ☎ a ✏ a ☎ ☎ 2 2 2 2♦✉222 b c a2 a .✁ m ✏ 2 2 b c 4❆♥❛❧♦❣❛♠❡♥t❡✱ ♦❜tê♠✲s❡ ♣❛r❛ ❛s ♠❡❞✐❛♠❛s m ❡ m ✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✿222222 a c b a b c22 e .✁ m b ✏ ✁ m c ✏ 2 2 4 2 2 4❙♦♠❛♥❞♦✲s❡ ❛s três ú❧t✐♠❛s ❡q✉❛çõ❡s✱ ♦❜t❡♠♦s222 3222 m m m b ca 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 BT11 T2 2 C . B T1 T2 A11 C❙❡ AT ✏ d ✱ ❛♣❧✐❝❛♥❞♦✱ t❛♠❜é♠✱ ♦ ❚❡♦r❡♠❛ ❞❡ ❙t❡✇❛rt ♥♦ tr✐â♥❣✉❧♦ ABC ♣❛r❛ ❛1 ❝❡✈✐❛♥❛ AT ✱ ♦❜t❡♠♦s2222 a 2a a 2a b 2c 2a222 b c11 .☎ ☎ ✁ d ☎ a ✏ a ☎ ☎ ô ✁ d ✏ 32 32 3 3 3 3 9❆♥❛❧♦❣❛♠❡♥t❡✱ s❡ AT ✏ d ✱ ❛♣❧✐❝❛♥❞♦✲s❡ ♦ ❚❡♦r❡♠❛ ❞❡ ❙t❡✇❛rt ♥♦ tr✐â♥❣✉❧♦ ABC2 ♣❛r❛ ❛ ❝❡✈✐❛♥❛ AT ✱ ♦❜t❡♠♦s✿2222b c 2a22 .✁ d ✏ 3 3 9❙♦♠❛♥❞♦✲s❡ ❛s ❞✉❛s ú❧t✐♠❛s ❡q✉❛çõ❡s✱ ♦❜t❡♠♦s22222 4a b c .12 ✁ d ✁ d ✏2 922 cP♦rt❛♥t♦✱ ♣❡❧♦ ❚❡♦r❡♠❛ ❞❡ P✐tá❣♦r❛s✱ ♦❜t❡♠♦s b ✏ a ✳ ▲♦❣♦✱2222 4a a ,12 ✁ d ✁ d ✏ 9♦✉ s❡❥❛✱222 5a d d .12 ✏ 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♦✳ ❆❞♦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 ✉♥✐❞❛❞❡s2 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❡❛ ❞♦ P❛r❛❧❡❧♦❣r❛♠♦ ❈♦♥s✐❞❡r❡♠♦s ♦ ♣❛r❛❧❡❧♦❣r❛♠♦ ABCD✱ ♥❛ ✜❣✉r❛ ❛❜❛✐①♦✱ ❝♦♠ ❜❛s❡ AB ✏ a ❡ ❛❧t✉r❛ h✳ D C ha 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 ha 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✳ A D ha 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❡ ♠❡♥♦rAD ✏ b ❡ ❛❧t✉r❛ h✱ ❝♦♠♦ ♥❛ ✜❣✉r❛✳ bA 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 S✏ b ☎ h ✶ ❈♦♠♦ 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 S1 S2 S1✏ S2 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 ✈ért✐❝❡ ♣❡r❝♦rr❡ ✉♠❛ r❡t❛ ♣❛r❛❧❡❧❛ à ❜❛s❡✳B A C A ♣a ✁ bq ☎ h Pr♦♣r✐❡❞❛❞❡ ✶❆ ár❡❛ ❞❡ ✉♠ tr✐â♥❣✉❧♦ ♥ã♦ s❡ ❛❧t❡r❛ q✉❛♥❞♦ s✉❛ ❜❛s❡ ♣❡r♠❛♥❡❝❡ ✜①❛ ❡ ♦ t❡r❝❡✐r♦ ✸✳✾✳✹ Pr♦♣r✐❡❞❛❞❡s ●❡♦♠étr✐❝❛s ■♠♣♦rt❛♥t❡s 2 . ♣a bq ☎ h 2✏ 2b☎ h a ☎ h ✁ b ☎ h 2✏ ❱❡❥❛♠♦s✱ ❛❣♦r❛✱ ✉♠❛ ❛♣❧✐❝❛çã♦ ❞❡ss❛ ♣r♦♣r✐❡❞❛❞❡ ❛tr❛✈és ❞❡ ✉♠ ❡①❡♠♣❧♦ s✐♠♣❧❡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♦❧✉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♦❜❧❡♠❛✳ Pr♦♣r✐❡❞❛❞❡ ✸❙❡ ❞♦✐s tr✐â♥❣✉❧♦s tê♠ ♠❡s♠❛ ❛❧t✉r❛✱ ❡♥tã♦ ❛ r❛③ã♦ ❡♥tr❡ s✉❛s ár❡❛s é ✐❣✉❛❧ ❛ r❛③ã♦ ❡♥tr❡ s✉❛s ❜❛s❡s✳❆ ❛✜r♠❛çã♦ t❡♠ ❝♦♠♣r♦✈❛çã♦ ✐♠❡❞✐❛t❛ ❛ ♣❛rt✐r ❞❛ ❢ór♠✉❧❛ q✉❡ ❞❡✜♥❡ ❛ ár❡❛ ❞♦ tr✐â♥❣✉❧♦✳ A B C S1 S2 a1 a2 hD ❋✐❣✉r❛ ✸✳✷✾✿ tr✐â♥❣✉❧♦s ABD ❡ ADC tê♠ ♠❡s♠❛ ❛❧t✉r❛ ❡ ❜❛s❡s ❞✐❢❡r❡♥t❡s✳❈♦♠♦ S1 ✏ a1 ☎ h 2❡ S2 ✏ a2 ☎ h 2✱ t❡♠♦s q✉❡ ❛ r❡❧❛çã♦ ❡♥tr❡ ❛s ár❡❛s S1 ❡ S2 é ❞❛❞❛ ♣♦r S1 S2✏ a1 ☎ h 2 a2☎ h 2✏ a1 a2 . ❆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 ❡ ❛❧t✉r❛s h1 ❡ h2 ✳❙❡♥❞♦ ♦s tr✐â♥❣✉❧♦s s❡♠❡❧❤❛♥t❡s✱ ❡♥tã♦ ❛ r❛③ã♦ ❡♥tr❡ ❛s ❜❛s❡s é ❛ ♠❡s♠❛ r❛③ã♦ ❡♥tr❡ D Ah2 h1 B C E F ab ❛s ❛❧t✉r❛s✳ ❊ss❛ r❛③ã♦ é ✉♠ ♥ú♠❡r♦ q✉❡ ❝❤❛♠❛♠♦s ❞❡ r❛③ã♦ ❞❡ s❡♠❡❧❤❛♥ç❛✳ a h1 ✏ ✏ k.1 b h22❋❛③❡♥❞♦ ❛ r❛③ã♦ ❡♥tr❡s ❛s ár❡❛s S ❡ S ❞❡ss❡s tr✐â♥❣✉❧♦s✱ t❡♠♦s a1 ☎ hS 1 a h12 2✏ ✏ ☎ ✏ k ☎ k ✏ k b2 S 2 b h2 ☎ 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✳ ❯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 ha O R n ❆ ár❡❛ S ❞❡ss❡ ♣♦❧í❣♦♥♦ é ❞❛❞♦ ♣♦r a pn☎ 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❡♥❞♦ nn ❛r❜✐tr❛r✐❛♠❡♥t❡ ❣r❛♥❞❡ t❡♠♦s q✉❡ ❛ ár❡❛ S ❛♣r♦①✐♠❛✲s❡ ❞❛ ár❡❛ S ❞♦ ❝ír❝✉❧♦ ❞❡ r❛✐♦R ✳ ❖✉ s❡❥❛✱2π2 ☎ 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❛✉sα2 S , B R Lα A O✭✐✐✮ q✉❛♥❞♦ ♦s â♥❣✉❧♦s sã♦ ♠❡❞✐❞♦s ❡♠ r❛❞✐❛♥♦s α2 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 L2 ☎ 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 ✶✳ ❙❡ 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✉❡ÐÑDE 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 βγ 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 ✆ ③B 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 ✆ ③BI 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 ✏ ♣ ✁ ♣ ✳ ✶✵✳ ❈❛❧❝✉❧❡ ❛ s♦♠❛ ❞♦s â♥❣✉❧♦s ♥♦s ✈ért✐❝❡s A, B, C, D, e E ❞❛ ❡str❡❧❛ ❞❡ ❝✐♥❝♦ ♣♦♥t❛s❞❛ ✜❣✉r❛ ❛❜❛✐①♦✳ A D C B E④ ④ BAC ACD ABD.✶✶✳ ◆❛ ✜❣✉r❛ ❛❜❛✐①♦✱ ♣r♦✈❡ q✉❡ α ✏ ③ C Bα 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✳ C Bδ βα γ12 A D✶✹✳ ❙❡❥❛♠ P ❡ P ♣♦♥t♦s ♥♦ ✐♥t❡r✐♦r ❞❡ ✉♠ s❡❣♠❡♥t♦ AB✱ t❛✐s q✉❡ AP1 AP2✏12 P1 B P2 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 xG ✆ ✆ 75 65 A D✶✼✳ ❆ ✜❣✉r❛ ♠♦str❛ ✉♠ r❡tâ♥❣✉❧♦ ❡ s✉❛s ❞✐❛❣♦♥❛✐s✳ ◗✉❛❧ ❛ r❡❧❛çã♦ ❡♥tr❡ ♦s â♠❣✉❧♦s x ❡ y ❞❛ ✜❣✉r❛❄ yx ✶✽✳ ◆❛ ✜❣✉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❡ ❡ ◗✉❛❧ 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 M2 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 B12 E ✭❛✮ ▼♦str❡ q✉❡ s❡ L é ❛ r❡t❛ q✉❡ ♣❛ss❛ ♣♦r AC ❡ L é ❛ r❡t❛ q✉❡ ♣❛ss❛ ♣♦r BO✱ ❡♥tã♦L12 ❡ L sã♦ ♣❛r❛❧❡❧❛s✳2✭❜✮ ❙❛❜❡♥❞♦ q✉❡ ❛ ár❡❛ ❞♦ q✉❛❞r❛❞♦ ABCD é ✐❣✉❛❧ ❛ 36 cm ✱ ❝❛❧❝✉❧❡ ❛ ár❡❛ ❞♦ tr✐â♥❣✉❧♦ ACO✳ ✷✹✳ ◆❛ ✜❣✉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✳ 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 ha 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❡rr❡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✐♠✿11 ❙❡ ✉♠❛ ❞❡❝✐sã♦ D ♣♦❞❡ s❡r t♦♠❛❞❛ ❞❡ m ♠♦❞♦s✱ ❡ q✉❛❧q✉❡r q✉❡ s❡❥❛ ❡ss❛ ❡s❝♦❧❤❛✱22n❛ ❞❡❝✐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 , n1 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 ❛ 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✳ ✶✳ ❊♠ ✉♠❛ 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❞❡❞♦r③❡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 1F 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♠✉❧❛222 n ☎ ♣n 1q ☎ ♣2n 1qP 2 . ♣nq : 1 ☎ ☎ ☎ n ✏ 6◆♦t❡ q✉❡ 2 ☎ ♣1 1q ☎ ♣2 ☎ 1 1q 1 P♣1q : 1 ✏ 6é ✈❡r❞❛❞❡✐r❛✳❙✉♣♦♥❤❛ q✉❡ ♣❛r❛ ❛❧❣✉♠ n P N✱ s❡ t❡♥❤❛ q✉❡ P ♣nq é ✈❡r❞❛❞❡✐r❛✱ ✐st♦ é✱222 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❛✳ ❉❡ ❢❛t♦✱ s♦♠❛♥❞♦ ♣n 1q2 ❛ ❛♠❜♦s ♦s ❧❛❞♦s ❞❛ ✐❣✉❛❧❞❛❞❡ ❛❝✐♠❛✱ t❡♠♦s q✉❡ 6✏ ♣n 1q 2☎ ☎ ☎ n ✏ n 1 ♣1q : 13 ✏ 12é ✈❡r❞❛❞❡✐r❛ ❡ q✉❡ ❛ ✐❣✉❛❧❞❛❞❡ P❡r❝❡❜❛ q✉❡P 23☎ ☎ ☎ n3 ✏ ♣1 2 ☎ ☎ ☎ nq2 . 13 ✭❜✮ ❉❛ ♠❡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 ♣n 1q ☎ r2 ☎ ♣n 1q2 ♣n 1q 2♣n 1q 1s 12 6✏ 2 q ♣n 1q ♣2n 2q 1s ♣n 1q ☎ r♣2n24n 6✏ ♣n 1q ☎ r♣2n2 n 6n6 qs 6✏ ♣n 1q ☎ rn ☎ ♣2n 1q 6 ☎ ♣n 1qs 6✏ ✏ n☎ ♣n 1q ☎ ♣2n 1q 6 ☎ ♣n 1q2 6♣n 1q2 ☎ ♣n 1q ☎ ♣2n 1q 22☎ ☎ ☎ n2 ♣n 1q2 ✏ n 2é ✈á❧✐❞❛✳❙✉♣♦♥❤❛✱ ❡♥tã♦ q✉❡ ♣❛r❛ ❛❧❣✉♠ n P N ❛ ❡①♣r❡ssã♦ P ♣nq s❡❥❛ ✈❡r❞❛❞❡✐r❛✳ ❙♦♠❛♥❞♦ ♣n 1q3 ❡♠ ❛♠❜♦s ♠❡♠❜r♦s ❞❛ ✐❣✉❛❧❞❛❞❡✱ t❡♠♦s q✉❡ ☎ ☎ n2 ✏ ♣✁1q n ✁1 ☎ n☎ ♣n 1q 2 .◆♦t❡ q✉❡ P♣1q : 1 ✏ ♣✁1q1 ✁1 1☎ ♣1 1q 32✁ ☎ ☎ ☎ ♣✁1q n 2é ✈❡r❞❛❞❡✐r❛✳❙✉♣♦♥❤❛ q✉❡ ♣❛r❛ ❛❧❣✉♠ n P N✱ s❡ t❡♥❤❛ q✉❡ P ♣nq é ✈❡r❞❛❞❡✐r❛✱ ✐st♦ é✱ 1✁ 22 32✁ ☎ ☎ ☎ ♣✁1q n ✁1 ☎ n2 ✏ ♣✁1q n ✁1 ☎ n☎ ♣n 1q ✁1 ♣nq : 1 ✁ 22 13 4✏ 23☎ ☎ ☎ n3 ♣n 1q3 ✏ ✒ n☎ ♣n 1q 2✚2 ♣n 1q3✏ n2 ☎ ♣n 1q2 4♣n 1q3 ♣n 1q2 ☎ rn2 ✭❝✮ ◗✉❡r❡♠♦s ✈❛❧✐❞❛r ❛ ❢ór♠✉❧❛P 4☎ ♣n 1qs 4✏ ♣n 1q2 ☎ ♣n 2q2 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❛✳ n 2 ❉❡ ❢❛t♦✱ s♦♠❛♥❞♦ ♣✁1q ☎ ♣n 1q ❛ ❛♠❜♦s ♦s ♠❡♠❜r♦s ❞❛ ✐❣✉❛❧❞❛❞❡✱ t❡♠♦s q✉❡2 n n ☎ ♣n 1q n2 n2 ✁1 1✁ 2 ☎ ☎ ☎ ♣✁1q ☎ ♣n 1q ✏ ♣✁1q ☎ ♣✁1q ☎ ♣n 1q n 2 n2♣✁1q ☎ ♣n 1q n ✏ ☎ ♣✁1q ☎ ♣n 1q 2♣✁1q ✒ ✚ n2 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✉❡3 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❛✳ ❉❡ ❢❛t♦✱ s✉❜st✐t✉✐♥❞♦ n ♣♦r n 1✱ t❡♠♦s q✉❡32♣n 1q ✁ ♣n 1q ✏ ♣n 1q ☎ r♣n 1q ✁ 1s2 2n✏ ♣n 1q ☎ rn s322 2n n 2n✏ n3222 3n n 3n✏ n ✁ n ✁ n3 ✏ ♣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♦❞♦ nP N✳ ✭❜✮ ◗✉❡r❡♠♦s ♣r♦✈❛r q✉❡ nP ♣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 k2 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❡♠♦s2 k222 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✐♠✱ ♣❡❧♦ nPr✐♥❝í♣✐♦ ❞❛ ■♥❞✉çã♦ ❋✐♥✐t❛ q✉❡ 5 ✁ 1 é ✉♠ ♠ú❧t✐♣❧♦ ❞❡ 24✱ ♣❛r❛ t♦❞♦ n P N✱ ❝♦♠ ✭❝✮ ◗✉❡r❡♠♦s ♣r♦✈❛r q✉❡ nP 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 k2 ✁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❡♠♦s2 k222 ✁1 2 12 k 1 ☎ 2 ☎ 2 ✏ 3 ☎ t ☎ 2 22 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❛✱ ♣♦✐s2 1 .2 ✏ 1❙✉♣♦♥❤❛ P ♣nq ✏ n ✈❡r❞❛❞❡ ♣❛r❛ ❛❧❣✉♠ n P N✱ ✐st♦ é✱2 1 3☎ ☎ ☎ ♣2n ✁ 1q ✏ n2 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♦ í♠♣❛r ❛♣ós 2n ✁ 1✱ t❡♠♦s q✉❡2 1 3☎ ☎ ☎ ♣2n ✁ 1q ♣2n 1q ✏ n ♣2n 1q2 ✏ ♣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 11 ✟ P . ♣1q : 1 ✏ ☎ 10 ✁ 1 ✁ 81 9❙✉♣♦♥❞♦ P ♣nq ✈❡r❞❛❞❡ ♣❛r❛ ❛❧❣✉♠ n P N✱ ♦✉ s❡❥❛ 10 nn 1 11 1☎ ☎ ☎ ☎ ☎ ☎ 1 ✏ ☎ ♣10 ✁ 1q ✁ ❧♦♦♠♦♦♥ n uns 81 9é ✈á❧✐❞❛✱ ❞❡✈❡♠♦s ♠♦str❛r q✉❡ P ♣n 1q é ✈❡r❞❛❞❡✐r❛✳ ❆❝r❡s❝❡♥❞♦ 11 ☎ ☎ ☎ 1❧♦♦♠♦♦♥ n 1 uns 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 ✏ 2é ✈❡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♦ ❛ ❛♠❜♦s ♦s ♠❡♠❜r♦s✱ t❡♠♦s q✉❡ 10 ✏ ♣n 1q uns ❧♦♦♠♦♦♥ 11☎ ☎ ☎ 1 9 81☎ ♣10 n ✁ 1q ✁ n ✏ 81☎ ♣10 n ✁ 1q ✁ n ♣n 1q uns ❧♦♦♠♦♦♥ 11☎ ☎ ☎ 1 ❧♦♦♠♦♦♥ n uns 1☎ ☎ ☎ 1 1☎ ☎ ☎ 10 9 9, ❱✐st♦ q✉❡ 10 n 1✁ 1 S✏ ✏ 1 10 ☎ ☎ ☎ 10 né ✉♠❛ ♣r♦❣r❡ssã♦ ❣❡♦♠étr✐❝❛ ❞❡ ♣r✐♠❡✐r♦ t❡r♠♦ ✐❣✉❛❧ ❛ ✶ ❡ r❛③ã♦ ✐❣✉❛❧ ❛ ✶✵✱ t❡♠♦s q✉❡ s✉❛ s♦♠❛ é ✐❣✉❛❧ ❛ ♣n 1q uns ❧♦♦♠♦♦♥ 11☎ ☎ ☎ 1 ☞ ✌. 1 ☞✌ ♣n 1q uns ❧♦♦♠♦♦♥ ☎✆ 11 ☎ ☎ ☎ 1 ☎✆9 ☎ 9☎ ♣♦✐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✉❡ ♣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 ✏ ✁ 12 2 n✁ 3n 2n ✁ 2 ✏2 2 n✁ n ✁ 2 ✏ 2♣n 1q ☎ ♣n ✁ 2q .✏ n1 2✼✳ ◆♦t❡ q✉❡ ❛ ❢ór♠✉❧❛ a ✏ a ♣n ✁ 1q ☎ r é ✈❡r❞❛❞❡✐r❛ ♣❛r❛ n ✏ 1✱ ♦♥❞❡ t❡♠♦s q✉❡ a11 ✏ a ♣1 ✁ 1q ☎ r. ❙✉♣♦♥❞♦✱ ❛❣♦r❛✱ q✉❡ ❛ ❢ór♠✉❧❛ s❡❥❛ ✈❡r❞❛❞❡ ♣❛r❛ ❛❧❣✉♠ n P N✱ s♦♠❛♥❞♦ r ❛ ❛♠❜♦s♦s ❧❛❞♦s ❞❛ ✐❣✉❛❧❞❛❞❡✱ s❡❣✉❡11 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 a1 2♣a q ☎ 1 P1 1 .♣1q : S ✏ ✏ a 2 1 a n n ♣a q ☎ n ❛ 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 ☎ nS a n 1 n 1 ✏ 2 a1 n n 1☎ n a ☎ n 2 ☎ a .✏ n n 11 1 n 2❙❡♥❞♦✱ a ✏ a ♣n ✁ 1q ☎ r ❡ a ✏ a ☎ r✱ s❡❣✉❡ q✉❡2 a111 2☎ n a ☎ n n ☎ r ✁ r ☎ n 2 ☎ a ☎ n ☎ r S n 1✏2 2 21 1 n☎ a ☎ n 2 ☎ a ☎ r n ☎ r ✏ 2 21☎ a ☎ ♣n 1q n ☎ r ☎ ♣n 1q ✏ 1 a 1 n 2♣a ☎ rq ☎ ♣n 1q ✏ 1 a n 1 2♣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✳ nn1 ✁1 ✾✳ ❖❜s❡r✈❡ q✉❡ P ♣1q é ✈❡r❞❛❞❡✐r❛ ♣❛r❛ a ✏ a ☎ q ✱ ♣♦✐s1 ✁1 a1 1 . ✏ a ☎ q nn1 ✁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 11 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♦❞♦ ✶✵✳ ❖❜s❡r✈❡ q✉❡ P ♣1q é ✈❡r❞❛❞❡✐r❛ ♣❛r❛ S ✏ ✱ ♣♦✐s q1 ✁ 1 a1☎ ♣q ✁ 1q S1 1 . ✏ ✏ a qn 11 n ✁ 1 1❖❜s❡r✈❡ q✉❡ a ✏ a ☎ q ❡ q✉❡ ♣❛r❛ ♦❜t❡r ❛ s♦♠❛ ❞♦s ♣n q t❡r♠♦s✱ t❡♠♦sS n 1 n a n 1 ✏ S ✳ ▲♦❣♦✱ n a1☎ ♣q ✁ 1q n S a n 11✏ ☎ q q ✁ 1 n n n a1 1 a11☎ q ✁ a ☎ q ☎ q ✁ a ☎ q ✏ qn 1 ✁ 1 a1☎ ♣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✳ ✷✳ ❚❡♥❞♦ 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 ♣❛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❛♠ ♣❡❧♦ ♠❡♥♦s1985 ✁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 ✶✻ ♠❛✲ ✈✐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 ➙ 1é ✉♠ ♠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, m2, ☎ ☎ ☎ , 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♦s0, 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 j1 ✏ 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 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❡♠♦s22 ✶ ,22 ♣3kq ✏ 9k ✏ 3k ✶ ♦♥❞❡ k ✏ 3k ✱ ❡ ♣3kq ❞❡✐①❛ r❡st♦ ③❡r♦ q✉❛♥❞♦ ❞✐✈✐❞✐❞♦ ♣♦r 3✳ ❊❧❡✈❛♥❞♦ ❛♦ q✉❛❞r❛❞♦ 3k 1✱ t❡♠♦s♣3k 1q2 ✏ 9k2 ✏ 3k24k 8k, 8k 1, 8k 2, 8k 3, 8k 4, 8k 5, 8k P♦rt❛♥t♦✱ t♦❞♦ ♥ú♠❡r♦ q✉❛❞r❛❞♦ ♣❡r❢❡✐t♦ ❞❡✐①❛ r❡st♦ 0 ♦✉ 1 q✉❛♥❞♦ ❞✐✈✐❞✐❞♦ ♣♦r 4✳❝✮ ❉❛ ♠❡s♠❛ ❢♦r♠❛✱ ❡❧❡✈❛✲s❡ ❛♦ q✉❛❞r❛❞♦ ❛s ❡①♣r❡ssõ❡s ♣4k 3q2 ✏ 16k2 24k 9, deixa resto 1 quando dividido por 4. ♣4k 2q2 ✏ 16k2 16k 4, deixa resto 0 quando dividido por 4. ♣4k 1q2 ✏ 16k2 8k 1, deixa resto 1 quando dividido por 4. ♣4kq2 ✏ 16k2 , 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 2q2 ❞❡✐①❛ r❡st♦ 1 q✉❛♥❞♦ ❞✐✈✐❞✐❞♦ ♣♦r 3✳ ✶ 6k 1,♦♥❞❡ k ✶ 4✏ 3k ♣3k 2q2 ✏ 9k212k ❡ ♣3k 1q2 ❞❡✐①❛ r❡st♦ 1 q✉❛♥❞♦ ❞✐✈✐❞✐❞♦ ♣♦r 3✳❊❧❡✈❛♥❞♦ ❛♦ q✉❛❞r❛❞♦ 3k 2✱ t❡♠♦s ✏ 3k22k ✶ 1,♦♥❞❡ k ✶ 1✏ 3k 6 ❡8k 7.♣8kq2 ✏ 64k2 , deixa resto 0 quando dividido por 8.♣8k 1q2 ✏ 64k2 16k 1, deixa resto 1 quando dividido por 8.♣8k 2q2 ✏ 64k2 32k 4, deixa resto 4 quando dividido por 8.♣8k 3q2 ✏ 64k2 48k 9, deixa resto 1 quando dividido por 8.♣8k 4q2 ✏ 64k2 64k 16, deixa resto 0 quando dividido por 8.♣8k 5q2 ✏ 64k2 80k 25, deixa resto 1 quando dividido por 8.♣8k 6q2 ✏ 64k2 96k 36, deixa resto 4 quando dividido por 8.♣8k 7q2 ✏ 64k2 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✳ ✹✳ ▼♦str❛r❡♠♦s✱ ♣r✐♠❡✐r♦✱ ❛ ✐❞❛✱ ♦✉ s❡❥❛ b121 1 r 1 e a2 2 r 2 , com r1 2 .⑤ ♣a ✁ a q ñ a ✏ b ☎ q ✏ b ☎ q ✏ r1 1 r12 2 r212 ❙✉♣♦♥❤❛ q✉❡ a ✏ b ☎ q ❡ a ✏ b ☎ q ✱ ❝♦♠ r ➔ b ❡ r ➔ b✱ ♠♦str❛r❡♠♦s q✉❡ r12✏ r ✳ ❙❡♥❞♦ a12 1 r1221212✁ a ✏ b ☎ q ✁ b ☎ q ✁ r ✏ b ☎ ♣q ✁ q q ♣r ✁ r q, ♠❛s b121212⑤ ♣a ✁ a q, b ⑤ b ☎ ♣q ✁ q q ❡ ⑤r ✁ r ⑤ ➔ b, ❧♦❣♦ r121 2 .12 ✁ r ✏ 0 ñ r ✏ rP♦rt❛♥t♦✱ a ❡ a ❞❡✐①❛♠ r❡st♦s ✐❣✉❛✐s ♥❛ ❞✐✈✐sã♦ ♣♦r b✳❘❡❝✐♣r♦❝❛♠❡♥t❡✱ ♠♦str❛r❡♠♦s ❛ ✈♦❧t❛✱ ♦✉ s❡❥❛ a1 1 r12 2 r 2 , com r1212 ✏ b ☎ q ❡ a ✏ b ☎ q ✏ r ñ b ⑤ ♣a ✁ a q. ❙❡♥❞♦✱ a1212121 2 ,✁ a ✏ b♣q ✁ q q r ✁ r ❡ r ✏ r ❡♥tã♦✱ a121212 ✁ a ✏ b♣q ✁ q q. P♦rt❛♥t♦✱ b ⑤ ♣a ✁ a q✳✺✳ ❙❡1 n 2 n2 α n a n a n n . ✁1 ✏ a ☎ ☎ ☎ a ✏ ♣a ☎ 10 q ☎ ☎ ☎ a ☎ 10 ☎ a ☎ ☎ ☎ a2❆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♦ ❛♦2222222 ♦ ❡❧❡✈❛r♠♦s ❛♦ q✉❛❞r❛❞♦✳ 0 ✏ 0, 1 ✏ 1, 2 ✏ 4, 3 ✏ 9, 4 ✏ 16, 5 ✏ 25, 6 ✏22236, 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♦ ❞❡ 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, ✂ ✡224998 mod 5 2✑ ✁1 ♣mod 5q ñ ♣2 q ✑ ✁1 ♣mod 5q ñ 2 ☎ 2 ✑ ✁1 ☎ 2 mdc99 ♣5, 2q2 ñ 2 ✑ ✁2 ♣mod 5q,9899 3✑ ✁1 ♣mod 5q ñ 3 ✑ ✁1 ♣mod 5q ñ 3 ✑ ✁3 ♣mod 5q,9899 4✑ ✁1 ♣mod 5q ñ 4 ✑ 1 ♣mod 5q ñ 4 ✑ 4 ♣mod 5q ❡99 5 ✑ 0 ♣mod 5q.❆❣♦r❛✱ t❡♠♦s9999999999 1 2 3 4 5999999 ✑ 1 ♣✁2q ♣✁3q 4 0 ♣mod 5q.9999 2 3 4 5 P♦rt❛♥t♦✱ ♦ ♥ú♠❡r♦ 1 é ❞✐✈✐sí✈❡❧ ♣♦r 5✳ ✽✳ ❙❡❥❛ S ✏ 110 110✑ 610 ✑ 1410 ✑ 1710 ✑ 3610 ✑ 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❛❞♦ ❡♠ 20052 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; 310✑ 71 ✑ 1810 ✑ 4210 ✑ 5010 ✑ ✁36 ♣mod 101q 510✑ 1610 ✑ 2210 ✑ 3010 ✑ 3110 ✑ 36 ♣mod 101q 910✑ 2110 ✑ 2510 ✑ 4710 ✑ 4910 ✑ ✁17 ♣mod 101q 810✑ 1110 ✑ 1510 ✑ 3510 ✑ 4810 ✑ 17 ♣mod 101q 1310✑ 1910 ✑ 2010 ✑ 2310 ✑ 3710 ✑ ✁14 ♣mod 101q 210✑ 1210 ✑ 2810 ✑ 2910 ✑ 3410 ✑ 14 ♣mod 101q 410✑ 2410 ✑ 3310 ✑ 4310 ✑ 4510 ✑ ✁6 ♣mod 101q 2610✑ 2710 ✑ 3810 ✑ 4010 ✑ 4610 ✑ 6 ♣mod 101q 1010✑ 3210 ✑ 3910 ✑ 4110 ✑ 4410 ✑ ✁1 ♣mod 101q ❡ ❛ss✐♠ ❜❛st❛ ♦❜s❡r✈❛♠♦s ❛s ❝♦♥❣r✉ê♥❝✐❛s ❞❡ 110 ❛té 15 ✳❖❜s❡r✈❛♠♦s q✉❡ 210☎ ☎ ☎ 10010 ♦✉ S ✏ 10010 5110✑ 5010 ♣mod 101q, ☎ ☎ ☎ 9910✑ 210 ♣mod 101q ✑ 110 ♣mod 101q ❆♣❧✐❝❛♥❞♦✲s❡ ♦ ❝♦♥❝❡✐t♦ ❞❡ ❝♦♥❣r✉ê♥❝✐❛✱ ♣❡r❝❡❜❡♠♦s q✉❡✿10010 110 q. 210 q ♣10010 9910 q ☎ ☎ ☎ ♣9910 10010 q ♣210 2S✏ ♣110 ✐❣✉❛❧❞❛❞❡s✱ ♦❜t❡♠♦s✿ 9910☎ ☎ ☎ 110 ✳ ❙♦♠❛♥❞♦✲s❡ ❛s ❞✉❛s ✑ 3 ♣mod 7q ñ 20052 ✑ 9 ♣mod 7q ñ 20052 ✑ 2 ♣mod 7q. 2 P♦rt❛♥t♦ 2001 ☎ 2002 ☎ 2003 ☎ 2004 2005 ❞❡✐①❛ r❡st♦ 2 q✉❛♥❞♦ ❞✐✈✐❞✐❞♦ ♣♦r 7✳ ✭❜✮ ❇❛st❛ s❡❣✉✐r50 450 ✑ 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♥❝✐❛1234 7 ; 75 ✏ 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♥❝✐❛✳ 7774 ☎194 1 ❚❡♠♦s q✉❡ 777 ✏ 4 ☎ 194 1 ❡ ♦ ♥ú♠❡r♦ 777 ✏ 777 t❡♠ 7 ♥♦ ú❧t✐♠♦ ❞í❣✐t♦✳❡1234 25 ✏ 2 ; 2 ✏ 4 ; 2 ✏ 8 ; 2 ✏ 16 2✏ 32 ; ☎ ☎ ☎50 ✳✳✳4 ☎12 2 ❉❛ ♠❡s♠❛ ❢♦r♠❛ 2 ✏ 2 ❡ ❞❡✐①❛ 4 ♥♦ ú❧t✐♠♦ ❞í❣✐t♦ ❞❛ ♣♦tê♥❝✐❛✳ 77750 2▲♦❣♦✱ 777 t❡♠ ❡♠ s❡✉ ú❧t✐♠♦ ❛❧❣❛r✐s♠♦ ♦ ♥ú♠❡r♦ 1✱ ♣♦✐s 7 4 ✏ 11✳22 ✭❝✮ ❋❛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✳22 ◗✉❛♥❞♦ ✈♦❝ê ❞✐str✐❜✉✐r✱ ♣❡r❝❡❜❡rá q✉❡ ♦ ú❧t✐♠♦ ❛❧❣❛r✐s♠♦ ❞❛ s♦♠❛ ❞❡ 2001 ❛ 201522 ✭❞✮ ❖❜s❡r✈❡ q✉❡12 43 ✏ 4 ; 4 ✏ 164 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✉❡222014 ✑ 5 ♣mod 7q ñ 2014 ✑ 5 ♣mod 7q2 ✑ 4 ♣mod 7q22015 ✑ 6 ♣mod 7q ñ 2015 ✑ 6 ♣mod 7q✑ 1 ♣mod 7q ▲♦❣♦✱22 2014 2015✑ 4 1 ♣mod 7q. ✶✹✳ ❙❡❥❛54 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✳ ✁1 ✁1 a4 ✑ 1 ♣mod pq ñ n ✑ 1 ♣mod 5q ñ n ✑ ✁4 ♣mod 5q ñ n ✑ 0 ♣mod 5q.5 4nP♦rt❛♥t♦✱ n é ❞✐✈✐sí✈❡❧ ♣♦r 5✱ ♣❛r❛ t♦❞♦ n P N✳33 ✶✺✳ ❙❡❣✉✐♥❞♦ ❛ ❞✐❝❛✱ ✐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✉❡32 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✉❡✱32 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 ✂ ✡ 20132 2 2015 mod 7 ñ 3 ☎ 3 ✑ ✁1 ☎ 3 ñ 3 ✑ 5 ♣mod 7q.2 mdc , 72015 ♣3 qP♦rt❛♥t♦✱ ♦ r❡st♦ ❞❛ ❞✐✈✐sã♦ ❞❡ 3 ♣♦r 7 é 5✳ ✶✼✳ ❯s❛♥❞♦ ♦ ❝♦♥❤❡❝✐♠❡♥t♦ ❞❡ ❝♦♥❣r✉ê♥❝✐❛✱ t❡♠♦s q✉❡3 3 1851 18512222 ✑ 3 ♣mod 7q ñ 2222 ✑ ✁1 ♣mod 7q ñ ♣2222 q ✑ ♣✁1q ♣mod 7q✂ ✡ 555322 mod7 ñ 2222 ☎ 2222 ✑ ♣✁1q ☎ 22222 mdc , 75555 ♣2222 q ñ 2222 ✑ ✁2 ♣mod 7q. ❆ss✐♠ ❝♦♠♦✱5555 22✡ 1✁ 1225 ✡✏ ✂ 1✁ 1 ☎✂ 9✡ 1✁ 1 32✡ ☎ ☎ ☎✂ 1✁ 1 ☎ ☎ ☎✂ 1 ✑ 4 ♣mod 7q ñ 55554 ✑ 1 ♣mod 7q ñ ♣55554 q 555 ✑ 1 ♣mod 7qñ 5555 2220 ☎ 55552 ✑ 1 ☎ 55552 ♣mod 7q ✏ ♣2n 1q ☎ 1 ✏ 2n 1.❯t✐❧✐③❡ ♦ ✐t❡♠ ♣dq ♣❛r❛ r❡s♦❧✈❡r ♦s ✐t❡♥s ♣aq ❡ ♣bq✳ 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♦ a2 ✁ b2 ✏ ♣a bq ☎ ♣a ✁ bq✳ ✭❝✮555.5552 ✁ 444.4442 ✏ ♣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. ✭❞✮♣n 1q2 ✁ n2 ✏ ♣n 1 nq ☎ ♣n 1 ✁ nq ✶✾✳ ❖ ❛❧❣❛r✐s♠♦ ✜♥❛❧ ❞❡ 98673 é ♦ ♠❡s♠♦ q✉❡ ♦ ❞❡ 73 ✱ ✐st♦ é✱ 3❀ ♦ ❛❧❣❛r✐s♠♦ ✜♥❛❧ ❞❡98672 1✁ é ♦ ♠❡s♠♦ q✉❡ ♦ ❞❡ 72 ✏ 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 a2 ✁ b2 ✱ ♦♥❞❡ a ✏ 1✳ ❚❡♠♦s q✉❡✂ 1✁ 1 4✡ ☎✂ 152✡ . 22 ❯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❧♦♦♠♦♦♥ ❧♦♦♠♦♦♥11 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✐♠♦s102015 ☎201 ❛❧❣❛r✐s♠♦s ❞❡ 9 sã♦ ✵✶✱ s❡❣✉❡ q✉❡ ♦s ❞♦✐s ú❧t✐♠♦s ❛❧❣❛r✐s♠♦s ❞❡ 9 sã♦ ♦s ❞♦✐s52015 1ú❧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✉❡✿99.990.0008 ↕ x ↕ 99.999.999.8 84 222 ❈♦♠♦ 10 ✏ ♣10 q ✏ 10.000 ✱ t❡♠♦s q✉❡ 99.990.000 ↕ 10.000 ✳ ❆❣♦r❛ ♦ ♠❛✐♦r2 q✉❛❞r❛❞♦ ♣❡r❢❡✐t♦ ♠❡♥♦r q✉❡ 10.000 é ✐❣✉❛❧ ❛2229.999 ✏ ♣10.000✁1q ✏ 10.000 ✁20.000 1 ✏ 100.000.000✁20.000 1 ✏ 99.980.001.22 ❈♦♠♦ 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❡♠♦s2 12 1✁2 ❄ 1 100✁ 0, 01 ✁ 1 100 n.✁ 1 → ✏ ✏ 20, 02 200 100❊❧❡✈❛♥❞♦ ♥♦✈❛♠❡♥t❡ ❛♦ q✉❛❞r❛❞♦ ♦s ❞♦✐s ❧❛❞♦s ✭♥ã♦ ♥❡❣❛t✐✈♦s✮ ❞❡st❛ ✐♥❡q✉❛çã♦✱♦❜t❡♠♦s✿22422 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 n1 .✁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✳ ✷✹✳ ■♥✐❝✐❛❧♠❡♥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 km ✭✐✮ ❙❡ 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 7m3 ❡ 3k 7m sã♦ ♠ú❧t✐♣❧♦s ❞❡ 7✳ ❈♦♥❝❧✉í♠♦s q✉❡ N é ♠ú❧t✐♣❧♦ ❞❡ 7 ✳2k 7m 2k 7mm ✭✐✐✮ ❙❡ 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 q5 ✁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, ❡ 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 ✺✳ ▲♦❣♦✱ 3é ❞✐✈✐sí✈❡❧ ♣♦r ✺✳3 ✷✽✳ ❯♠ ❝✉❜♦ ♣❡r❢❡✐t♦ é ✉♠ ♥ú♠❡r♦ ❞❛ ❢♦r♠❛ a ✱ ♦♥❞❡ a é ✉♠ ♥ú♠❡r♦ ♥❛t✉r❛❧✳ ❈♦♠♦424883236 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❪❛♦ ❡❧❡ é3❞❛ ❢♦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✳333 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✿33123 a b3 c9k 1 9k 1 9k 1 9k 39k1 1 9k2 1 9k3 8 9k 19k11 1 9k22 8 9k33 8 9k 83 9k3 8 9k3 8 9k 8 9k 6 b cP♦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✳33 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✿333 a b c333 ✑ 1 1 1 ✑ 3 ♣mod 9q a b c333 ✑ 1 1 8 ✑ 1 ♣mod 9q a b c333 ✑ 1 8 8 ✑ 8 ♣mod 9q a b c333 ✑ 8 8 8 ✑ 6 ♣mod 9q ✸✵✳ ❯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 b1 ;1✏ r ☎ 5 r ➔ r ➔ r r1 2 ; 021 ✏ r ☎ 3 r ➔ r ➔ r r12 3 ; 032✏ r ☎ 3 r ➔ r ➔ r r23 4 ; 043 ✏ r ☎ 1 r ➔ r ➔ r r3 r44 ✏ ☎ 33 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✐❛ ❊✉❝❧✐❞✐❛♥❛ ✶✳ ❙❡❥❛ 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 CAPAP 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♠❛ ❢♦r♠❛✱ ♣♦❞❡✲s❡ ♦❜s❡r✈❛r q✉❡ ♦s tr✐â♥❣✉❧♦s AMB ❡ AMC sã♦ ❝♦♥❣r✉❡♥t❡s

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