❉❡♣❛rt❛♠❡♥t♦ ❞❡ ▼❛t❡♠át✐❝❛
▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧ ❡♠ ▼❛t❡♠át✐❝❛
❡♠ ❘❡❞❡ ◆❛❝✐♦♥❛❧ P❘❖❋▼❆❚
▼❛t❡♠át✐❝❛ ❉✐s❝r❡t❛✿
▼é❞✐❛s ❡ Pr✐♥❝í♣✐♦ ❞❛s ●❛✈❡t❛s✳
♣♦r
❈❆❘▲❖❙ ❆▲❇❊❘❚❖ ▼❯◆■❩ ❏Ú◆■❖❘
s♦❜ ❛ ♦r✐❡♥t❛çã♦ ❞♦ Pr♦❢✳ ❉r✳ ❈❛r❧♦s ❇♦❝❦❡r ◆❡t♦
❉✐ss❡rt❛çã♦ ❛♣r❡s❡♥t❛❞❛ ❛♦ ❈♦r♣♦ ❉♦✲ ❝❡♥t❡ ❞♦ ▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧ ❡♠ ▼❛✲ t❡♠át✐❝❛ ❡♠ ❘❡❞❡ ◆❛❝✐♦♥❛❧ P❘❖❋▼❆❚✲ ❈❈❊◆✲❯❋P❇✱❝♦♠♦ r❡q✉✐s✐t♦ ♣❛r❝✐❛❧ ♣❛r❛ ♦❜t❡♥çã♦ ❞♦ tít✉❧♦ ❞❡ ▼❡str❡ ❡♠ ▼❛t❡✲ ♠át✐❝❛✳
M966m Muniz Júnior, Carlos Alberto.
Matemática discreta: médias e princípio das gavetas / Carlos Alberto Muniz Júnior.- João Pessoa, 2016.
63f.
Orientador: Carlos Bocker Neto Dissertação (Mestrado) - UFPB/CCEN
1. Matemática. 2. Médias. 3. Desigualdade das Médias. 4. Princípio das Gavetas.
❆ ❉❡✉s✱ ♣r✐♠❡✐r❛♠❡♥t❡✱ q✉❡ s❡ ❢❡③ ♣r❡s❡♥t❡ ❡♠ t♦❞❛s ❛s ❡t❛♣❛s ❞❡ss❛ ❝❛♠✐♥❤❛❞❛✳ ❆♦s ♠❡✉s ♣❛✐s✱ ❈❛r❧♦s ❡ ▼❛r❧❡♥❡✱ ♣♦r t❡r❡♠ ♠❡ ❛♣♦✐❛❞♦ ❞✐❛♥t❡ ❞❛s ❞✐✜❝✉❧❞❛❞❡s✱ ❡ ♠❡ ✐♥❝❡♥t✐✈❛❞♦ ♥❛s ❤♦r❛s ❞✐❢í❝❡✐s❀
❆ t♦❞♦s ♠❡✉s ❢❛♠✐❧✐❛r❡s ♣❡❧❛ ❝♦♠♣r❡❡♥sã♦ ♥♦s ♠♦♠❡♥t♦s ❡♠ q✉❡ ❡st✐✈❡ ❛✉s❡♥t❡❀ ❆ t♦❞♦s ♦s ❝♦❧❡❣❛s ❞❡ t✉r♠❛✱ ♣❡❧♦ ✈í♥❝✉❧♦ ❞❡ ❛♠✐③❛❞❡ q✉❡ ❝♦♥str✉í♠♦s ❞✉r❛♥t❡ ❛s ❛✉❧❛s ❡ ♦s ❞✐❛s ❡ ♥♦✐t❡s ❞❡ ❡st✉❞♦s q✉❡ ♥♦s ♣r♦♣♦r❝✐♦♥❛r❛♠ ❛♣r❡♥❞✐③❛❞♦ ❡ ❢♦rt❛✲ ❧❡❝❡r❛♠ ♦s ❧❛ç♦s ❞❡ ❛♠✐③❛❞❡❀
❆♦s ❝♦❧❡❣❛s ❞❡ tr❛❜❛❧❤♦✱ q✉❡ ♠❡s♠♦ ❞❡ ❧♦♥❣❡ ♠❡ ❞❡r❛♠ s✉♣♦rt❡ q✉❛♥❞♦ ❢♦✐ ♥❡✲ ❝❡ssár✐♦ ♠❡ ❛✉s❡♥t❛r✱ ♦✉ q✉❛♥❞♦ ❡st✐✈❡ ❡♥✈♦❧✈✐❞♦ ❝♦♠ ♦✉tr❛s t❛r❡❢❛s ❡ ❢✉✐ s✉❜st✐t✉í❞♦❀ ❆♦ ♠❡✉ ♦r✐❡♥t❛❞♦r✱ Pr♦❢✳ ❉r✳ ❈❛r❧♦s ❇♦❝❦❡r ◆❡t♦✱ ♣❡❧❛ ♣❛❝✐ê♥❝✐❛ ❡ ❞❡❞✐❝❛çã♦ ♥♦ ❞❡s❡♥✈♦❧✈✐♠❡t♦ ❞❡ss❡ tr❛❜❛❧❤♦❀
❆♦s ♣r♦❢❡ss♦r❡s✱ ❛♦ ❝♦♦r❞❡♥❛❞♦r✱ ❡ t♦❞❛ ❡q✉✐♣❡ ❞❛ ❯❋P❇ ♣❡❧❛ ❞❡❞✐❝❛çã♦ ❞✉r❛♥t❡ ❝❛❞❛ ❞✐s❝✐♣❧✐♥❛✱ ❞♦❝✉♠❡♥t❛çã♦ s♦❧✐❝✐t❛❞❛✱ ♦✉ ♣❡rí♦❞♦ ❞❡ ♠❛trí❝✉❧❛✱ ♣♦r ❡st❛r❡♠ s❡♠✲ ♣r❡ ♥♦s ♠♦t✐✈❛♥❞♦ ❡ ❛♣♦✐❛♥❞♦✳
❉❡❞✐❝♦ ❛♦s ♠❡✉s ❢❛♠✐❧✐❛r❡s ♣♦r ❡st❛r❡♠ s❡♠♣r❡ ♣r❡s❡♥t❡s ❡♠ t♦❞❛s ❛s ❡t❛♣❛s ❞❡ ♠✐♥❤❛ ✈✐❞❛✱ ❡♠ ❡s♣❡❝✐❛❧ ❛♦s ♠❡✉s ♣❛✐s✳ ❊ t❛♠❜é♠ ❛♦s ❛♠✐❣♦s ❞❡ ❝✉rs♦✱ q✉❡ ❡st✐✈❡r❛♠ ❥✉♥t♦ ❛ ♠✐♠ ❡♠ t♦❞♦ ❡ss❡ ♣❡r❝✉ss♦✳
✧❖ q✉❡ ♣r♦♣♦r❝✐♦♥❛ ♦ ♠á①✐♠♦ ❞❡ ♣r❛③❡r ♥ã♦ é ♦ ❝♦♥❤❡❝✐♠❡♥t♦ ❡ s✐♠ ❛ ❛♣r❡♥❞✐③❛❣❡♠✳✧ ✲ ●❛✉ss
◆❡st❡ tr❛❜❛❧❤♦ ❛♣r❡s❡♥t❛♠♦s ♦s ♣r✐♥❝✐♣❛✐s ❝♦♥❝❡✐t♦s ❞❡ ♠é❞✐❛s✿ ❆r✐t♠ét✐❝❛✱ ●❡♦✲ ♠étr✐❝❛✱ ❍❛r♠ô♥✐❝❛ ❡ ◗✉❛❞rát✐❝❛ ❡ t❛♠❜é♠ ♦ Pr✐♥❝í♣✐♦ ❞❛s ●❛✈❡t❛s ❞❡ ❉✐r✐❝❤❧❡t✳ ❉❡st❛❝❛♠♦s ❝♦♠♦ ♣r✐♥❝✐♣❛✐s ❞✐r❡çõ❡s ❞❡st❡ tr❛❜❛❧❤♦✱ ❛s ❛♣❧✐❝❛çõ❡s ❞❡st❡s ❝♦♥❝❡✐t♦s ♥❛s ❞✐✈❡rs❛s ár❡❛s ❞❛ ♠❛t❡♠át✐❝❛ ❡ ❛ ♣♦ss✐❜✐❧✐❞❛❞❡ ❞❡ s❡ tr❛❜❛❧❤❛r t❛✐s ❝♦♥t❡ú❞♦s ♥♦ ❊♥s✐♥♦ ▼é❞✐♦✳ ❘❡ss❛❧t❛♠♦s t❛♠❜é♠ ♦s ♣r✐♥❝✐♣❛✐s t❡♦r❡♠❛s ❛❜♦r❞❛❞♦s✱ ♦s q✉❛✐s sã♦✱ ❛ ❉❡s✐❣✉❛❧❞❛❞❡ ❞❛s ▼é❞✐❛s ❡ ♦s ❚❡♦r❡♠❛s ❞❡ ❘❛♠s❡② ❡ ❞❡ ❉✐r✐❝❤❧❡t q✉❡ sã♦ ❛♣❧✐❝❛çõ❡s ♥ã♦ tr✐✈✐❛✐s ❞♦ Pr✐♥❝í♣✐♦ ❞❛s ●❛✈❡t❛s✳
P❛❧❛✈r❛s✲❝❤❛✈❡✿ ▼é❞✐❛s✱ ❉❡s✐❣✉❛❧❞❛❞❡ ❞❛s ▼é❞✐❛s✱ Pr✐♥❝í♣✐♦ ❞❛s ●❛✈❡t❛s✳
❲❡ ♣r❡s❡♥t t❤❡ ♠❛✐♥ ❝♦♥❝❡♣ts ♦❢ ❛✈❡r❛❣❡s✿ ❆r✐t❤♠❡t✐❝✱ ●❡♦♠étr✐❝❛✱ ❍❛r♠♦♥✐❝❛ ❛♥❞ ◗✉❛❞r❛t✐❝ ❛♥❞ ❛❧s♦ t❤❡ ❉✐r✐❝❤❧❡t✬s ❉r❛✇❡r Pr✐♥❝✐♣❧❡✳ ❲❡ ❤✐❣❤❧✐❣❤t ❛s t❤❡ ♠❛✐♥ ❞✐✲ r❡❝t✐♦♥s ♦❢ t❤✐s ✇♦r❦✱ t❤❡ ❛♣♣❧✐❝❛t✐♦♥ ♦❢ t❤❡s❡ ✐♥ ✈❛r✐♦✉s ❛r❡❛s ♦❢ ♠❛t❤❡♠❛t✐❝s ❛♥❞ t❤❡ ❛❜✐❧✐t② t♦ ✇♦r❦ ✇✐t❤ s✉❝❤ ❝♦♥t❡♥t ✐♥ ❤✐❣❤ s❝❤♦♦❧✳ ❲❡ ❛❧s♦ ❡♠♣❤❛s✐③❡ t❤❡ ♠❛✐♥ t❤❡♦r❡♠s ❛♣♣r♦❛❝❤❡❞✱ ✇❤✐❝❤ ❛r❡ t❤❡ ■♥❡q✉❛❧✐t② ♦❢ ▼❡❞✐✉♠ ❛♥❞ t❤❡♦r❡♠s ❘❛♠s❡② ❛♥❞ ❉✐r✐❝❤❧❡t t❤❛t ❛r❡ ♥♦♥tr✐✈✐❛❧ ❛♣♣❧✐❝❛t✐♦♥s ♦❢ t❤❡ ❉r❛✇❡r Pr✐♥❝✐♣❧❡✳
❑❡②✇♦r❞s✿ ▼❡❞✐✉♠✱ ■♥❡q✉❛❧✐t② ♦❢ ▼❡❛♥s✱ Pr✐♥❝✐♣❧❡ ♦❢ ❉r❛✇❡rs✳
✶ ▼é❞✐❛s ✶ ✶✳✶ ❉❡✜♥✐çõ❡s ❞❡ ▼é❞✐❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶ ✶✳✶✳✶ ▼é❞✐❛ ❆r✐t♠ét✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷ ✶✳✶✳✷ ▼é❞✐❛ P♦♥❞❡r❛❞❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹ ✶✳✶✳✸ ▼é❞✐❛ ●❡♦♠étr✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻ ✶✳✶✳✹ ▼é❞✐❛ ❍❛r♠ô♥✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾ ✶✳✶✳✺ ▼é❞✐❛ ◗✉❛❞rát✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶ ✶✳✷ ❉❡s✐❣✉❛❧❞❛❞❡ ❞❛s ▼é❞✐❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸ ✶✳✸ ❘❡♣r❡s❡♥t❛çã♦ ●❡♦♠étr✐❝❛ ♣❛r❛ ❛s ▼é❞✐❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵ ✶✳✸✳✶ ❘❡♣r❡s❡♥t❛çã♦ ❞❛ ▼é❞✐❛ ❆r✐t♠ét✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵ ✶✳✸✳✷ ❘❡♣r❡s❡♥t❛çã♦ ❞❛ ▼é❞✐❛ ●❡♦♠étr✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵ ✶✳✸✳✸ ❘❡♣r❡s❡♥t❛çã♦ ❞❛ ▼é❞✐❛ ❍❛r♠ô♥✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶ ✶✳✸✳✹ ❘❡♣r❡s❡♥t❛çã♦ ❞❛ ▼é❞✐❛ ◗✉❛❞rát✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷ ✶✳✸✳✺ ❈♦♥❝❧✉sã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸
✷ Pr✐♥❝í♣✐♦ ❞❛s ●❛✈❡t❛s ✷✹
✷✳✶ Pr✐♥❝í♣✐♦ ❞❛s ●❛✈❡t❛s ❞❡ ❉✐r✐❝❤❧❡t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺ ✷✳✷ ❆ ●❡♥❡r❛❧✐③❛çã♦ ❞♦ Pr✐♥❝í♣✐♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺ ✷✳✸ ❊①❡r❝í❝✐♦s ❘❡s♦❧✈✐❞♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✽ ✷✳✹ ❊①❡r❝í❝✐♦s ❆♣❧✐❝❛❞♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✷ ✷✳✹✳✶ ❆♣❧✐❝❛❞♦ ❛ ❆r✐t♠ét✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✷ ✷✳✹✳✷ ❆♣❧✐❝❛❞♦ ❛ ❆♥á❧✐s❡ ❈♦♠❜✐♥❛tór✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✷ ✷✳✹✳✸ ❆♣❧✐❝❛❞♦ ❛ ●❡♦♠❡tr✐❛ P❧❛♥❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✸ ✷✳✹✳✹ ❆♣❧✐❝❛❞♦ ❛ ●❡♦♠❡tr✐❛ ❊s♣❛❝✐❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✸ ✷✳✹✳✺ ❆♣❧✐❝❛❞♦ ❛ ●❡♦♠❡tr✐❛ ❆♥❛❧ít✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✹ ✷✳✹✳✻ ❆♣❧✐❝❛❞♦ ❛ ❋✉♥çõ❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✹
✸ ❚❡♦r❡♠❛s ✸✺
✸✳✶ ❚❡♦r❡♠❛ ❞❡ ❉✐r✐❝❤❧❡t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✺ ✸✳✷ ❚❡♦r❡♠❛ ❞❡ ❘❛♠s❡② ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✽
✹✳✸ ❘❡s✉❧t❛❞♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✾
❘❡❢❡rê♥❝✐❛s ❇✐❜❧✐♦❣rá✜❝❛s ✺✶
✶✳✶ ❆ ❛❧t✉r❛ é ❛ ▼é❞✐❛ ●❡♦♠étr✐❝❛ ❞❛s ♣r♦❥❡çõ❡s ❞♦s ❝❛t❡t♦s s♦❜r❡ ❛ ❤✐♣♦t❡♥✉s❛✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼ ✶✳✷ Pr✐s♠❛ ❞❡ ❞✐♠❡♥sõ❡s ❛✱❜✱❝ ❡ ❈✉❜♦ ❞❡ ❛r❡st❛ ❧✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽ ✶✳✸ ❚r❛♣é③✐♦ ❆❇❈❉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻ ✶✳✹ P❛r❛❧❡❧❡♣í♣❡❞♦ ❞❡ ❛r❡st❛sx✱ x ❡ h ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽
✶✳✺ ▲❛t❛ ❞❡ ❩✐♥❝♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾ ✶✳✻ ❘❡♣r❡s❡♥t❛çã♦ ❞❛ ▼é❞✐❛ ❆r✐t♠ét✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵ ✶✳✼ ❘❡♣r❡s❡♥t❛çã♦ ❞❛ ▼é❞✐❛ ●❡♦♠étr✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶ ✶✳✽ ❘❡♣r❡s❡♥t❛çã♦ ❞❛ ▼é❞✐❛ ❍❛r♠ô♥✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶ ✶✳✾ ❘❡♣r❡s❡♥t❛çã♦ ❞❛ ▼é❞✐❛ ◗✉❛❞rát✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷ ✶✳✶✵ ❘❡♣r❡s❡♥t❛çã♦ ❞❛s ▼é❞✐❛s ◗✱ ❆✱ ●✱ ❍ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸ ✸✳✶ ❉✐r✐❝❤❧❡t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✺ ✸✳✷ P❛rt✐çã♦ ❞❡ ✵ ❛ ✶ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✻ ✸✳✸ ❘❛♠s❡② ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✽ ✸✳✹ ❱ért✐❝❡s ❞♦ ❍❡①á❣♦♥♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✽ ✸✳✺ ❙❡❣♠❡♥t♦AD ❝♦♥tí♥✉♦ ❡ ❙❡❣♠❡♥t♦AC tr❛❝❡❥❛❞♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✾
✸✳✻ ❙❡❣♠❡♥t♦s ❝♦♥tí♥✉♦s AB, AC, AD, AE, AF ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✾
✸✳✼ ❙❡❣♠❡♥t♦sAB, AD, AF ❝♦♥tí♥✉♦s ❡ ❙❡❣♠❡♥t♦s AC, AE tr❛❝❡❥❛❞♦s ✳ ✳ ✹✵
✸✳✽ ❙❡❣♠❡♥t♦ ❝♦♥tí♥✉♦ BD ❢♦r♠❛♥❞♦ ♦ tr✐â♥❣✉❧♦ ABD ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✵
✸✳✾ BD, DF , F B ❢♦r♠❛♥❞♦ ♦ tr✐â♥❣✉❧♦ BDF ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✶
✹✳✶ ●rá✜❝♦ ❞❡ ❛❝❡rt♦s ❞♦ ❚❊❙❚❊ ■ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✻ ✹✳✷ ●rá✜❝♦ ❞❡ ❛❝❡rt♦s ❞♦ ❚❊❙❚❊ ■■ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✽ ✹✳✸ ●rá✜❝♦ ❞❛ ▼é❞✐❛ ❞❡ ❆❝❡rt♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✾ ✹✳✹ ●rá✜❝♦ ❈♦♠♣❛r❛t✐✈♦ ❞❡ ❛❝❡rt♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✵
✶✳✶ ❚❛❜❡❧❛ ❞♦s ❙❛❧ár✐♦s ❞♦s ❋✉♥❝✐♦♥ár✐♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺ ✶✳✷ ❚❛❜❡❧❛ ❞❡ ❘❡♥❞✐♠❡♥t♦s ❆♥✉❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽
◆♦ ♣r❡s❡♥t❡ tr❛❜❛❧❤♦✱ tr❛t❛r❡♠♦s s♦❜r❡ ❛s ▼é❞✐❛s✱ s✉❛s ❉❡s✐❣✉❛❧❞❛❞❡s ❡ ♦ Pr✐♥✲ ❝í♣✐♦ ❞❛s ●❛✈❡t❛s ❞❡ ❉✐r✐❝❤❧❡t✳ ❆s ♠é❞✐❛s sã♦ ❜❛st❛♥t❡ ❝♦♥❤❡❝✐❞❛s ❡ tr❛❜❛❧❤❛❞❛s ♥♦ ❊♥s✐♥♦ ❇ás✐❝♦✱ ❥á ❛s ❞❡s✐❣✉❛❧❞❛❞❡s ♥❡♠ t❛♥t♦✱ ❛♣❡s❛r ❞❡ t❛♥t❛s ♣♦ss✐❜✐❧✐❞❛❞❡s ❞❡ ❛♣❧✐❝❛çõ❡s✱ ❝♦♠♦ ♠♦str❛r❡♠♦s ♥❡ss❡ tr❛❜❛❧❤♦✳ ◆♦ ❡♥t❛♥t♦✱ ♦ Pr✐♥❝í♣✐♦ ❞❛s ●❛✈❡t❛s✱ é ✉♠ t❡♠❛ ♣♦✉❝♦ ♦✉ q✉❛s❡ ♥✉♥❝❛ ❧❡♠❜r❛❞♦ ♣❡❧♦s ♣r♦❢❡ss♦r❡s ❞❡ ❊♥s✐♥♦ ▼é❞✐♦✳ ❊ ❛q✉✐✱ tr❛t❛♠♦s ❞❡❧❡ ❝♦♠♦ ❢❡rr❛♠❡♥t❛ ❜❛st❛♥t❡ ❡✜❝✐❡♥t❡ ♥❛ r❡s♦❧✉çã♦ ❞❡ ♣r♦❜❧❡♠❛s✱ ❥á q✉❡ é ♣♦ssí✈❡❧ ❛♣❧✐❝❛r t❛❧ ♣r✐♥❝í♣✐♦ ❡♠ ❞✐✈❡rs❛s ár❡❛s ❞❛ ▼❛t❡♠át✐❝❛✳ ◆❡ss❡ s❡♥✲ t✐❞♦✱ ❞✐✈✐❞✐♠♦s ♦ tr❛❜❛❧❤♦ ❡♠ q✉❛tr♦ ❝❛♣ít✉❧♦s✳
❖ ♣r✐♠❡✐r♦ ❝❛♣ít✉❧♦ tr❛t❛ ❞❛s ▼é❞✐❛s ❆r✐t♠ét✐❝❛✱ ●❡♦♠étr✐❝❛✱ ❍❛r♠♦♥✐❛ ❡ ✐♥✲ ❝❧✉s✐✈❡ ❛ ◗✉❛❞rát✐❝❛ q✉❡ ♣♦✉❝♦ é ✈✐st❛ ♥♦ ❊♥s✐♥♦ ▼é❞✐♦✳ ❉❡✜♥✐♠♦s ❝❛❞❛ ✉♠ ❞❡❧❛s✱ tr❛③❡♥❞♦ ❛✐♥❞❛ ❛❧❣✉♠❛s ❛♣❧✐❝❛çõ❡s ❞❡ ❝❛❞❛ t✐♣♦ ❞❡ ♠é❞✐❛ ❡ ❡①❡♠♣❧✐✜❝❛♥❞♦ ❝♦♠ ♣r♦❜❧❡♠❛s ✈♦❧t❛❞♦s ♣❛r❛ ♦ ❊♥s✐♥♦ ▼é❞✐♦✱ ❛ ♣❛rt✐r ❞❡ ❬✶❪✳ ❆ss✐♠✱ ❝♦♠♦ t❛♠❜é♠ tr❛✲ t❛♠♦s ❞❛s ❞❡s✐❣✉❛❧❞❛❞❡s ❡♥tr❡ ❛s ♠é❞✐❛s✱ ❞❡♠♦♥str❛♥❞♦ ❡ss❛ r❡❧❛çã♦ ❜❛st❛♥t❡ út✐❧ ♥❛ r❡s♦❧✉çã♦ ❞❡ ♣r♦❜❧❡♠❛s✱ ❡ ❛♣r❡s❡♥t❛♥❞♦ ❞✐✈❡rs♦s ❡①❡r❝í❝✐♦s ♦♥❞❡ ♣♦❞❡♠♦s ❛♣❧✐❝❛r t❛✐s ❞❡s✐❣✉❛❧❞❛❞❡s ❡♠ ❞✐✈❡rs❛s ár❡❛s ❞❛ ♠❛t❡♠át✐❝❛✱ ❝♦♠ ♠❛✐s ✐♥❢♦r♠❛çõ❡s ❡♠ ❬✷❪✳ ❆❧é♠ ❞❡ ❢❛③❡r ❛ r❡♣r❡s❡♥t❛çã♦ ❞❡ t❛✐s ♠é❞✐❛s ❣❡♦♠❡tr✐❝❛♠❡♥t❡✱ ✉t✐❧✐③❛♥❞♦ ❞✐✈❡rs❛s ❝♦♥str✉çõ❡s ❡ ❛♣❧✐❝❛♥❞♦ r❡❧❛çõ❡s ❡ ❝♦♥❝❡✐t♦s ♠❛t❡♠át✐❝♦s✳ P❡r♠✐t✐♥❞♦ ❛ ♣❡r❝❡♣çã♦ ❞❡ ❢♦r♠❛ ❧ú❞✐❝❛ ❞❛s ❞❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❡♥tr❡ ❛s ♠é❞✐❛s✳
◆♦ s❡❣✉♥❞♦ ❝❛♣ít✉❧♦✱ ♦ Pr✐♥❝í♣✐♦ ❞❛s ●❛✈❡t❛s ❞❡ ❉✐r✐❝❤❧❡t é ❛♣r❡s❡♥t❛❞♦ ❡ ❞❡✲ ♠♦♥str❛❞♦✱ ❛ss✐♠ ❝♦♠♦ s✉❛ ❣❡♥❡r❛❧✐③❛çã♦✱ ❛ ✜♠ ❞❡ ❡✈✐❞❡♥❝✐❛r s✉❛ ❡✜❝á❝✐❛ ❝♦♠♦ ❢❡r✲ r❛♠❡♥t❛ ♠❛t❡♠át✐❝❛✱ ✉♠❛ ✈❡③ q✉❡ ❞✐✈❡rs♦s ♣r♦❜❧❡♠❛s ♣♦❞❡♠ s❡r r❡s♦❧✈✐❞♦s ✉s❛♥❞♦ ❡st❡ ♣r✐♥❝í♣✐♦ ❡♠ ❞✐✈❡rs❛s ár❡❛s ❞❛ ♠❛t❡♠át✐❝❛✳ ❆♣r❡s❡♥t❛♠♦s ❞✐✈❡rs♦s ❡①❡♠♣❧♦s ❞❛ s✉❛ ❛♣❧✐❝❛çã♦✱ ✐♥❝❧✉s✐✈❡ ❞❡ ❝❛rát❡r ❧ó❣✐❝♦ ❡ ✉♠❛ s❡çã♦ ❝♦♠ ❡①❡r❝í❝✐♦s r❡s♦❧✈✐❞♦s ♦♥❞❡ ❥á ♣♦❞❡♠♦s ♣❡r❝❡❜❡r s✉❛ ✐♠♣♦rtâ♥❝✐❛ ♥❛ r❡s♦❧✉çã♦ ❞❡ ♣r♦❜❧❡♠❛s ♠❛t❡♠át✐❝♦s✱ ❜❛s❡❛❞♦s ❡♠ ❬✸❪✱ ❬✹❪✱ ❬✺❪✱ ❬✻❪✱ ❬✼❪ ❡ ❬✽❪ ✳ P♦r ✜♠✱ r❡❧❛❝✐♦♥❛♠♦s ❛❧❣✉♠❛s ár❡❛s ❞❛ ♠❛t❡♠át✐❝❛ ❝♦♠ ♦ ♣r✐♥❝í♣✐♦ ❞❛s ❣❛✈❡t❛s✱ r❡ss❛❧t❛♥❞♦ ♦s ❝♦♥❤❡❝✐♠❡♥t♦s ♣ré✈✐♦s ♥❛ r❡s♦❧✉çã♦✱ ❝♦♠♦ ♣♦❞❡ s❡r ✈✐st♦ ❡♠ ❬✷❪✳
❞♦ Pr✐♥❝í♣✐♦ ❞❛s ●❛✈❡t❛s ❡ ✉♠ ❝♦♥✈✐t❡ ❛ s❡ ❛♣r♦❢✉♥❞❛r ♠❛✐s ♥♦ ❛ss✉♥t♦ ❛tr❛✈és ❞❛ ❧❡✐t✉r❛ ❞❡ ❬✷❪ ✱ ❬✾❪ ❡ ❬✶✵❪✳
◆♦ q✉❛rt♦ ❡ ú❧t✐♠♦ ❝❛♣ít✉❧♦✱ ❛♣r❡s❡♥t❛♠♦s ❛ ❡①♣❡r✐ê♥❝✐❛ ❡♠ s❛❧❛ ❝♦♠ ❛ ❛♣❧✐❝❛çã♦ ❞❡ t❡st❡s ❡ s❡✉s r❡s♣❡❝t✐✈♦s r❡s✉❧t❛❞♦s✳ ❋♦r❛♠ ❛♣❧✐❝❛❞♦s ❞♦✐s t❡st❡ ✈❡rs❛♥❞♦ s♦❜r❡ ♦ Pr✐♥❝í♣✐♦ ❞❛s ●❛✈❡t❛s✳ ❖ ♣r✐♠❡✐r♦ t❡st❡ ❢♦✐ ❛♣❧✐❝❛❞♦ ❛♦s ❛❧✉♥♦s s❡♠ ❛♣r❡s❡♥t❛r ♦ Pr✐♥❝í♣✐♦ ❞❛s ●❛✈❡t❛s✳ ❊✱ ❛♥t❡s ❞❛ ❛♣❧✐❝❛çã♦ ❞♦ s❡❣✉♥❞♦ t❡st❡✱ ♦ Pr✐♥❝í♣✐♦ ❞❛s ●❛✈❡t❛s ❢♦✐ ❛♣r❡s❡♥t❛❞♦ ❛♦s ❛❧✉♥♦s✱ ❞❡ ❢♦r♠❛ ❡①♣♦s✐t✐✈❛✱ ❡ ❛✐♥❞❛ ❢♦✐ ❢❡✐t❛ ❛ r❡s♦❧✉çã♦ ❞♦ ♣r✐♠❡✐r♦ t❡st❡✳ ❖s r❡s✉❧t❛❞♦s sã♦ ❛♣r❡s❡♥t❛❞♦s ♣❡❧♦s ♣❡r❝❡♥t✉❛✐s ❞❡ ❛❝❡rt♦s ❡♠ ❛♠❜♦s ♦s t❡st❡s✱ ❛❧é♠ ❞❡ ❣rá✜❝♦s q✉❡ ❢❛③❡♠ ✉♠ ❝♦♠♣❛r❛t✐✈♦ ❡♥tr❡ ❝❛❞❛ t❡st❡✳
P❡r❝❡❜❡♥❞♦ ♦ ✐♥t❡r❡ss❡ ❡ ❛ ❝✉r✐♦s✐❞❛❞❡ ❞♦s ❛❧✉♥♦s ❞♦ ❊♥s✐♥♦ ▼é❞✐♦✱ ❡♠ ✉t✐❧✐③❛r ♦ Pr✐♥❝í♣✐♦ ❞❛s ●❛✈❡t❛s✱ s✉❣❡r✐♠♦s ❛♦s ♣r♦❢❡ss♦r❡s q✉❡ ♣r♦❝✉r❡♠ ❛❜♦r❞❛r ♦ t❡♠❛ q✉❛♥❞♦ ♣♦ssí✈❡❧✱ ❡♠ s✉❛s ❛✉❧❛s✳ ❊ss❡ é ♦ ♣r✐♥❝✐♣❛❧ ♦❜❥❡t✐✈♦ ❞❡ss❡ tr❛❜❛❧❤♦✱ ♠♦str❛r q✉❡ ♣♦❞❡♠♦s ❛♣❧✐❝❛r ♦s t❡♠❛s ❛q✉✐ ❛♣r❡s❡♥t❛❞♦s ♣❛r❛ t♦r♥❛r ♦ ❡♥s✐♥♦ ❞❡ ♠❛t❡♠át✐❝❛ ♠❛✐s ❛tr❛t✐✈♦ ♣❛r❛ ♦s ❛❧✉♥♦s✳
▼é❞✐❛s
◆❡st❡ ❝❛♣ít✉❧♦ ❢❛③❡♠♦s ✉♠❛ ❜r❡✈❡ ✐♥tr♦❞✉çã♦ ❞❛s ♣r✐♥❝✐♣❛✐s ♠é❞✐❛s✳ ❆s ♠é❞✐❛s ❡stã♦ ❛ss♦❝✐❛❞❛s ❛ ✐❞❡✐❛ ❞❡ s✉❜st✐t✉✐r ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ ♥ú♠❡r♦s ♣♦r ✉♠ ú♥✐❝♦ q✉❡ r❡♣r❡s❡♥t❡ t♦❞❛ s❡q✉ê♥❝✐❛✳ ❖❜s❡r✈❛♠♦s q✉❡✱ ❣❡r❛❧♠❡♥t❡✱ ❛s ♠é❞✐❛s ❡stã♦ ❛ss♦❝✐❛❞❛s ❛ ✉♠❛ ❞❡t❡r♠✐♥❛❞❛ ♦♣❡r❛çã♦ s♦❜r❡ ❛ s❡q✉ê♥❝✐❛ ❞♦s ♥ú♠❡r♦s✳ ❆♣ós ❞❡✜♥✐r ❝❛❞❛ ♠é❞✐❛ ❛ ♣❛rt✐r ❞❡ ❬✶❪✱ ❢❛③❡♠♦s ❛♣❧✐❝❛çõ❡s ❞❛ ✉t✐❧✐③❛çã♦ ❝♦♠ ❛❧❣✉♥s ❡①❡♠♣❧♦s ❡ ♣r♦✲ ♣♦♥❞♦ ❛❧❣✉♥s ❡①❡r❝í❝✐♦s ♣❛r❛ q✉❡ ❤❛❥❛ ✉♠❛ ♠❡❧❤♦r ✜①❛çã♦ ❞❛s ✐❞❡✐❛s✱ t♦♠❛♥❞♦ ❝♦♠♦ r❡❢❡rê♥❝✐❛s ❬✷❪ ❡ ❬✻❪✳
❆s ▼é❞✐❛s sã♦ ❡ss❡♥❝✐❛✐s ♣❛r❛ ❢❛③❡r ❡st✐♠❛t✐✈❛s ❞❡ t❡♥❞ê♥❝✐❛s ❞❡ ❝r❡s❝✐♠❡♥t♦ ♣♦♣✉❧❛❝✐♦♥❛❧✱ ❞❡ t❛①❛s ❞❡ r❡♥❞✐♠❡♥t♦ ❡♠ ✐♥✈❡st✐♠❡♥t♦s ❛♦ ❧♦♥❣♦ ❞❡ ✉♠ ❞❛❞♦ t❡♠♣♦✱ ✈❡❧♦❝✐❞❛❞❡ ♠é❞✐❛ ♦✉✱ ❛té ♠❡s♠♦✱ ♣❛r❛ ❛♣❧✐❝❛r ♥❛ ❣❡♦♠❡tr✐❛ ♣❧❛♥❛ ❡ ❡s♣❛❝✐❛❧✳ ❆♣❡s❛r ❞♦ ❝♦♥❝❡✐t♦ ❞❡ ▼é❞✐❛ s❡r ❡①tr❡♠❛♠❡♥t❡ s✐♠♣❧❡s✱ é ✐♠♣♦rt❛♥t❡ s❛❜❡r ✐❞❡♥t✐✜❝❛r ❛s s✐t✉❛çõ❡s ❛❞❡q✉❛❞❛s ♣❛r❛ ✉♠❛ ❛♣❧✐❝❛çã♦ ❝♦rr❡t❛ ❞❡ ❝❛❞❛ t✐♣♦ ❞❡ r❡❧❛çã♦ ❡♥✈♦❧✈❡♥❞♦ ♦s ❝♦♥❝❡✐t♦s ❞❡ ▼é❞✐❛✱ ♣♦✐s ✉♠❛ ❛♣❧✐❝❛çã♦ ✐♥❝♦rr❡t❛ ♣♦❞❡ ❣❡r❛r ❡rr♦s r❡❧❡✈❛♥t❡s ❡ ❡st✐♠❛t✐✈❛s ❞✐s❝r❡♣❛♥t❡s ❝♦♠ ❛ r❡❛❧✐❞❛❞❡✳
✶✳✶ ❉❡✜♥✐çõ❡s ❞❡ ▼é❞✐❛s
P❛r❛ ❛s ♠é❞✐❛s q✉❡ tr❛❜❛❧❤❛♠♦s ♣♦❞❡♠♦s ❞❛r ✉♠❛ ❝♦♥❝❡✐t✉❛❧✐③❛çã♦ ❣❡r❛❧✳ ❆ ✐❞❡✐❛ ❝❤❛✈❡ é ❛ ❞❛ s✉❜st✐t✉✐çã♦ ❞❡ ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ ✈❛❧♦r❡s ♣♦r ✉♠ ✈❛❧♦r q✉❡ r❡♣r❡✲ s❡♥t❡ t♦❞♦s✳
❉❡✜♥✐çã♦ ✶✳✶ ❈♦♥s✐❞❡r❡ ✉♠❛ s❡q✉ê♥❝✐❛ ✜♥✐t❛ ❞❡ ♥ú♠❡r♦s r❡❛✐s (x1, x2, . . . , xn) ❡
⋆ ✉♠❛ ♦♣❡r❛çã♦ s♦❜r❡ ♦s ♠❡♠❜r♦s ❞❛ s❡q✉ê♥❝✐❛✳ ❯♠❛ ♠é❞✐❛ ❞♦s ❡❧❡♠❡♥t♦s ❞❛
s❡q✉ê♥❝✐❛ ❝♦♠ r❡s♣❡✐t♦ à ♦♣❡r❛çã♦ ⋆ é ✉♠ ♥ú♠❡r♦ r❡❛❧ M ❝♦♠ ❛ ♣r♦♣r✐❡❞❛❞❡ ❞❡
s✉❜st✐t✉✐r t♦❞♦s ♦s ❡❧❡♠❡♥t♦s ❞❛ s❡q✉ê♥❝✐❛ ♥♦ q✉❡ ❞✐③ r❡s♣❡✐t♦ ❛ ♦♣❡r❛çã♦ ⋆ ✱ ✐st♦ é
x1⋆ x2⋆ . . . ⋆ xn=M ⋆ M ⋆ . . . ⋆ M
| {z }
n t❡r♠♦s
.
❖❜s❡r✈❛çã♦ ✶✳✶ ❖ ❝♦♥❝❡✐t♦ ❣❡r❛❧ ❞❡ ♠é❞✐❛ ❞❡s❝r✐t♦ ❛❝✐♠❛ é ❛❜str❛t♦✳ P♦rt❛♥t♦✱ ❞❡✈❡♠♦s ❡s♣❡❝✐❛❧✐③á✲❧♦ ♣❛r❛ ❡♥❝♦♥tr❛r ✐♠♣♦rt❛♥t❡s t✐♣♦s ✉s✉❛✐s ❞❡ ♠é❞✐❛✳ ◆♦s ❝❛s♦s q✉❡ tr❛❜❛❧❤❛♠♦s ❛ ♠é❞✐❛ é✱ ❞❡ ❢❛t♦✱ ✉♠ ♥ú♠❡r♦ ✐♥t❡r♠❡❞✐ár✐♦✱ ❡♥tr❡ ♦ ♠❡♥♦r ❡ ♦ ♠❛✐♦r ❡❧❡♠❡♥t♦ ❞❛ s❡q✉ê♥❝✐❛✳ ■st♦ é✱
min{xi} ≤M ≤max{xi}
❈❧❛r❛♠❡♥t❡✱ s❡ ♦ ♠❡♥♦r ❡ ♦ ♠❛♦✐r ♥ú♠❡r♦s sã♦ ✐❣✉❛✐s✱ ❛ ♠é❞✐❛ é ✐❣✉❛❧ ❛ ❡st❡s ♥ú♠❡r♦s✳
❖❜s❡r✈❛çã♦ ✶✳✷ ◗✉❛♥❞♦ ❢♦r ❞✐t♦ q✉❡ ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ t❡r♠♦s ❡stá ❡♠ ♦r❞❡♠ ❝r❡s❝❡♥t❡ q✉❡r❡♠♦s ❞✐③❡r q✉❡ x1 ≤ x2 ≤ . . . ≤ xn✳ ◗✉❛♥❞♦ ❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❢♦r
❡str✐t❛✱ x1 < x2 < . . . < xn ❞✐r❡♠♦s q✉❡ ❛ s❡q✉ê♥❝✐❛ ❞❡ t❡r♠♦s é ❡str✐t❛♠❡♥t❡
❝r❡s❝❡♥t❡✳P❛r❛ ❛s ♠é❞✐❛s q✉❡ tr❛❜❛❧❤❛♠♦s ❡ ♥❛ ♠❛✐♦r✐❛ ❞♦s r❡s✉❧t❛❞♦s ♦❜t✐❞♦s ❛ ♦r❞❡♠ ❞♦s t❡r♠♦s ♥ã♦ é r❡❧❡✈❛♥t❡✱ ♣♦✐s ❡st❛♠♦s ❧✐❞❛♥❞♦ ❝♦♠ ♦♣❡r❛çõ❡s ❝♦♠✉t❛t✐✈❛s ❝♦♠♦ ❛❞✐çã♦ ❡ ♠✉❧t✐♣❧✐❝❛çã♦ ❡♥tr❡ ♥ú♠❡r♦s r❡❛✐s✳
✶✳✶✳✶ ▼é❞✐❛ ❆r✐t♠ét✐❝❛
❉❡✜♥✐çã♦ ✶✳✷ ❆ ♠é❞✐❛ ❛r✐t♠ét✐❝❛ ✭s✐♠♣❧❡s✮ ❞❡ ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ ♥ú♠❡r♦s r❡❛✐s
(x1, x2, . . . , xn) é ♦ ♥ú♠❡r♦ ❆ ❝♦♠ r❡s♣❡✐t♦ ❛ ♦♣❡r❛çã♦ ❞❡ ❛❞✐çã♦✱ ❞❡st❛ ❢♦r♠❛
(x1+x2+. . .+xn) = A+A+. . .+A
| {z }
n t❡r♠♦s
=n.A.
P♦rt❛♥t♦✱
A= x1+x2+. . .+xn
n .
❆♣❧✐❝❛çã♦ ✶✳✶ ❈❛❞❛ t❡r♠♦ ❞❡ ✉♠❛ ♣r♦❣r❡ssã♦ ❛r✐t♠ét✐❝❛ ✭P❆✮✱ ❡①❝❡t♦ ♦s ❡①tr❡♠♦s✱ ♣♦❞❡ s❡r ♦❜t✐❞♦ ♣❡❧❛ ♠é❞✐❛ ❛r✐t♠ét✐❝❛ ❞♦s t❡r♠♦s ❡q✉✐❞✐st❛♥t❡s✳ ❊♠ ♣❛rt✐❝✉❧❛r❀
ai =
ai−1 +ai+1
2 ♣❛r❛ i= 2,3, . . . , n−1⇔ ai+1−ai =ai−ai−1 =r
♦♥❞❡ ✭✳✳✳✱ ai−1, ai, ai+1✱ ✳✳✳✮ ❡stã♦ ❡♠ P❆ ❝✉❥❛ r❛③ã♦ é r✳
❊①❡♠♣❧♦ ✶✳✶ ❈♦♥s✐❞❡r❡ ✉♠❛ P❆ t❛❧ q✉❡ a3 = 7 ❡ a19= 55✳ ❉❡t❡r♠✐♥❡ a7✳
❙♦❧✉çã♦✿ ❇❛st❛ ♣❡r❝❡❜❡r q✉❡
a11=
a3+a19
2 =
7 + 55 2 = 31 ❡
a7 =
a3 +a11
2 =
7 + 31 2 = 19.
⋄
❊①❡♠♣❧♦ ✶✳✷ ❊♠ s❡✐s ♣r♦✈❛s✱ ♦♥❞❡ ❛s ♥♦t❛s ❛tr✐❜✉✐❞❛s ✈❛r✐❛♠ ❞❡ ✵ ❛ ✶✵✵✱ ✉♠ ❡st✉❞❛♥t❡ ♦❜t❡✈❡ ♠é❞✐❛ ✽✸✳ ❙❡ ❛ ♠❡♥♦r ♥♦t❛ ❢♦r ❞❡s♣r❡③❛❞❛ ❛ s✉❛ ♠é❞✐❛ s♦❜❡ ♣❛r❛ ✽✼✳ ◗✉❛❧ ❢♦✐ ❛ ♠❡♥♦r ♥♦t❛ ♦❜t✐❞❛ ♥❛s ✻ ♣r♦✈❛s❄
❙♦❧✉çã♦✿ ❈♦♥s✐❞❡r❡ x4 ❛ ♠❡♥♦r ♥♦t❛✳ ❊♥tã♦✱
x1+x2+x3+x4+x5 +x6
6 = 83 logo x1+x2+x3+x4+x5+x6 = 498 ✭✶✳✶✮
❡
x1+x2+x3+x5+x6
5 = 87 logo x1 +x2+x3+x5+x6 = 435. ✭✶✳✷✮
❙✉❜tr❛✐♥❞♦ 1.2❞❡ 1.1✱ t❡♠♦s
x4 = 498−435 ⇒x4 = 63
P♦rt❛♥t♦ ❛ ♠❡♥♦r ♦❜t✐❞❛ ♥♦t❛ ❢♦✐ 63 ✳ ⋄
❊①❡♠♣❧♦ ✶✳✸ ❙✉♣♦♥❤❛ q✉❡ ❛ ♠é❞✐❛ ❛r✐t♠ét✐❝❛ ❞❡ ✉♠❛ t✉r♠❛ ❞❡ 20♣❡ss♦❛s s❡❥❛ 7 ♥✉♠❛ ❞❡t❡r♠✐♥❛❞❛ ♣r♦✈❛✳ ◗✉❛❧ s❡rá ❛ ♠é❞✐❛ ❞❛s 19 ♣❡ss♦❛s r❡st❛♥t❡s s❡ r❡t✐r❛r♠♦s ❞❛ t✉r♠❛ ♦ ú♥✐❝♦ ❛❧✉♥♦ q✉❡ t✐r♦✉ ♥♦t❛ 10✳
❙♦❧✉çã♦✿ ❚❡♠♦s ♣❡❧❛ ♠é❞✐❛ ❛rt♠ét✐❝❛ q✉❡
x1+x2+. . .+x20
20 = 7 =⇒x1+x2+. . .+x20= 140.
❖✉ s❡❥❛✱ ❛ s♦♠❛ ❞❛s ♥♦t❛s é ✶✹✵✳ ❈♦♥s✐❞❡r❡ xn = 10 ♦ ❛❧✉♥♦ ❝♦♠ ♥♦t❛ 10✳ ❉❡st❛
❢♦r♠❛✱
x1+x2+. . .+x20−xn
19 =
140−10 19 =
130
19 ≈6,84
P♦rt❛♥t♦✱ ❛ ♠é❞✐❛ ❞❛s 19 ♣❡ss♦❛s s❡r✐❛ 6,84✳ ⋄ ▲❡♠❛ ✶✳✶ Pr♦♣r✐❡❞❛❞❡ ❞❛ ▼é❞✐❛ ❆rt♠ét✐❝❛ ❙❡ ❛ ♠é❞✐❛ ❛r✐t♠ét✐❝❛ ❞♦s ♥ú♠❡✲
r♦s x1, x2, ..., xn é ✐❣✉❛❧ ❛ x✱ ♣❡❧♦ ♠❡♥♦s✱ ✉♠ ❞♦s ♥ú♠❡r♦s x1, x2, ..., xn é ♠❛✐♦r q✉❡
♦✉ ✐❣✉❛❧ ❛ x✳ P♦❞❡♥❞♦ ♦ ❧❡✐t♦r ❝♦♥s✉❧t❛r ❬✷❪ ♣❛r❛ ♠❛✐s ❞❡t❛❧❤❡s✳
❉❡♠♦♥str❛çã♦✿ ❙✉♣♦r✱ ♣♦r ❝♦♥tr❛❞✐çã♦✱ q✉❡ xi < x♣❛r❛ t♦❞♦ i= 1,2, ..., n.
❖✉ s❡❥❛✱ x1 < x, x2 < x, ..., xn < x ✳ ❆ss✐♠✱ x1 +x2 +...+xn < n.x✱ ❡ ❞✐✈✐❞✐♥❞♦
❛ ❞❡s✐❣✉❛❧❞❛❞❡ ♣♦r n✱ t❡♠♦s✱ x1+x2 +...+xn
n < x✳ ❊ ♣♦rt❛♥t♦ x < x✱ ♦ q✉❡ é
❛❜s✉r❞♦✳ ▲♦❣♦✱ ❡①✐st❡ i∈1,2, ..., n t❛❧ q✉❡ xi ≥x✳
❆ ♣♦ss✐❜✐❧✐❞❛❞❡ ❞❡ ♦❝♦rr❡r ✈ár✐♦sxi ✐❣✉❛✐s ✐♥s♣✐r❛ ❛ ❞❡✜♥✐çã♦ ❞❡ ✉♠❛ ♠é❞✐❛ ❛r✐t✲
♠ét✐❝❛ ♦♥❞❡ ❛s ❣r❛♥❞❡③❛s ♣♦ss❛♠ t❡r ♣❡s♦s ❛ ❡❧❛s ❛ss♦❝✐❛❞♦s✱ ♣❡s♦s ❡st❡s q✉❡ ❞❡ ❛❧❣✉♠❛ ❢♦r♠❛ ❞❡❡♠ ✉♠❛ ✐❞❡✐❛ ❞❡ ♠✉❧t✐♣❧✐❝✐❞❛❞❡✳ ❊♥tã♦✱ s❡ ❛❣r✉♣❛r♠♦s ♦s t❡r♠♦s ✐❣✉❛✐s ❡ ♠✉❧t✐♣❧✐❝❛r♠♦s ♣❡❧❛ ❢r❡q✉ê♥❝✐❛ ❞❡ ❝❛❞❛ ✉♠ ❞❡❧❡s t❡r❡♠♦s ❛ ❝♦♥❤❡❝✐❞❛ ♠é❞✐❛ ❛r✐t♠ét✐❝❛ ♣♦♥❞❡r❛❞❛✳
✶✳✶✳✷ ▼é❞✐❛ P♦♥❞❡r❛❞❛
❉❡✜♥✐çã♦ ✶✳✸ ❆ ♠é❞✐❛ ❛r✐t♠ét✐❝❛ ✭♣♦♥❞❡r❛❞❛✮ ❞❡ ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ ♥ú♠❡r♦s
(x1, x2, . . . xn) ❡ ❞❡ ♣❡s♦s (p1, p2, . . . pn)✱ ❝♦♠ ♦s pis >0✱ é ♦ ♥ú♠❡r♦ P ❝♦♠ r❡s♣❡✐t♦
à ♦♣❡r❛çã♦ ❞❡ ❛❞✐çã♦ ❝♦♠ ♣❡s♦s✱ ❞❡st❛ ❢♦r♠❛✱
p1x1+p2x2+. . .+pnxn =p1P +p2P +. . .+pnP.
P♦rt❛♥t♦✱
P = p1x1+p2x2+. . .+pnxn
p1+p2+. . .+pn
.
❊①❡♠♣❧♦ ✶✳✹ ❊♠ ✉♠ ❣r✉♣♦ ❞❡ ♣❡ss♦❛s✱ 70% ❞❛s ♣❡ss♦❛s sã♦ ❛❞✉❧t♦s ❡ 30% sã♦ ❝r✐❛♥ç❛s✳ ❆ ♠❛ss❛ ♠é❞✐❛ ❞♦s ❛❞✉❧t♦s é 70❦❣ ❡ ❛ ♠❛ss❛ ♠é❞✐❛ ❞❛s ❝r✐❛♥ç❛s é ❞❡ 40 ❦❣✳ ◗✉❛❧ ❛ ♠❛ss❛ ♠é❞✐❛ ❞♦ ❣r✉♣♦❄ ❘❡t✐r❛❞♦ ❞❡ ❬✻❪✳
❙♦❧✉çã♦✿ ❆ ♠❛ss❛ ♠é❞✐❛ s❡rá ❛ ♠é❞✐❛ ❛r✐t♠ét✐❝❛ ♣♦♥❞❡r❛❞❛ ❞♦s ❞♦✐s s✉❜❣r✉♣♦s✱ ❝♦♠ ♣❡s♦s r❡❧❛t✐✈♦s ❞❡ 0,7 ❡ 0,3✳
P = 0,7.70 + 0,3.40 0,7 + 0,3 =
49 + 12 1 = 61
▲♦❣♦✱ ♦ ❣r✉♣♦ t❡♠ 61 ❦❣ ❞❡ ♠❛ss❛ ♠é❞✐❛✳ ⋄
▼é❞✐❛ ❆r✐t♠ét✐❝❛ ❙✐♠♣❧❡s ❡ ▼é❞✐❛ ❆r✐t♠ét✐❝❛ P♦♥❞❡r❛❞❛
❱❡r✐✜❝❛✲s❡ q✉❡ ❛ ▼é❞✐❛ ❆r✐t♠ét✐❝❛ ❙✐♠♣❧❡s ♥ã♦ tr❛❞✉③ ♣r❡❝✐s❛♠❡♥t❡ ❞✐❢❡r❡♥ç❛s ❞❡ ❞❡s❡♠♣❡♥❤♦✱ ❝r❡s❝✐♠❡♥t♦ ♣♦♣✉❧❛❝✐♦♥❛❧ ❡t❝✳✱ ♣♦r ❝♦♥s✐❞❡r❛r q✉❡ t♦❞♦s ♦s ❡❧❡♠❡♥✲ t♦s ❝♦♠♣♦♥❡♥t❡s ♣♦ss✉❡♠ ♦ ♠❡s♠♦ ♣❡s♦✱ ♦✉ s❡❥❛✱ ❛ ▼é❞✐❛ ❆r✐t♠ét✐❝❛ ❙✐♠♣❧❡s ♥ã♦ ❝♦♥s✐❞❡r❛ r❡♣❡t✐çõ❡s ❞♦s ❡❧❡♠❡♥t♦s✱ t❛♠♣♦✉❝♦ ❛s ✈❛r✐❛çõ❡s ❞❡st❡s ♠❡s♠♦s ❡❧❡♠❡♥✲ t♦s ❛♦ ❧♦♥❣♦ ❞♦ t❡♠♣♦✳ P♦r ✐ss♦✱ ❛ ▼é❞✐❛ ❆r✐t♠ét✐❝❛ é ♠❛✐s ♣r❡❝✐s❛ ♣❛r❛ ♠♦str❛r r❡t♦r♥♦s ♥✉♠ér✐❝♦s ❞❡ ♣r♦❜❧❡♠❛s q✉❡ ♥ã♦ ❡♥✈♦❧✈❛♠ r❡♣❡t✐çõ❡s ❞♦s ❡❧❡♠❡♥t♦s ❝♦♥s✲ t✐t✉✐♥t❡s ♦✉ ❣r❛♥❞❡s ✈❛r✐❛çõ❡s ❡♥tr❡ ♦s ✈❛❧♦r❡s ❞❡st❡s ❡❧❡♠❡♥t♦s ❛♦ ❧♦♥❣♦ ❞♦ t❡♠♣♦✳ ◆❡st❡s ❝❛s♦s✱ ❛ ▼é❞✐❛ ❆r✐t♠ét✐❝❛ P♦♥❞❡r❛❞❛ ♠♦str❛ r❡s✉❧t❛❞♦s ♠❛✐s ♣r❡❝✐s♦s✳ ❊①❡♠♣❧♦ ✶✳✺ ❊♠ ✉♠ ❞❡♣❛rt❛♠❡♥t♦ ❞❡ ✉♠❛ ❡♠♣r❡s❛ q✉❛❧q✉❡r✱ ✉♠ ❢✉♥❝✐♦♥ár✐♦ r❡✲ ❝❡❜❡ ✉♠ s❛❧ár✐♦ ❞❡ ❘✩ ✶✳✵✵✵✱✵✵ ♣♦r ♠ês✱ ❡♥q✉❛♥t♦ ♦✉tr♦ r❡❝❡❜❡ ❘✩ ✶✷✳✺✵✵✱✵✵ ♣♦r ♠ês✳ ◗✉❛❧ é ❛ ♠é❞✐❛ s❛❧❛r✐❛❧ ♠❡♥s❛❧ ❞❡st❡s ❢✉♥❝✐♦♥ár✐♦s❄
❙♦❧✉çã♦✿ ❈♦♠♦ t❡♠♦s q✉❡ x1 = 1000, x2 = 12500 ❡ q✉❡ n = 2✱ ♦ ♥ú♠❡r♦ ❢✉♥❝✐♦✲
♥ár✐♦s✳ ❊♥tã♦✱
A= 1000 + 12500
2 ⇒A= 6750
▲♦❣♦✱ ❛ ♠é❞✐❛ s❛❧❛r✐❛❧ ♠❡♥s❛❧ s❡rá ❞❡ ❘✩ ✻✳✼✺✵✱✵✵✳ ⋄
❱❡r✐✜❝❛✲s❡ q✉❡ ♦ ✈❛❧♦r ♦❜t✐❞♦ ♣♦r ♠❡✐♦ ❞❛ ▼é❞✐❛ ❆r✐t♠ét✐❝❛ ❙✐♠♣❧❡s ♥ã♦ ♣♦ss✉✐ ✉♠❛ ❝♦rr❡s♣♦♥❞ê♥❝✐❛✱ q✉❡ ♣❛r❡ç❛ ✈❡r❞❛❞❡✐r❛✱ ❝♦♠ ♦s s❛❧ár✐♦s ❛♣r❡s❡♥t❛❞♦s✳
❱❛♠♦s ✈❡r✐✜❝❛r✱ ♥♦ ♣ró①✐♠♦ ❡①❡♠♣❧♦✱ s❡ ❤❛✈❡rá ❡ss❛ ❞✐s❝r❡♣â♥❝✐❛ ❡♥tr❡ ♦s ✈❛❧♦r❡s ❛♣r❡s❡♥t❛❞♦s ❡ ❛ ♠é❞✐❛✿
❊①❡♠♣❧♦ ✶✳✻ ❱❡r✐✜q✉❡ ❛ t❛❜❡❧❛ ❛ s❡❣✉✐r ❡✱ ❝♦♠ ❜❛s❡ ♥♦s ❞❛❞♦s ♥❡❧❛ ❝♦♥t✐❞♦s✱ ❝❛❧✲ ❝✉❧❡ ❛ ♠é❞✐❛ s❛❧❛r✐❛❧ ♠❡♥s❛❧✿
◗✉❛♥t✐❞❛❞❡ ❞❡ ❋✉♥❝✐♦♥ár✐♦s ❙❛❧ár✐♦s ✴ ♠ês ✭❡♠ ❘✩✮
✶✺ ✽✵✵✱✵✵
✸ ✸✳✵✵✵✱✵✵
✷ ✺✳✷✺✵✱✵✵
✶ ✶✷✳✶✵✵✱✵✵
❚❛❜❡❧❛ ✶✳✶✿ ❚❛❜❡❧❛ ❞♦s ❙❛❧ár✐♦s ❞♦s ❋✉♥❝✐♦♥ár✐♦s
❙♦❧✉çã♦✿ ❈♦♠♦ ❤á r❡♣❡t✐çõ❡s ❞♦ ♠❡s♠♦ ✈❛❧♦r s❛❧❛r✐❛❧✱ ♦✉ s❡❥❛✱ ♠❛✐s ❞❡ ✉♠ ❢✉♥❝✐♦✲ ♥ár✐♦ r❡❝❡❜❡ ♦ ♠❡s♠♦ s❛❧ár✐♦✱ ♦ ✉s♦ ❞❛ ▼é❞✐❛ ❆r✐t♠ét✐❝❛ P♦♥❞❡r❛❞❛ é ♠❛✐s ✐♥❞✐❝❛❞♦✳ ❈♦♠♦ t❡♠♦s ♦s ✈❛❧♦r❡s
x1 = 800, x2 = 3000, x3 = 5250 e x4 = 12.100
❡ ♦s ♣❡s♦s
p1 = 15, p2 = 3, p3 = 2 e p4 = 1
❊♥tã♦✱ ❛♣❧✐❝❛♥❞♦ ❛ ▼é❞✐❛ P♦♥❞❡r❛❞❛ t❡♠♦s
P = (15.800) + (3.3000) + (2.5250) + (1.12100) 15 + 3 + 2 + 1
P = 12000 + 9000 + 10500 + 12100
21 ⇒P = 2076,19
P♦rt❛♥t♦ ❛ ♠é❞✐❛ s❛❧❛r✐❛❧ ♠❡♥s❛❧ s❡r✐❛ ❞❡ ❘✩ ✷✵✼✻✱✶✾✳ ⋄
❙❡ ♦s ❢✉♥❝✐♦♥ár✐♦s ❝♦♥❢r♦♥t❛ss❡♠ s❡✉s s❛❧ár✐♦s ❡ ❛s ♠é❞✐❛s ♠❡♥s❛✐s ❞♦s s❡✉s s❛❧á✲ r✐♦s ❝♦♠ ♦s ♦✉tr♦s ❢✉♥❝✐♦♥ár✐♦s✱ ❝❡rt❛♠❡♥t❡✱ ♥✐♥❣✉é♠ ❝♦♥❝♦r❞❛r✐❛ ❝♦♠ t❛✐s ✈❛❧♦r❡s✱ t❛♥t♦ ♦s q✉❡ ❣❛♥❤❛♠ ♠❛✐s q✉❛♥t♦ ♦s q✉❡ ❣❛♥❤❛♠ ♠❡♥♦s✳ P♦r ❡ss❛ r❛③ã♦✱ ❝♦♥s✐❞❡✲ r❛♠♦s ❛s ▼é❞✐❛s ❆r✐t♠ét✐❝❛s✭s✐♠♣❧❡s ♦✉ ♣♦♥❞❡r❛❞❛s✮ ❛♣❡♥❛s ❝♦♠♦ ✉♠❛ t❡♥t❛t✐✈❛ ❞❡ ♠✐♥✐♠✐③❛r ❛s r❡❧❛çõ❡s ❡♥tr❡ ❞✉❛s ♦✉ ♠❛✐s ♠❡❞✐❞❛s✱ ♥ã♦ t❡♥❞♦ ♠✉✐t❛ ✉t✐❧✐❞❛❞❡ ♣rát✐❝❛✱ ❛ ♥ã♦ s❡r ❡♠ s✐t✉❛çõ❡s ♥❛s q✉❛✐s ❡①✐st❛ ✉♠❛ ❣r❛♥❞❡ q✉❛♥t✐❞❛❞❡ ❞❡ ❡❧❡♠❡♥✲ t♦s ❛ ♠❡❞✐r ❡ s❡ ❢❛③ ♥❡❝❡ssár✐♦ ❞❡t❡r♠✐♥❛r ❛♣❡♥❛s ✉♠❛ ❛♠♦str❛ ♣❛r❛ ❧✐❞❛r ❝♦♠ ♦ t❡♠❛ ❛❜♦r❞❛❞♦✳ P♦r ❝♦♥s❡q✉ê♥❝✐❛✱ ❛s ▼é❞✐❛s ●❡♦♠étr✐❝❛s ❡ ❛s ▼é❞✐❛s ❍❛r♠ô♥✐❝❛s ♣♦ss✉❡♠ ♠❛✐s ✉t✐❧✐❞❛❞❡ ♣rát✐❝❛✳
✶✳✶✳✸ ▼é❞✐❛ ●❡♦♠étr✐❝❛
❉❡✜♥✐çã♦ ✶✳✹ ❆ ♠é❞✐❛ ❣❡♦♠étr✐❝❛ ❞❡ ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ ♥ú♠❡r♦s r❡❛✐s ♣♦s✐t✐✈♦s
(x1, x2, . . . , xn)é ♦ ♥ú♠❡r♦ G❝♦♠ r❡s♣❡✐t♦ à ♦♣❡r❛çã♦ ❞❡ ♠✉❧t✐♣❧✐❝❛çã♦✱ ❞❡st❛ ❢♦r♠❛
x1.x2. . . . .xn=G.G. . . . .G
| {z }
n t❡r♠♦s
=Gn.
P♦rt❛♥t♦✱
G= √n
x1.x2. . . . .xn.
❆♣❧✐❝❛çã♦ ✶✳✷ ❈❛❞❛ t❡r♠♦ ❞❡ ✉♠❛ ♣r♦❣r❡ssã♦ ❣❡♦♠étr✐❝❛ ✭P●✮✱ ❡①❝❡t♦ ♦s ❡①tr❡✲ ♠♦s✱ ♣♦❞❡ s❡r ♦❜t✐❞♦ ♣❡❧❛ ♠é❞✐❛ ❣❡♦♠étr✐❝❛✱ ❡♠ ♠ó❞✉❧♦✱ ❞♦s t❡r♠♦s ❡q✉✐❞✐st❛♥t❡s✳ ❊♠ ♣❛rt✐❝✉❧❛r
|ai|=√ai−1.ai+1 para i= 2,3, . . . , n−1⇔
ai
ai−1
= ai+1
ai
=q
♦♥❞❡ (..., ai−1, ai, ai+1, ...) ❡stã♦ ❡♠ P● ❝✉❥❛ r❛③ã♦ é q✳
❊①❡♠♣❧♦ ✶✳✼ ❈♦♥s✐❞❡r❡ ✉♠❛ P● t❛❧ q✉❡ a8 = 1 ❡ a16= 625✳ ❉❡t❡r♠✐♥❡ a10✳
❙♦❧✉çã♦✿ ❇❛st❛ ♣❡r❝❡❜❡r q✉❡ a12 é ❛ ♠é❞✐❛ ❣❡♦♠étr✐❝❛ ❡♥tr❡ a8 ❡ a16✱ ♣♦rt❛♥t♦
a12=√a8.a16=
√
1.625 =√625 = 25
❡ q✉❡ a10 é ❛ ♠é❞✐❛ ❣❡♦♠étr✐❝❛ ❡♥tr❡ a8 ❡ a12✱ ♣♦rt❛♥t♦
a10=√a8.a12 =
√
1.25 =√25 = 5
⋄
❆♣❧✐❝❛çã♦ ✶✳✸ ❖ ♠ó❞✉❧♦ ❞♦ ♣r♦❞✉t♦ ❞♦s n ♣r✐♠❡✐r♦s t❡r♠♦s ❞❡ ✉♠❛ P● é✿ |Pn|= (a1an)
n
2
❉❡ ❢❛t♦✱ s❡ ❝♦♥s✐❞❡r❛r♠♦s ♦ ♣r♦❞✉t♦ Pn = a1.a2. . . . .an−a.an ❡ r❡❡s❝r❡✈❡r♠♦s ❡ss❡
♣r♦❞✉t♦ ♥❛ ♦r❞❡♠ ✐♥✈❡rs❛ ❞♦s t❡r♠♦s✱ ♦✉ s❡❥❛✱ Pn =an.an−a. . . . .a2.a1✳ ❆ ♠✉❧t✐♣❧✐✲ ❝❛çã♦ ❞❛s ❞✉❛s ❡①♣r❡ss♦❡s t❡r♠♦ ❛ t❡r♠♦✱ ♥❛ ♦r❞❡♠ s❡rá
P2
n = (a1.an).(a2.an−1). . .(an−1.a2).(an.a1)
s❛❜❡♠♦s q✉❡ a1.an=a2.an−1 =. . .=an−1.a2 =an.a1✱ ❧♦❣♦
P2
n = (a1.an)n ⇔ |Pn|= (a1an)
n
2
❊①❡♠♣❧♦ ✶✳✽ ❉❡t❡r♠✐♥❡ ♦ ♣r♦❞✉t♦s ❞♦s ✹✵ ♣r✐♠❡✐r♦s t❡r♠♦s ❞❡ ✉♠❛ P● ❝✉❥♦ ♣r✐✲ ♠❡✐r♦ ❡ ♦ q✉❛❞r❛❣és✐♠♦ t❡r♠♦ sã♦ ✶ ❡ ✷ r❡s♣❡❝t✐✈❛♠❡♥t❡✳
❙♦❧✉çã♦✿
|P40|= (1.2)
40 2 = 220
❈♦♠♦ a1 ❡ a40 sã♦ ❛♠❜♦s ♣♦s✐t✐✈♦s✱ ❡♥tã♦ ❛ P● é ♣♦s✐t✐✈❛✱ ❧♦❣♦✱ P40= 220 ⋄
❆♣❧✐❝❛çã♦ ✶✳✹ ➱ ♠✉✐t♦ ❝♦♠✉♠ s❡ ❢❛③❡r ✉s♦ ❞❛s ▼é❞✐❛s ●❡♦♠étr✐❝❛s ❡♠ ❣❡♦♠❡tr✐❛ ♣❧❛♥❛ ❡ ❡s♣❛❝✐❛❧✿
❊①❡♠♣❧♦ ✶✳✾ ❆ ❛❧t✉r❛ ❞❡ ✉♠ tr✐â♥❣✉❧♦ r❡tâ♥❣✉❧♦ ❡♠ r❡❧❛çã♦ à ❤✐♣♦t❡♥✉s❛ é ❛ ♠é❞✐❛ ❣❡♦♠étr✐❝❛ ❞❛s ♣r♦❥❡çõ❡s ♦rt♦❣♦♥❛✐s ❞♦s ❝❛t❡t♦s s♦❜r❡ ❛ ❤✐♣♦t❡♥✉s❛✳
❙♦❧✉çã♦✿ ❖❜s❡r✈❡ ❛ ❝♦♥str✉çã♦ ❛❜❛✐①♦✱ ❛♣❧✐❝❛♥❞♦ ❛ r❛③ã♦ ❞❡ s❡♠❡❧❤❛♥ç❛ ♥♦s tr✐â♥❣✉❧♦s s❡♠❡❧❤❛♥t❡s AHC ❡ BHA t❡r❡♠♦s✿
❋✐❣✉r❛ ✶✳✶✿ ❆ ❛❧t✉r❛ é ❛ ▼é❞✐❛ ●❡♦♠étr✐❝❛ ❞❛s ♣r♦❥❡çõ❡s ❞♦s ❝❛t❡t♦s s♦❜r❡ ❛ ❤✐♣♦✲ t❡♥✉s❛✳
AH
CH =
BH
AH ⇔
h
m =
n
h ⇔h=
√
m×n.
⋄
❊①❡♠♣❧♦ ✶✳✶✵ P♦❞❡♠♦s ✐♥t❡r♣r❡t❛r ❛ ▼é❞✐❛ ●❡♦♠étr✐❝❛ ❞❡ três ♥ú♠❡r♦s ❛✱ ❜ ❡ ❝ ❝♦♠♦ ❛ ♠❡❞✐❞❛ ❧ ❞❛ ❛r❡st❛ ❞❡ ✉♠ ❝✉❜♦✱ ❝✉❥♦ ✈♦❧✉♠❡ é ♦ ♠❡s♠♦ ❞❡ ✉♠ ♣r✐s♠❛ r❡t❛♥❣✉❧❛r r❡t♦✱ ❞❡s❞❡ q✉❡ ❡st❡ t❡♥❤❛ ❛r❡st❛s ♠❡❞✐♥❞♦ ❡①❛t❛♠❡♥t❡ ❛✱ ❜ ❡ ❝✳
❙♦❧✉çã♦✿ ❙❛❜❡♠♦s q✉❡ ♦ ✈♦❧✉♠❡ ❞❡ ✉♠ Pr✐s♠❛ ❞❡ ❞✐♠❡ñsõ❡s a, b, c é ❞❛❞❛ ♣♦r
V = a.b.c ❡ ♦ ✈♦❧✉♠❡ ❞❡ ✉♠ ❝✉❜♦ ❞❡ ❛r❡st❛ l é ❞❛❞❛ ♣♦r V = l3✳ ■❣✉❛❧❛♥❞♦ ♦s
✈♦❧✉♠❡s✿
a.b.c=l3
→l =√3
a.b.c
P♦rt❛♥t♦ t❡♠♦s q✉❡ l é ❛ ▼é❞✐❛ ●❡♦♠étr✐❝❛ ❡♥tr❡a✱ b ❡ c✳ ⋄
❋✐❣✉r❛ ✶✳✷✿ Pr✐s♠❛ ❞❡ ❞✐♠❡♥sõ❡s ❛✱❜✱❝ ❡ ❈✉❜♦ ❞❡ ❛r❡st❛ ❧✳
❆♣❧✐❝❛çã♦ ✶✳✺ ❆ ▼é❞✐❛ ●❡♦♠étr✐❝❛ é ❢r❡q✉❡♥t❡♠❡♥t❡ ✉s❛❞❛ ♥❛ ▼❛t❡♠át✐❝❛ ❋✐♥❛♥✲ ❝❡✐r❛ q✉❛♥❞♦ ❞✐s❝✉t✐♠♦s t❛①❛s ❞❡ r❡♥❞✐♠❡♥t♦ ❡♠ ✐♥✈❡st✐♠❡♥t♦s✱ ♦✉ ❛✐♥❞❛ ❡♠ ❥✉r♦s s✉❝❡ss✐✈♦s✳
❊①❡♠♣❧♦ ✶✳✶✶ ❯♠ ✐♥✈❡st✐♠❡♥t♦ r❡♥❞❡✉ ❛♥✉❛❧♠❡♥t❡ ❝♦♥❢♦r♠❡ ❛ s❡❣✉✐♥t❡ t❛❜❡❧❛✿ ◗✉❛❧ ❛ ♠é❞✐❛ ❛♥✉❛❧ ❞❡ r❡♥❞✐♠❡♥t♦ ❞❡ss❡ ✐♥✈❡st✐♠❡♥t♦❄
✷✵✶✸ ✷✵✶✹ ✷✵✶✺
✶✺✪ ✺✪ ✼✪
❚❛❜❡❧❛ ✶✳✷✿ ❚❛❜❡❧❛ ❞❡ ❘❡♥❞✐♠❡♥t♦s ❆♥✉❛✐s ❙♦❧✉çã♦✿ ◗✉❡r❡♠♦s ❡♥❝♦♥tr❛r ✉♠❛ ❞❡t❡r♠✐♥❛❞❛ t❛①❛ i t❛❧ q✉❡
(1 +i)3
= (1 + 0,15).(1 + 0,05).(1 + 0,07).
▲♦❣♦✱
(1 +i) = p3
1,15.1,05.1,07≈1,0891.
q✉❡ ♥♦s ❞á ✉♠❛ t❛①❛ ♠é❞✐❛ ❞❡ ❛♣r♦①✐♠❛❞❛♠❡♥t❡ 9%✳ ❆ss✐♠✱ t❡♠♦s q✉❡i= 9%✳ ⋄
❆♣❧✐❝❛çã♦ ✶✳✻ ◆✉♠❛ ❛♣❧✐❝❛çã♦ ❛ ❥✉r♦s ❝♦♠♣♦st♦s ♦ ❢❛t♦r ❞❡ ❛✉♠❡♥t♦ ♠é❞✐♦ é ❛ ♠é❞✐❛ ❣❡♦♠étr✐❝❛ ❞♦s ❢❛t♦r❡s ❞❡ ❛✉♠❡♥t♦ ✐♥❞✐✈✐❞✉❛✐s✳
❊①❡♠♣❧♦ ✶✳✶✷ ❈♦♥s✐❞❡r❡ q✉❡ ❛ t❛①❛ ❞❡ r❡♥❞✐♠❡♥t♦ ❞❡ ✉♠ ❢✉♥❞♦ ❞❡ r❡♥❞❛ ✜①❛ t❡♥❤❛♠ s✐❞♦ 10% ♥♦ ♣r✐♠❡✐r♦ q✉❛❞r✐♠❡str❡✱ 20% ♥♦ s❡❣✉♥❞♦ ❡ 15% ♥♦ t❡r❝❡✐r♦✳ ❉❡t❡r♠✐♥❡ ❛ t❛①❛ ♠é❞✐❛ ❞❡ r❡♥❞✐♠❡♥t♦s ❛♥✉❛✐s ❛❞♠✐t✐♥❞♦ r❡❣✐♠❡ ❞❡ ❝❛♣✐t❛❧✐③❛çã♦ ❝♦♠♣♦st❛ ❡♥tr❡ ♦s q✉❛❞r✐♠❡str❡s✳
❙♦❧✉çã♦✿ ◗✉❡r❡♠♦s ❡♥❝♦♥tr❛r ✉♠❛ ❞❡t❡r♠✐♥❛❞❛ t❛①❛ i✱ t❛❧ q✉❡✿
(1 +i)3 = (1 + 0,1).(1 + 0,2).(1 + 0,15).
▲♦❣♦✱
(1 +i) = p3
1,1.1,2.1,15≈1,1493
q✉❡ ♥♦s ❞❛ ✉♠❛ t❛①❛ ♠❡❞✐❛ ❞❡ ❛♣r♦①✐♠❛❞❛♠❡♥t❡ 14,93%✳ ⋄
✶✳✶✳✹ ▼é❞✐❛ ❍❛r♠ô♥✐❝❛
❉❡✜♥✐çã♦ ✶✳✺ ❆ ♠é❞✐❛ ❤❛r♠ô♥✐❝❛ ❞❡ ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ ♥ú♠❡r♦s r❡❛✐s ♥ã♦ ♥✉❧♦s
(x1, x2, . . . , xn) é ♦ ♥ú♠❡r♦ H ❝♦♠ r❡s♣❡✐t♦ ❛ ♦♣❡r❛çã♦ ❞❡ s♦♠❛ ❞♦s ✐♥✈❡rs♦s✱ ❞❡st❛
❢♦r♠❛
1
x1
+ 1
x2
+. . .+ 1
xn
= 1
H +
1
H +. . .+
1
H
| {z }
n t❡r♠♦s
= n
H.
P♦rt❛♥t♦✱
H = n
1
x1
+ 1
x2
+. . .+ 1
xn
.
❖❜s❡r✈❛çã♦ ✶✳✸ ❆❧❣✉♥s ❞♦s ♣r♦❜❧❡♠❛s ♣rát✐❝♦s ♠❛✐s ✐♥t❡r❡ss❛♥t❡s s♦❜r❡ ♠é❞✐❛s ❡stã♦ r❡❧❛❝✐♦♥❛❞♦s à ♠é❞✐❛ ❤❛r♠ô♥✐❝❛✳ ➱ ✐♠♣♦rt❛♥t❡ q✉❡ s❛✐❜❛♠♦s r❡❝♦♥❤❡❝❡r ❡s✲ s❡s ♣r♦❜❧❡♠❛s✳ ❆ s❡❣✉✐r ❝♦❧♦❝❛♠♦s ❛❧❣✉♥s ❡①❡♠♣❧♦s ♦♥❞❡ s✉r❣❡ ❛ ✐❞❡✐❛ ❞❛ ♠é❞✐❛ ❤❛r♠ô♥✐❝❛✳ ◆❡ss❡s ♣r♦❜❧❡♠❛s ♦ q✉❡ ❣❡r❛❧♠❡♥t❡ ♦❝♦rr❡ é ♦ ❢♦r♥❡❝✐♠❡♥t♦ ❞❡ t❛①❛s ❞❡ ✈❛r✐❛çã♦ ✭✈❡❧♦❝✐❞❛❞❡s✱ ♣❡rí♦❞♦s✱ ✈❛③õ❡s ❡t❝✮ ❡ s❡ ♣❡❞❡ ❛❧❣♦ r❡❧❛t✐✈♦ ❛ t❛①❛ ❞❡ ✈❛r✐❛çã♦ ♠é❞✐❛✳
❆♣❧✐❝❛çã♦ ✶✳✼ ❯♠ ❛✉t♦♠ó✈❡❧ ✈❛✐ ❞❛ ❝✐❞❛❞❡ A ♣❛r❛ B ❝♦♠ ✉♠❛ ✈❡❧♦❝✐❞❛❞❡ ♠é❞✐❛
❞❡ v1 ❡ ✈♦❧t❛✱ ♣❡❧♦ ♠❡s♠♦ ❝❛♠✐♥❤♦✱ ❞❡ B ♣❛r❛ A ❝♦♠ ✉♠❛ ✈❡❧♦❝✐❞❛❞❡ ♠é❞✐❛ ❞❡ v2✳
❆ ✈❡❧♦❝✐❞❛❞❡ ♠é❞✐❛ ❡♠ t♦❞♦ ♣❡r❝✉rs♦ s❡rá ❛ ♠é❞✐❛ ❤❛r♠ô♥✐❝❛ ❞❡ v1 ❡ v2.
❙♦❧✉çã♦✿ ❉❡ ❢❛t♦✱ s❡♥❞♦ d❛ ❞✐stâ♥❝✐❛ ❡♥tr❡ A ❡ B ✱ t❡♠♦s q✉❡ t1 =
d v1
✱ t❡♠♣♦ ❞❡ ✐❞❛✱ ❡ t2 =
d v2
t❡♠♣♦ ❞❡ ✈♦❧t❛✳ ❙❡ v é ❛ ✈❡❧♦❝✐❞❛❞❡ ♠é❞✐❛ ❡♠ t♦❞♦ ♣❡r❝✉rs♦✱ ❡♥tã♦
v = 2d
t1+t2
✱ ❞♦♥❞❡
2d
v =t1+t2 =
d v1 + d v2 ❡ ♣♦rt❛♥t♦✱ 2 v = 1 v1 + 1
v2 ⇒
v = 1 2
v1
+ 1
v2
P♦rt❛♥t♦✱ t❡♠♦s q✉❡ v é ❛ ♠é❞✐❛ ❤❛r♠ô♥✐❝❛ ❡♥tr❡ v1 ❡v2✳
◆♦t❡ q✉❡ ❛ ❛r❣✉♠❡♥t❛çã♦ ♥ã♦ s❡ ❛❧t❡r❛ s❡ t✐✈❡r♠♦s n ❞❡s❧♦❝❛♠❡♥t♦s ✐❣✉❛✐s ❝♦♠
✈❡❧♦❝✐❞❛❞❡s ♠é❞✐❛s ❡♠ ❝❛❞❛ ♣❛rt❡ v1, v2, . . . , vn ✳ ❖✉ s❡❥❛✱ s❡v é ❛ ✈❡❧♦❝✐❞❛❞❡ ♠é❞✐❛
❡♠ t♦❞♦ ♣❡r❝✉rs♦✱ t❡♠♦s
n v = 1 v1 + 1 v2
+. . .+ 1
vn ⇒
v = 1 n
v1
+ 1
v2
+. . .+ 1
vn
.
⋄
❊①❡♠♣❧♦ ✶✳✶✸ ❯♠ ✈❡í❝✉❧♦ ❢❛③ ♠❡t❛❞❡ ❞❛ ❞✐stâ♥❝✐❛ ❞❡ ✉♠ tr❛❥❡t♦ q✉❛❧q✉❡r ❛ ✾✵ ❦♠✴❤ ❡ ❛ ♦✉tr❛ ♠❡t❛❞❡ ❛ ✺✵ ❦♠✴❤✱ ❛ ✈❡❧♦❝✐❞❛❞❡ ♠é❞✐❛ ❞♦ tr❛❥❡t♦ s❡rá✿
❙♦❧✉çã♦✿ ❈♦♠♦ t❡♠♦s q✉❡ x1 = 90 km/h , x2 = 50 km/h ✱ ❡ ❞✉❛s ♣❛rt❡s
❞♦ tr❛❥❡t♦✱ ❧♦❣♦ n = 2✳ ❆ ✈❡❧♦❝✐❞❛❞❡ ♠é❞✐❛ s❡rá ❛ ♠é❞✐❛ ❤❛r♠ô♥✐❝❛ ❡♥tr❡ x1 ❡ x2✱
♣♦rt❛♥t♦
H = 2
1 90+
1 50
= 2 5 + 9
450
= 2 14 450
= 900
14 = 64,3.
❆ ✈❡❧♦❝✐❞❛❞❡ ♠é❞✐❛ é ♣♦rt❛♥t♦ ❞❡ 64,3 km/h. ⋄
❆♣❧✐❝❛çã♦ ✶✳✽ ❙❡ ✉♠ t❛♥q✉❡ ♣♦❞❡ s❡r ❡♥❝❤✐❞♦ ✐♥❞✐✈✐❞✉❛❧♠❡♥t❡ ♣♦r ✉♠❛ t♦r♥❡✐r❛ 1 ❡♠ ✉♠ t❡♠♣♦ T1 ✱ ♣♦r ✉♠❛ t♦r♥❡✐r❛ 2 ❡♠ ✉♠ t❡♠♣♦ T2 ✱ ❛ss✐♠ s✉❝❡ss✐✈❛♠❡♥t❡
❛té ✉♠❛ t♦r♥❡✐r❛ n ❡♠ ✉♠ t❡♠♣♦ Tn ✱ ❡♥tã♦ s❡ ♣✉s❡r♠♦s t♦❞❛s ❛s t♦r♥❡✐r❛s s✐♠✉❧✲
t❛♥❡❛♠❡♥t❡ ♣❛r❛ ❡♥❝❤❡r ♦ t❛♥q✉❡✱ ♦ ✐♥✈❡rs♦ ❞♦ t❡♠♣♦ q✉❡ ❧❡✈❛rã♦ é ❛ s♦♠❛ ❞♦s ✐♥✈❡rs♦s ❞♦s t❡♠♣♦s ❞❡❧❛s s❡♣❛r❛❞❛s✳ ◆♦t❡ q✉❡ ❝❛❞❛ ✉♠❛ ❞❛s t♦r♥❡✐r❛s ♣♦❞❡ s❡r s✉❜st✐t✉í❞❛ ♣♦r t♦r♥❡✐r❛s ❞❡ ♠❡s♠❛ ✈❛③ã♦✱ ❞❡ ♠♦❞♦ q✉❡ ♦ t❡♠♣♦ ♥❡❝❡ssár✐♦ ♣❛r❛ q✉❡ ❡st❛ t♦r♥❡✐r❛ s✉❜st✐t✉t❛ ❡♥❝❤❛ ♦ t❛♥q✉❡ é ❛ ♠é❞✐❛ ❤❛r♠ô♥✐❝❛ ❞♦s t❡♠♣♦s ✐♥❞✐✈✐❞✉❛✐s✳ ❙♦❧✉çã♦✿ ❆ r❛③ã♦ 1
Ti
❝♦rr❡s♣♦♥❞❡ ❛ ❢r❛çã♦ ❞♦ t❛♥q✉❡ q✉❡ é ❝❤❡✐❛ ❡♠ ✉♠❛ ✉♥✐❞❛❞❡ ❞❡ t❡♠♣♦ ♣❡❧❛ t♦r♥❡✐r❛ i ✱ ♦✉ s❡❥❛✱ ❛ ✈❛③ã♦ ❞❛ t♦r♥❡✐r❛ i ✳ ▲♦❣♦✱ s❡ T ❢♦r ♦ t❡♠♣♦
♥❡❝❡ssár✐♦ ♣❛r❛ ❛s t♦r♥❡✐r❛s ❥✉♥t❛s ❡♥❝❤❡r❡♠ t♦❞♦ t❛♥q✉❡ ❡ 1
t ❛ ✈❛③ã♦ ❞❡n t♦r♥❡✐r❛s
✐❞ê♥t✐❝❛s ❡♥❝❤❡r❡♠ ❥✉♥t❛s ♦ t❛♥q✉❡✱ t❡r❡♠♦s✿ n
t = 1 T = 1 T1 + 1 T2
+. . .+ 1
Tn
. ⋄
❊①❡♠♣❧♦ ✶✳✶✹ ❚rês t♦r♥❡✐r❛s ❧✐❣❛❞❛s s♦③✐♥❤❛s ❡♥❝❤❡♠ ✉♠ t❛♥q✉❡ ❡♠ ✸ ❤✱ ✹ ❤ ❡ ✻ ❤ r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ▲✐❣❛♥❞♦ ❛s tr❡s t♦r♥❡✐r❛s s✐♠✉❧t❛♥❡❛♠❡♥t❡✱ q✉❛♥t♦ t❡♠♣♦ ❧❡✈❛rã♦ ♣❛r❛ ❡♥❝❤❡r ♦ t❛♥q✉❡ s❛❜❡♥❞♦ q✉❡ ❤á ✉♠ ✈❛③❛♠❡♥t♦ ❝❛♣❛③ ❞❡ ❡s✈❛③✐❛r ♦ t❛♥q✉❡ ❡♠ ✶✷ ❤✳
❙♦❧✉çã♦✿ ❖❜s❡r✈❡ q✉❡ ♦ ♣r♦❜❧❡♠❛ t❡♠ ✉♠ ✈❛③❛♠❡♥t♦✱ q✉❡ é ❝♦♥s✐❞❡r❛❞♦ ❝♦♠♦ ✉♠❛ t♦r♥❡✐r❛ q✉❡ ❡♥❝❤❡ ♦ t❛♥q✉❡ ❡♠ t❡♠♣♦ ♥❡❣❛t✐✈♦✳ ❇❛st❛ ❢❛③❡r✱
1 T = 1 3 + 1 4 + 1 6 − 1 12 = 4 12+ 3 12+ 2 12− 1 12 = 8 12 ❖❜t❡♥❞♦ ❛ss✐♠ T = 3/2✳
P♦rt❛♥t♦✱ ♣❛r❛ ❡♥❝❤❡r ♦ t❛♥q✉❡ s❡rá ♣r❡❝✐s♦ 3
2❤✱ ♦✉ s❡❥❛✱ ✶❤ ✸✵♠✐♥✳ ⋄
▼é❞✐❛s ❍❛r♠ô♥✐❝❛s sã♦ ✉s❛❞❛s q✉❛♥❞♦ t❡♠♦s q✉❡ ❧✐❞❛r ❝♦♠ ✉♠❛ sér✐❡ ❞❡ ✈❛❧♦r❡s ✐♥✈❡rs❛♠❡♥t❡ ♣r♦♣♦r❝✐♦♥❛✐s ❝♦♠♦ ✉♠ ❝á❧❝✉❧♦ ❞❡ ✉♠❛ ✈❡❧♦❝✐❞❛❞❡ ♠é❞✐❛✱ r❡❧❛çõ❡s ❡♥tr❡ ✈❡❧♦❝✐❞❛❞❡ ❡ t❡♠♣♦✱ ✉♠ ❝✉st♦ ♠é❞✐♦ ❞❡ ❝♦♠♣r❛s ❝♦♠ ✉♠❛ t❛①❛ ✜①❛ ❞❡ ❥✉r♦s ❡ r❡s✐stê♥❝✐❛s ❡❧étr✐❝❛s ❡♠ ♣❛r❛❧❡❧♦✱ ♣♦r ❡①❡♠♣❧♦✳
P♦❞❡♠♦s ❡①❡♠♣❧✐✜❝❛r ❡ss❛ r❡♣r❡s❡♥t❛çã♦ ♠♦str❛♥❞♦ r❡❧❛çã♦ ❡♥tr❡ ❛ r❡s✐stê♥❝✐❛ t♦t❛❧✱ RT✱ ❞❡ ✉♠ s✐st❡♠❛ ❡♠ ♣❛r❛❧❡❧♦ ❡ ❛ s♦♠❛ ❞❛s s✉❛s r❡s✐stê♥❝✐❛s✱ R1 ❡ R2✱ ♣♦r
❡①❡♠♣❧♦✳ ❚❡♠♦s✿ 1
RT
= ( 1
R1
+ 1
R2
)✱ ✉♠❛ r❡❧❛çã♦ ❝♦♠ ♦ ✐♥✈❡rs♦ ❞❛s r❡s✐stê♥❝✐❛s✳
❊①❡♠♣❧♦ ✶✳✶✺ Pr♦✈❡ q✉❡ ♠é❞✐❛ ❣❡♦♠❡tr✐❝❛ ❡♥tr❡ ❞♦✐s t❡r♠♦s é ♠é❞✐❛ ❣❡♦♠❡tr✐❝❛ ❡♥tr❡ ❛s ♠é❞✐❛s ❤❛r♠ô♥✐❝❛ ❡ ❛r✐t♠ét✐❝❛ ❞❡ss❡s ❞♦✐s t❡r♠♦s✳
❙♦❧✉çã♦✿ ❉❛❞♦s x ❡ y r❡❛✐s ♣♦s✐t✐✈♦s✱ t❡♠♦s✿
G=√x.y, A= x+y
2 e H = 2xy
x+y.
❧♦❣♦✱
√
A.H =
r
(x+y 2 ).(
2xy
x+y) =
√x.y =G.
⋄
✶✳✶✳✺ ▼é❞✐❛ ◗✉❛❞rát✐❝❛
❉❡✜♥✐çã♦ ✶✳✻ ❆ ♠é❞✐❛ q✉❛❞rát✐❝❛ ❞❡ ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ ♥ú♠❡r♦s r❡❛✐s ♥ã♦ ♥✉❧♦s
(x1, x2, . . . , xn)é ♦ ♥ú♠❡r♦Q ❝♦♠ r❡s♣❡✐t♦ ❛ ♦♣❡r❛çã♦ ❞❡ s♦♠❛ ❞♦s q✉❛❞r❛❞♦s✱ ❞❡st❛
❢♦r♠❛
x2 1+x
2
2+. . .+x
2
n=Q
2
+Q2
+. . .+Q2
| {z }
n t❡r♠♦s
=nQ2
.
P♦rt❛♥t♦
Q=
r
x2 1 +x
2
2+. . .+x
2
n
n .
❊①❡♠♣❧♦ ✶✳✶✻ ❉❡t❡r♠✐♥❡ ❛ ▼é❞✐❛ ◗✉❛❞rát✐❝❛ ❡♥tr❡ ♦s ♥ú♠❡r♦s ✶ ❡ ✼✳ ❆❞❛♣t❛❞♦ ❞♦ ❬✻❪✳
❙♦❧✉çã♦✿ ❉❡ ❢♦r♠❛ ❞✐r❡t❛ ✿ Q=
r
12
+ 72
2 =
r
1 + 49 2 =
r
50 2 =
√
25 = 5. ⋄
❆♣❧✐❝❛çã♦ ✶✳✾ ❊st❛t✐st✐❝❛♠❡♥t❡✱ ❛ ❢♦r♠❛ ♠❛✐s ♥❛t✉r❛❧ ❞❡ ♠❡❞✐r ♦ q✉❛♥t♦ ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ ♥ú♠❡r♦s s❡ ❞✐s♣❡rs♦✉ ❞❛ ♠é❞✐❛ ❛r✐t♠ét✐❝❛ é ❛tr❛✈és ❞♦ ❞❡s✈✐♦ ♣❛✲ ❞rã♦ ✭σ✮✳ ❖ ❞❡s✈✐♦ ♣❛❞rã♦ é ❛ ♠é❞✐❛ q✉❛❞rát✐❝❛ ❞♦s ❞❡s✈✐♦s ✐♥❞✐✈✐❞✉❛✐s✳
❖✉ s❡❥❛✱ ❞❛❞♦s (x1, x2, . . . , xn) ❡ ❞❡♥♦t❛♥❞♦ ♣♦r x ❛ ♠é❞✐❛ ❛r✐t♠ét✐❝❛ ❞♦s x
′
is
t❡♠♦s
σ =
r
(x1−x)2+ (x2 −x)2+. . .+ (xn−x)2
n
❊①❡♠♣❧♦ ✶✳✶✼ ❈♦♥s✐❞❡r❡ ❛ ❢✉♥çã♦ r❡❛❧ ❞❡ ✈❛r✐á✈❡❧ r❡❛❧✿
σ(x) =
r
(x1 −x)2+ (x2−x)2+. . .+ (xn−x)2
n .
Pr♦✈❡ q✉❡ x ✱ ❛ ♠é❞✐❛ ❛r✐t♠ét✐❝❛ ❞♦s x′
is é ♦ ✈❛❧♦r q✉❡ ♠✐♥✐♠✐③❛ ♦ ❞❡s✈✐♦ ♣❛❞rã♦✳
❙♦❧✉çã♦✿ P❛r❛ ♠✐♥✐♠✐③❛r σ ✱ ❜❛st❛ ♠✐♥✐♠✐③❛r ❛ ❢✉♥çã♦ σ(x)2✳ ❉❡s❡♥✈♦❧✈❡♥❞♦
σ(x)2✱ t❡♠♦s✱
σ(x)2
=x2
−2.(x1+. . .+xn
n ).x+
x2
1+. . .+x
2
n
n .
❈♦♠♦ ❡ss❛ ❢✉♥çã♦ é ✉♠❛ ❢✉♥çã♦ q✉❛❞rát✐❝❛✱ ❞❛ ❢♦r♠❛y=ax2
+bx+c✱ ❝♦♠a >0✱
♦ ♠í♥✐♠♦ ♦❝♦rr❡ ♥♦ ✈❡rt✐❝❡✱ ♦♥❞❡ xv = −
b
2a ✳ ▲♦❣♦✱
xv =
x1+. . .+xn
n ⇒xv =x
⋄
✶✳✷ ❉❡s✐❣✉❛❧❞❛❞❡ ❞❛s ▼é❞✐❛s
◆❡st❛ s❡çã♦ ❢❛③✲s❡ ✉♠❛ ❝♦♠♣❛r❛çã♦ ❡♥tr❡ ❛s ✈ár✐❛s ♠é❞✐❛s✱ r❡s✉❧t❛♥❞♦ ♥✉♠❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❢✉♥❞❛♠❡♥t❛❧ ❡♥tr❡ ❛s ♠é❞✐❛s ❛r✐t♠ét✐❝❛✱ ❣❡♦♠étr✐❛✱ ❤❛r♠ô♥✐❝❛ ❡ q✉❛✲ ❞rát✐❝❛✱ ❛ ♣❛rt✐r ❞❛s r❡❢❡rê♥❝✐❛s ❬✶❪ ❡ ❬✷❪✳ ❱❡r✐✜❝❛♠♦s ❞✐✈❡rs❛s ❛♣❧✐❝❛çõ❡s ❡♥✈♦❧✈❡♥❞♦ ❛s ❞❡s✐❣✉❛❧❞❛❞❡s ❞❛s ♠é❞✐❛s ❡ ❛ ✐♠♣♦rtâ♥❝✐❛ ❞❡ t❛✐s ❞❡s✐❣✉❛❧❞❛❞❡s ♥♦ ❞❡s❡♥✈♦❧✈✐✲ ♠❡♥t♦ ❞❛s ❝♦♠♣❡tê♥❝✐❛s✱ ❡♠ ❞✐✈❡rs❛s ár❡❛s ❞❛ ♠❛t❡♠át✐❝❛✱ ❞♦s ❛❧✉♥♦s ❞♦ ❊♥s✐♥♦ ▼é❞✐♦✳
❚❡♦r❡♠❛ ✶✳✶ ❉❡s✐❣✉❛❧❞❛❞❡s ❞❛s ▼é❞✐❛s ❙❡❥❛♠ (x1, x2, . . . , xn) ♥ú♠❡r♦s r❡❛✐s
♣♦s✐t✐✈♦s ❡ ❞❡♥♦t❡♠♦s ♣♦r H, G, A, Q r❡s♣❡❝t✐✈❛♠❡♥t❡ ❛s ♠é❞✐❛s ❤❛r♠ô♥✐❝❛✱ ❣❡♦♠é✲
tr✐❝❛✱ ❛r✐t♠ét✐❝❛ ❡ q✉❛❞rát✐❝❛ ❞❡ss❡s ♥ú♠❡r♦s✱ ❡♥tã♦ t❡♠♦s ❛s s❡❣✉✐♥t❡s ❞❡s✐❣✉❛❧❞❛✲ ❞❡s✿
H ≤G≤A≤Q.
❆❧é♠ ❞✐ss♦✱ ❛ ✐❣✉❛❧❞❛❞❡ ❡♠ q✉❛❧q✉❡r ♣♦♥t♦ ❞❛s ❞❡s✐❣✉❛❧❞❛❞❡s ❛❝✐♠❛ é ♣♦ssí✈❡❧ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ (x1 = x2 = . . . = xn) ❡✱ ♥❡st❛s ❝♦♥❞✐çõ❡s✱ t❡r❡♠♦s ♥❡❝❡ss❛r✐❛♠❡♥t❡ ❛
✐❣✉❛❧❞❛❞❡ ❞❡ t♦❞❛s ❛s ♠é❞✐❛s✳ ❇❛s❡❛❞♦ ❡♠ ❬✻❪✳
❉❡♠♦♥str❛çã♦✿ ❱❛♠♦s ❝♦♠❡ç❛r ♠♦str❛♥❞♦ q✉❡ ❛ ❞❡s✐❣✉❛❧❞❛❞❡ G ≤ A ✐♠♣❧✐❝❛ ❛
❞❡s✐❣✉❛❧❞❛❞❡ H ≤ G✳ ❉❡ ❢❛t♦✱ ❛♣❧✐❝❛♥❞♦ ❛ ❞❡s✐❣✉❛❧❞❛❞❡ G ≤ A ♣❛r❛ (1
x1, . . . ,
1
xn)✱ t❡♠♦s
n
r
1
x1
. . . 1
xn ≤
1
x1 +· · ·+
1
xn
n .
❖ q✉❡ é ❡q✉✐✈❛❧❡♥t❡ ❛
n
1
x1 +· · ·+
1
xn
≥ √n
x1. . . xn
✐st♦ é✱ ❛ ❞❡s✐❣✉❛❧❞❛❞❡ H ≤G ✈❛❧❡ ♣❛r❛ ❛ ♥✲✉♣❧❛ (x1. . . xn)✳
P❛r❛ ❞❡♠♦♥str❛r ❛ ❞❡s✐❣✉❛❧❞❛❞❡ A≤Q✱ ❝♦♥s✐❞❡r❛♠♦s ❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❡❧❡♠❡♥t❛r
(x1−A) 2
+· · ·+ (xn−A)
2
≥0.
❉❡ ♦♥❞❡ ♦❜s❡r✈❛♠♦s q✉❡ (x1 −A)2+· · ·+ (xn−A)2 = 0 s❡✱ ❡ s♦♠❡♠t❡ s❡✱ x1 =
· · ·=xn=A✳ ❉❡s❡♥✈♦❧✈❡♥❞♦ ❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❛❝✐♠❛✱ t❡♠♦s
x2
1+· · ·+x
2
n−2A(x1+· · ·+xn) +nA
2
≥0.
▲❡♠❜r❛♥❞♦ q✉❡ x1+· · ·+xn=nA✱ ❞❛ ú❧t✐♠❛ ❞❡s✐❣✉❛❧❞❛❞❡ t❡♠♦s
x2
1+· · ·+x
2
n≥nA
2 . P♦rt❛♥t♦✱ A≤ r x2
1+· · ·+x
2
n
n =Q.
❖❝♦rr❡♥❞♦ ❛ ✐❣✉❛❧❞❛❞❡ s♦♠❡♥t❡ q✉❛♥❞♦ x1 = · · · = xn. P❛r❛ ❝♦♥❝❧✉✐r ❛ ♣r♦✈❛ ❞♦
t❡♦r❡♠❛ ❢❛❧t❛ ♠♦str❛r q✉❡G≤A❡ q✉❡ ❛ ✐❣✉❛❧❞❛❞❡ só ♦❝♦rr❡ q✉❛♥❞♦x1 =· · ·=xn✳
■st♦ s❡❣✉✐rá ❞♦s ❞♦✐s ♣ró①✐♠♦s ❧❡♠❛s✳
❆♥t❡s ❞❡ ❞❛r♠♦s ♦s ♣ró①✐♠♦s ❞♦✐s ❧❡♠❛s q✉❡ ♥♦s ♣❡r♠✐t✐rã♦ ❛ ❝♦♥❝❧✉sã♦ ❞♦ ♣r❡❝❡❞❡♥t❡ t❡♦r❡♠❛✱ ✈❛♠♦s ✐♥tr♦❞✉③✐r ❛ s❡❣✉✐♥t❡ ❞❡✜♥✐çã♦✳
❉❡✜♥✐çã♦ ✶✳✼ ❉✐③❡♠♦s q✉❡ ❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❞❛s ♠é❞✐❛s ✭A ≥ G✮ ✈❛❧❡ ♣❛r❛ n ∈ N s❡ ♣❛r❛ t♦❞♦ ❧✐st❛ x1, . . . , xn ❞❡ n ♥ú♠❡r♦s r❡❛✐s ♣♦s✐t✐✈♦s ✈❛❧❡ ❛ ❞❡s✐❣✉❛❧❞❛❞❡
n
√
x1. . . xn ≤
x1+· · ·+xn
n ,
♦❝♦rr❡♥❞♦ ❛ ✐❣✉❛❧❞❛❞❡ s♦♠❡♥t❡ q✉❛♥❞♦ x1 =· · ·=xn✳
▲❡♠❛ ✶✳✷ ❆ ❞❡s✐❣✉❛❧❞❛❞❡ ❞❛s ♠é❞✐❛s ✭A ≥G✮ ✈❛❧❡ ♣❛r❛ n = 2k ♣❛r❛ k ∈N✳
❉❡♠♦♥str❛çã♦✿ ❆ ♣r♦✈❛ s❡rá ❢❡✐t❛ ♣♦r ✐♥❞✉çã♦ s♦❜r❡ k✳ P❛r❛ k = 1✱ n = 21
= 2✱ t❡♠♦s✿
A−G= x1+x2
2 −
√
x1x2 =
(√x1−√x2) 2
2 ≥0
♦❝♦rr❡♥❞♦ ❛ ✐❣✉❛❧❞❛❞❡ s♦♠❡♥t❡ q✉❛♥❞♦x1 =x2✳ ❙✉♣♦♥❞♦✱ ♣♦r ❤✐♣ót❡s❡ ❞❡ ✐♥❞✉çã♦✱
q✉❡ ♦ r❡s✉❧t❛❞♦ ✈❛❧❤❛ ♣❛r❛ n = 2k✱ ♣r♦✈❡♠♦s q✉❡ ♦ r❡s✉❧t❛❞♦ t❛♠❜é♠ ✈❛❧❡ ♣❛r❛
2n = 2k+1✳ ❉❡ ❢❛t♦✱ ❛♣❧✐❝❛♥❞♦ ♦ ❝❛s♦
k = 1✱ ♣❛r❛ ♦s ♥ú♠❡r♦s (x1+· · ·+xn)
n ❡
(xn+1+· · ·+x2n)
n ✱ t❡♠♦s
x1+· · ·+xn+xn+1· · ·+x2n
2n ≥
r
(x1+· · ·+xn)
n
(xn+1+· · ·+x2n)
n .
❆❣♦r❛✱ ❛♣❧✐❝❛♥❞♦ ❛ ❤✐♣ót❡s❡ ❞❡ ✐♥❞✉çã♦✱ ♣❛r❛ (x1+· · ·+xn)
n ❡
(xn+1+· · ·+x2n)
n ✱
❛ ú❧t✐♠❛ ✐❣✉❛❧❞❛❞❡ ♥♦s ❞á
x1+· · ·+xn+xn+1· · ·+x2n
2n ≥
q
n
√
x1. . . xn
n
p
(xn+1. . . x2n).
♦ q✉❡ é ❡q✉✐✈❛❧❡♥t❡ ❛
x1+· · ·+xn+xn+1· · ·+x2n
2n ≥
2√nx
1. . . xnxn+1. . . x2n,
♦❝♦rr❡♥❞♦ ❛ ✐❣✉❛❧❞❛❞❡ s♦♠❡♥t❡ q✉❛♥❞♦ x1 =· · ·=xn✱xn+1 =· · ·=x2n ❡x1+· · ·+
xn = xn+1+· · ·+x2n✱ ♦ q✉❡ ✐♠♣❧✐❝❛ q✉❡ ❛ ✐❣✉❛❧❞❛❞❡ só ♦❝♦rr❡ q✉❛♥❞♦ x1 = · · ·=
xn =xn+1 =· · ·=x2n✳ P♦rt❛♥t♦✱ ♣❡❧♦ ♣r✐♥❝í♣✐♦ ❞❡ ✐♥❞✉çã♦ ✜♥✐t❛✱ ♦ ❧❡♠❛ s❡❣✉❡✳
▲❡♠❛ ✶✳✸ ❙❡ ❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❞❛s ♠é❞✐❛s A≥Gé ✈á❧✐❞❛ ♣❛r❛ ❛❧❣✉♠n ∈N✭n ≥3✮✱ ❡♥tã♦ ❡❧❛ t❛♠❜é♠ ✈❛❧❡ ♣❛r❛ ♣❛r❛ n−1✳
❉❡♠♦♥str❛çã♦✿ ❙❡❥❛♠ x1, . . . , xn−1 ♥ú♠❡r♦s r❡❛✐s ♣♦s✐t✐✈♦s✳ ❉❡✜♥❛ xn = A =
x1+· · ·+xn−1
n−1 ✳ ◆♦t❡ q✉❡
x1+· · ·+xn−1+xn
n =
(n−1)A
n +
A
n =A✳ ❊✱ ♣♦rt❛♥t♦✱
♣♦r ❤✐♣ót❡s❡✱
A≥ pn
x1. . . xn−1A
♦❝♦rr❡♥❞♦ ❛ ✐❣✉❛❧❞❛❞❡ s♦♠❡♥t❡ q✉❛♥❞♦ x1 =· · ·=xn−1 =A✳
❈♦♠♦ ❛ ú❧t✐♠❛ ❞❡s✐❣✉❛❧❞❛❞❡ é ❡q✉✐✈❛❧❡♥t❡ ❛ An ≥x
1. . . xn−1A✱ ❝♦♥❝❧✉í♠♦s ❞❛í
q✉❡
A≥ n−√1
x1. . . xn−1 =G,
♦❝♦rr❡♥❞♦ ❛ ✐❣✉❛❧❞❛❞❡ s♦♠❡♥t❡ q✉❛♥❞♦ x1 =· · ·=xn−1✳
❈♦r♦❧ár✐♦ ✶✳✶ ❙❡ ❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❞❛s ♠é❞✐❛s A ≥ G é ✈á❧✐❞❛ ♣❛r❛ ❛❧❣✉♠ n ∈ N ✭n ≥3✮✱ ❡♥tã♦ ❡❧❛ t❛♠❜é♠ ✈❛❧❡ ♣❛r❛ t♦❞♦ k ∈N t❛❧ q✉❡ 2≤k ≤n✳
❉❡♠♦♥str❛çã♦✿ ❙❡❣✉❡ ✐♠❡❞✐❛t❛♠❡♥t❡ ❛♣❧✐❝❛♥❞♦ ♦ ▲❡♠❛ ✶✳✸✱ r❡❝✉rs✐✈❛♠❡♥t❡ n−2
✈❡③❡s✳
❈♦♥❝❧✉sã♦ ❞❛ ♣r♦✈❛ ❞♦ ❚❡♦r❡♠❛ ✶✳✶
❙❡❥❛n∈N✱n ≥2✳ P❡❧♦ ▲❡♠❛ ✶✳✷✱ ✈❛❧❡ ❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❞❛s ♠é❞✐❛s ✭A≥G✮ ♣❛r❛ 2n✳ ❈♦♠♦ n < 2n✱ s❡❣✉❡ ❞♦ ❈♦r♦❧ár✐♦ ✶✳✶ q✉❡ ❛ ❞❡s✐❣✉❛❧❞❛❞❡ A ≥ G ✈❛❧❡ ♣❛r❛ n✳
■ss♦ ❝♦♥❝❧✉✐ ❛ ♣r♦✈❛ ❞♦ ❚❡♦r❡♠❛ ✶✳✶
❊①❡♠♣❧♦ ✶✳✶✽ ▼♦str❡ q✉❡✱ ❡♥tr❡ t♦❞♦s ♦s r❡tâ♥❣✉❧♦s ❞❡ ♣❡r✐♠❡tr♦ 2p✱ ♦ q✉❛❞r❛❞♦
é ♦ ❞❡ ♠❛✐♦r ár❡❛✳ ❘❡t✐r❛❞♦ ❞❡ ❬✻❪✳
❙♦❧✉çã♦✿ ❙❡♥❞♦ x ❡ y ♦s ❧❛❞♦s ❞♦ r❡tâ♥❣✉❧♦✱ t❡♠♦s q✉❡ x+y =p ✱❧♦❣♦ ❛ ♠é❞✐❛
❛r✐t♠ét✐❝❛ ❡♥tr❡ x❡ y é x+y
2 =
p
2✳
❊ ❛✐♥❞❛ t❡♠♦s q✉❡ ❛ ár❡❛ ❞♦ r❡tâ♥❣✉❧♦ A =x.y✱ ❞❛í✿ √
A=√x.y ≤ x+y
2 =
p
2. ♣♦rt❛♥t♦✱
A≤ p
2
4 ❡ ❛ ✐❣✉❛❧❞❛❞❡ só é ♦❜t✐❞❛ q✉❛♥❞♦ x=y✳
P♦rt❛♥t♦✱ ♦ r❡tâ♥❣✉❧♦ ❞❡ ♠❛✐♦r ár❡❛ é ♦ q✉❛❞r❛❞♦ ❝✉❥❛ ár❡❛ A= p
2
4 ⋄
❊①❡♠♣❧♦ ✶✳✶✾ ◆♦ tr❛♣é③✐♦ ABCD✱ ❞❛ ✜❣✉r❛ ✶✳✸✱ M ❡ N sã♦ ♣♦♥t♦s ♠é❞✐♦s ❞♦s
❧❛❞♦s AD❡ BC✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ❙❡❥❛ P é ♦ ♣♦♥t♦ ❞❡ ✐♥t❡rs❡çã♦ ❞❛s ❞✐❛❣♦♥❛✐sAC
❡ DB✳ ❚❡♠♦s q✉❡ XY é ♣❛r❛❧❡❧♦ ❛ AB ❡ ♣❛ss❛ ♣❡❧♦ ♣♦♥t♦P✳
❋✐❣✉r❛ ✶✳✸✿ ❚r❛♣é③✐♦ ❆❇❈❉
▼♦str❡ q✉❡ XY é ❛ ♠é❞✐❛ ❤❛r♠ô♥✐❝❛ ❡ q✉❡M N ❡ ❛ ♠é❞✐❛ ❛r✐t♠ét✐❝❛ ❞♦s ❧❛❞♦s
AB ❡ CD✳
❙♦❧✉çã♦✿ ❈♦♠♦ ♦ △ABD∼ △XP D✱ t❡♠♦s AD
AB =
XD
XP ⇒XD=
AD.XP
AB . ✭✶✳✸✮
P❡❧❛ s❡♠❡❧❤❛♥❝❛ △AXP ∼ △ADC✱ t❡♠♦s AD
DC =
AX
P X ⇒AX =
AD.XP
DC . ✭✶✳✹✮
❙♦♠❛♥❞♦ ❛s ❡①♣r❡ssõ❡s ✶✳✹ ❡ ✶✳✸ ♦❜t❡♠♦s
AD=AX+XD = AD.XP
DC +
AD.XP
AB . ✭✶✳✺✮
❉✐✈✐❞✐♥❞♦ ✶✳✺ ♣♦r AD✱ ♦❜t❡♠♦s
1 = XP
DC + XP AB ⇔ 1 XP = 1 DC + 1 AB.
❉❡ ♠♦❞♦ ❛♥á❧♦❣♦✱ ❝♦♠♦ △ABC ∼ △P Y C ❡ △BDC ∼ △BP Y✱ ♦❜t❡♠♦s q✉❡
1
P Y =
1
DC +
1
AB.▲♦❣♦ ❝♦♥❝❧✉í♠♦s q✉❡ XP ❂P Y ✱ ❡ ♣♦rt❛♥t♦ ❛ ❡①♣r❡ssã♦ ❛❝✐♠❛
♣♦❞❡ s❡r r❡❡s❝r✐t❛ ❝♦♠♦✱ 2
XY =
1
DC+
1
AB.Pr♦✈❛♥❞♦ q✉❡XY é ❛ ▼é❞✐❛ ❍❛r♠ô♥✐❝❛
❞❡ AB ❡ CD✳
