FUVEST 2009

Questão 44971

(FUVEST - 2009 - 1 FASE) Os comprimentos dos lados de um triângulo ABC formam uma PA . Sabendo-se também que o perímetro de ABC vale 15 e que o ângulo  mede 120º, então o produto dos comprimentos dos lados é igual a

Questão 44972

(FUVEST - 2009 - 1 FASE)  A figura representa sete hexágonos regulares de lado 1 e um hexágono maior, cujos vértices coincidem com os centros de seis dos hexágonos menores. Então, a área do pentágono hachurado é igual a

Questão 44973

(FUVEST - 2009 - 1 FASE) O ângulo formado por dois planos e é tal que . O ponto P pertence a e a distância de P a vale 1. Então, a distância de P à reta intersecção de e é igual a

Questão 44974

(FUVEST - 2009 - 2 fase - Questão 1)

Na figura ao lado, a reta r tem equação no plano cartesiano Oxy. Além disso, os pontos B₀, B₁, B₂, B₃ estão na reta r, sendo B₀=(0,1) . Os pontos A₀, A₁, A₂, A₃ estão no eixo Ox, com A₀ = O = (0,0) . O ponto D_i pertence ao segmento A_iB_i , para . Os segmentos A₁B₁ , A₂B₂ , A₃B₃ são paralelos ao eixo Oy, os segmentos B₀D₁ , B₁D₂ , B₂D₃ são paralelos ao eixo Ox, e a distância entre B_i e B_i+1 é igual a 9, para .

Nessas condições:

a) Determine as abscissas de A₁, A₂, A₃.

b) Sendo R_i o retângulo de base A_i A_{i +1} e altura , A_{i +1} D_{i +1} para , calcule a soma das áreas dos retângulos R₀ , R₁ e R₂ .

Questão 44988

(FUVEST - 2009 - 1 FASE)

Two in every three people on the planet–some 4 billion in total–are “excluded from the rule of law.” In many cases, this begins with the lack of official recognition of their birth: around 40% of the developing world’s five-yearold children are not registered as even existing.

Later, people will find that the home they live in, the land they farm, or the business that they start, is not protected by legally enforceable property rights. Even in the rare cases when they can afford to go to court, the service is poor. India, for example, has only 11 judges for every 1million people.

These alarming statistics are contained in a report from a commission on the legal empowerment of the poor, released on June 3rd at the United Nations. It argues that not only are such statistics evidence of grave injustice, they also reflect one of the main reasons why so much of humanity remains mired in poverty. Because they are outside the rule of law, the vast majority of poor people are obliged to work (if they work at all) in the informal economy, which is less productive than the formal, legal part of the economy.

The Economist, June 7th 2008.

De acordo com o texto,

Questão 44989

(FUVEST - 2009 - 1ª FASE)

Two in every three people on the planet–some 4 billion in total–are “excluded from the rule of law.” In many cases, this begins with the lack of official recognition of their birth: around 40% of the developing world’s five-yearold children are not registered as even existing.

Later, people will find that the home they live in, the land they farm, or the business that they start, is not protected by legally enforceable property rights. Even in the rare cases when they can afford to go to court, the service is poor. India, for example, has only 11 judges for every 1million people.

These alarming statistics are contained in a report from a commission on the legal empowerment of the poor, released on June 3rd at the United Nations. It argues that not only are such statistics evidence of grave injustice, they also reflect one of the main reasons why so much of humanity remains mired in poverty. Because they are outside the rule of law, the vast majority of poor people are obliged to work (if they work at all) in the informal economy, which is less productive than the formal, legal part of the economy.

The Economist, June 7^th 2008.

O relatório citado no texto observa que 

Questão 44990

(FUVEST - 2009 - 1ª FASE)

In 1993, the dawn of the Internet age, the liberating anonymity of the online world was captured in a wellknown New Yorker cartoon. One dog, sitting at a computer, tells another: “On the Internet, nobody knows you’re a dog.” Fifteen years later, that anonymity is gone.

Technology companies have long used “cookies,” little bits of tracking software slipped onto your computer, and other means, to record the Web sites you visit, the ads you click on, even the words you enter in search engines – information that some hold onto forever. They’re not telling you they’re doing it, and they’re not asking permission. Internet service providers (I.S.P.’s) are now getting into the act. Because they control your connection, they can keep track of everything you do online, and there have been reports that I.S.P.’s may have started to sell the information they collect.

The driving force behind this prying is commerce. The big growth area in online advertising right now is “behavioral targeting.” Web sites can charge a premium if they are able to tell the maker of an expensive sports car that its ads will appear on Web pages clicked on by upperincome, middle-aged men.

The New York Times, April 5th 2008.

ISP = Provedores de serviço de internet.

As personagens dos quadrinhos, mencionadas no texto, se vangloriam de 

Questão 44991

(FUVEST - 2009 - 1 FASE)

In 1993, the dawn of the Internet age, the liberating anonymity of the online world was captured in a wellknown New Yorker cartoon. One dog, sitting at a computer, tells another: “On the Internet, nobody knows you’re a dog.” Fifteen years later, that anonymity is gone.

Technology companies have long used “cookies,” little bits of tracking software slipped onto your computer, and other means, to record the Web sites you visit, the ads you click on, even the words you enter in search engines – information that some hold onto forever. They’re not telling you they’re doing it, and they’re not asking permission. Internet service providers (I.S.P.’s) are now getting into the act. Because they control your connection, they can keep track of everything you do online, and there have been reports that I.S.P.’s may have started to sell the information they collect.

The driving force behind this prying is commerce. The big growth area in online advertising right now is “behavioral targeting.” Web sites can charge a premium if they are able to tell the maker of an expensive sports car that its ads will appear on Web pages clicked on by upperincome, middle-aged men.

The New York Times, April 5th 2008.

ISP = Provedores de serviço de internet.

Segundo o texto, os provedores de internet

Questão 44992

(FUVEST - 2009 - 1 FASE)

In 1993, the dawn of the Internet age, the liberating anonymity of the online world was captured in a wellknown New Yorker cartoon. One dog, sitting at a computer, tells another: “On the Internet, nobody knows you’re a dog.” Fifteen years later, that anonymity is gone.

Technology companies have long used “cookies,” little bits of tracking software slipped onto your computer, and other means, to record the Web sites you visit, the ads you click on, even the words you enter in search engines – information that some hold onto forever. They’re not telling you they’re doing it, and they’re not asking permission. Internet service providers (I.S.P.’s) are now getting into the act. Because they control your connection, they can keep track of everything you do online, and there have been reports that I.S.P.’s may have started to sell the information they collect.

The driving force behind this prying is commerce. The big growth area in online advertising right now is “behavioral targeting.” Web sites can charge a premium if they are able to tell the maker of an expensive sports car that its ads will appear on Web pages clicked on by upperincome, middle-aged men.

The New York Times, April 5th 2008.

ISP = Provedores de serviço de internet.

De acordo com o texto, a evolução da internet nos últimos quinze anos permite concluir que 

Questão 45033

(FUVEST - 2009 - 2 fase - Questão 2)

Na figura, estão representadas a circunferência C, de centro O e raio 2, e os pontos A, B, P e Q , de tal modo que:

1. O ponto O pertence ao segmento .

2. OP=1, OQ= . 

3. A e B são pontos da circunferência,  e 

Assim sendo, determine:

a) A área do triângulo APO .

b) Os comprimentos dos arcos determinados por A e B em C .

c) A área da região hachurada.