Seja !$ P_n !$ um polígono convexo regular de !$ n !$ lados, com !$ n \ge 3 !$. Considere as afirmações a seguir:

I. !$ P_n !$ é inscritível numa circunferência.

II. !$ P_n !$ é circunscritível a uma circunferência.

III. Se !$ ℓn !$ é o comprimento de um lado de !$ P_n !$ e !$ a_n !$ é o comprimento de um apótema de !$ P_n !$, então !$ { \large {a_n \over ℓ_n}} \le 1 !$ para todo !$ n \ge 3 !$.

É (são) verdadeira(s)

Seja !$ p !$ o polinômio dado por !$ p(x) = x^8 + x^m - 2x^n !$, em que os expoentes !$ 8,m,n !$ formam, nesta ordem, uma progressão geométrica cuja soma dos termos é igual a 14. Considere as seguintes afirmações:

I. !$ x = 0 !$ é uma raiz dupla de !$ p !$.

II. !$ x = 1 !$ é uma raiz dupla de !$ p !$.

III. !$ p !$ tem quatro raízes com parte imaginária não nula.

Destas, é (são) verdadeira(s)

Um líquido condutor (metal fundido) flui no interior de duas chapas metálicas paralelas, interdistantes de !$ 2,0 \,cm !$, formando um capacitor plano, conforme a figura. Toda essa região interna está submetida a um campo homogêneo de indução magnética de !$ 0,01 \,T !$, paralelo aos planos das chapas, atuando perpendicularmente à direção da velocidade do escoamento. Assinale a opção com o módulo dessa velocidade quando a diferença de potencial medida entre as placas for de !$ 0,40 \,mV !$.

Brazil’s business Belindia – Why the country produces fewer world-class companies than it should

BRAZILIANS make up almost 3% of the planet’s population and produce about 3% of its output. Yet of the firms in Fortune magazine’s 2014 “Global 500” ranking of the biggest companies by revenue only seven, or 1.4%, were from Brazil, down from eight in 2013. And on Forbes’s list of the 2,000 most highly valued firms worldwide just 25, or 1.3%, were Brazilian. The country’s biggest corporate “star”, Petrobras, is mired in scandals, its debt downgraded to junk status. In 1974 Edmar Bacha, an economist, described its economy as “Belindia”, a Belgium-sized island of prosperity in a sea of India-like poverty. Since then Brazil has done far better than India in alleviating poverty, but in business terms it still has a Belindia problem: a handful of world-class enterprises in a sea of poorly run ones.

Brazilian businesses face a litany of obstacles: bureaucracy, complex tax rules, shoddy infrastructure and a shortage of skilled workers—to say nothing of a stagnant economy. But a big reason for Brazilian firms’ underperformance is less well rehearsed: poor management. Since 2004 John van Reenen of the London School of Economics and his colleagues have surveyed 11,300 midsized firms in 34 countries, grading them on a five-point scale based on how well they monitor their operations, set targets and reward performance. Brazilian firms’ average score, at 2.7, is similar to that of China’s and a bit above that of India’s. But Brazil ranks below Chile (2.8) and Mexico (2.9); America leads the pack with 3.3. The best Brazilian firms score as well as the best American ones, but its long tail of badly run ones is fatter.

Part of the explanation is that medium and large firms tend to be better-organised than small ones, and not only because well-run ones are likelier to grow. Brazil offers incentives aplenty to stay bitty, such as preferential tax treatment for firms with a turnover of no more than 3.6m reais ($1.3m). As they expand, many firms split rather than face increased scrutiny from the taxman. According to the World Bank, a midsized Brazilian firm spends 2,600 hours filing taxes each year. In Mexico, it is 330 hours.

Ownership patterns play a part too. Many Brazilian concerns are controlled by an individual shareholder, or one or two families. Two-thirds of those with sales of more than $1 billion a year are family-owned, notes Heinz-Peter Elstrodt of McKinsey, a consulting firm. That is less than in Mexico (96%) or South Korea (84%) but more than in America or Europe. Mr Van Reenen’s research shows that where family owners plump for outside chief executives, their firms do no worse than similarly sized ones with more diverse shareholders. But all too often they pick kin over professional managers—and performance suffers. This is particularly true in “low-trust” societies like Brazil, where bosses hire relatives instead of better-qualified strangers to avoid being robbed or sued for falling foul of overly worker-friendly labour laws.

Decades of economic turmoil—which ended when hyperinflation was vanquished in 1994—meant that companies were managed from crisis to crisis. This forced Brazilian firms to be nimble. But it also encouraged short-termism, which management consultants and academics finger as Brazilian managers’ number-one sin. Faced with a record drought in 2014, and a subsequent spike in energy prices in a hydropower-dependent country, Usiminas, a steelmaker, stopped smelting and started selling power it had bought on cheap long-term contracts. Energy sales made up most of its operating profits that year. Such short-term stunts are hardly the path to long-term greatness.

Worse, crisis management all too often consists of going cap in hand to the government. Brazilian bosses continue to waste hours in meetings with politicians that could be better spent improving their businesses. In January 2014, as vehicle sales flagged, the automotive industry’s reflex reaction was to descend on the capital, Brasília, and demand an extension of its costly tax breaks. Thanks to lifelines cast by the state, feeble firms stay afloat rather than sink and make room for more agile competitors. Shielded from competition by tariffs, subsidies and local-content rules, they have little reason to innovate. A locally invented gizmo which lets cars run on both petrol and biodiesel is nifty. But, asks Marcos Lisboa of Insper, a business school, does that really justify six decades of public support for the motor industry?

The dead hand of government

Indeed, a glance at the “Belgian” end of Brazil’s corporate landscape suggests that successful firms cluster in sectors the state has not tried desperately to help, such as retail or finance. Bradesco, a big lender, is internationally praised as a pioneer of automated banking. Each month Arezzo creates 1,000 new models of women’s shoes, and picks 170-odd to sell in its shops.

Brazil’s other world-beaters are in industries like agriculture and aerospace, which are free to compete at home and abroad, and in which the government sticks to its proper role. In 1990 farms were allowed to consolidate and to buy foreign machines, pesticides and fertiliser. Efforts by Brazil’s trade negotiators opened up export markets. JBS, a meat giant, can slaughter 100,000 head of cattle a day, selling more beef than any rival worldwide. Thanks in part to Embrapa, the national agriculture-research agency, Brazilian farms have been raising productivity by about 4% a year for two decades. Similarly, a supply of skilled engineers and know-how from the government’s Technological Institute of Aeronautics has helped turn Embraer, privatised in 1994, into one of the world’s most successful aircraft-makers.

The success of businesses such as these offers a lesson for the state. The best way to make Brazil’s underperforming firms more competitive would be to make them compete more. Coddling by the state can be more a curse than a blessing. Ronald Reagan’s dictum that the nine most terrifying words in the English language are, “I’m from the government and I’m here to help,” translates well into Flemish, Hindi and Brazilian Portuguese.

By Schumpeter. In: The Economist. Feb 28th,2015.

Sejam !$ λ !$ uma circunferência de raio 4 cm e !$ \overline{PQ} !$ uma corda em !$ λ !$ de comprimento 4 cm. As tangentes a !$ λ !$ em !$ P !$ e em !$ Q !$ interceptam-se no ponto !$ R\ !$ exterior a !$ λ !$. Então, a área do triângulo !$ PQR\ !$, em !$ cm^2 !$, é igual a

Brazil’s business Belindia – Why the country produces fewer world-class companies than it should

BRAZILIANS make up almost 3% of the planet’s population and produce about 3% of its output. Yet of the firms in Fortune magazine’s 2014 “Global 500” ranking of the biggest companies by revenue only seven, or 1.4%, were from Brazil, down from eight in 2013. And on Forbes’s list of the 2,000 most highly valued firms worldwide just 25, or 1.3%, were Brazilian. The country’s biggest corporate “star”, Petrobras, is mired in scandals, its debt downgraded to junk status. In 1974 Edmar Bacha, an economist, described its economy as “Belindia”, a Belgium-sized island of prosperity in a sea of India-like poverty. Since then Brazil has done far better than India in alleviating poverty, but in business terms it still has a Belindia problem: a handful of world-class enterprises in a sea of poorly run ones.

Brazilian businesses face a litany of obstacles: bureaucracy, complex tax rules, shoddy infrastructure and a shortage of skilled workers—to say nothing of a stagnant economy. But a big reason for Brazilian firms’ underperformance is less well rehearsed: poor management. Since 2004 John van Reenen of the London School of Economics and his colleagues have surveyed 11,300 midsized firms in 34 countries, grading them on a five-point scale based on how well they monitor their operations, set targets and reward performance. Brazilian firms’ average score, at 2.7, is similar to that of China’s and a bit above that of India’s. But Brazil ranks below Chile (2.8) and Mexico (2.9); America leads the pack with 3.3. The best Brazilian firms score as well as the best American ones, but its long tail of badly run ones is fatter.

Part of the explanation is that medium and large firms tend to be better-organised than small ones, and not only because well-run ones are likelier to grow. Brazil offers incentives aplenty to stay bitty, such as preferential tax treatment for firms with a turnover of no more than 3.6m reais ($1.3m). As they expand, many firms split rather than face increased scrutiny from the taxman. According to the World Bank, a midsized Brazilian firm spends 2,600 hours filing taxes each year. In Mexico, it is 330 hours.

Ownership patterns play a part too. Many Brazilian concerns are controlled by an individual shareholder, or one or two families. Two-thirds of those with sales of more than $1 billion a year are family-owned, notes Heinz-Peter Elstrodt of McKinsey, a consulting firm. That is less than in Mexico (96%) or South Korea (84%) but more than in America or Europe. Mr Van Reenen’s research shows that where family owners plump for outside chief executives, their firms do no worse than similarly sized ones with more diverse shareholders. But all too often they pick kin over professional managers—and performance suffers. This is particularly true in “low-trust” societies like Brazil, where bosses hire relatives instead of better-qualified strangers to avoid being robbed or sued for falling foul of overly worker-friendly labour laws.

Decades of economic turmoil—which ended when hyperinflation was vanquished in 1994—meant that companies were managed from crisis to crisis. This forced Brazilian firms to be nimble. But it also encouraged short-termism, which management consultants and academics finger as Brazilian managers’ number-one sin. Faced with a record drought in 2014, and a subsequent spike in energy prices in a hydropower-dependent country, Usiminas, a steelmaker, stopped smelting and started selling power it had bought on cheap long-term contracts. Energy sales made up most of its operating profits that year. Such short-term stunts are hardly the path to long-term greatness.

Worse, crisis management all too often consists of going cap in hand to the government. Brazilian bosses continue to waste hours in meetings with politicians that could be better spent improving their businesses. In January 2014, as vehicle sales flagged, the automotive industry’s reflex reaction was to descend on the capital, Brasília, and demand an extension of its costly tax breaks. Thanks to lifelines cast by the state, feeble firms stay afloat rather than sink and make room for more agile competitors. Shielded from competition by tariffs, subsidies and local-content rules, they have little reason to innovate. A locally invented gizmo which lets cars run on both petrol and biodiesel is nifty. But, asks Marcos Lisboa of Insper, a business school, does that really justify six decades of public support for the motor industry?

The dead hand of government

Indeed, a glance at the “Belgian” end of Brazil’s corporate landscape suggests that successful firms cluster in sectors the state has not tried desperately to help, such as retail or finance. Bradesco, a big lender, is internationally praised as a pioneer of automated banking. Each month Arezzo creates 1,000 new models of women’s shoes, and picks 170-odd to sell in its shops.

Brazil’s other world-beaters are in industries like agriculture and aerospace, which are free to compete at home and abroad, and in which the government sticks to its proper role. In 1990 farms were allowed to consolidate and to buy foreign machines, pesticides and fertiliser. Efforts by Brazil’s trade negotiators opened up export markets. JBS, a meat giant, can slaughter 100,000 head of cattle a day, selling more beef than any rival worldwide. Thanks in part to Embrapa, the national agriculture-research agency, Brazilian farms have been raising productivity by about 4% a year for two decades. Similarly, a supply of skilled engineers and know-how from the government’s Technological Institute of Aeronautics has helped turn Embraer, privatised in 1994, into one of the world’s most successful aircraft-makers.

The success of businesses such as these offers a lesson for the state. The best way to make Brazil’s underperforming firms more competitive would be to make them compete more. Coddling by the state can be more a curse than a blessing. Ronald Reagan’s dictum that the nine most terrifying words in the English language are, “I’m from the government and I’m here to help,” translates well into Flemish, Hindi and Brazilian Portuguese.

By Schumpeter. In: The Economist. Feb 28th,2015.

A respeito de reações químicas descritas pela equação de Arrhenius, são feitas as seguintes proposições:

I. Para reações bimoleculares, o fator pré-exponencial na equação de Arrhenius é proporcional à frequência de colisões, efetivas ou não, entre as moléculas dos reagentes.

II. O fator exponencial na equação de Arrhenius é proporcional ao número de moléculas cuja energia cinética relativa é maior ou igual à energia de ativação da reação.

III. Multiplicando-se o negativo da constante dos gases (−R) pelo coeficiente angular da reta ln k versus 1/T obtém-se o valor da energia de ativação da reação.

IV. O fator pré-exponencial da equação de Arrhenius é determinado pela intersecção da reta ln k versus 1/T com o eixo das abscissas.

Das proposições acima, está(ão) ERRADA(S)

No sistema de sinalização de trânsito urbano chamado de “onda verde”, há semáforos com dispositivos eletrônicos que indicam a velocidade a ser mantida pelo motorista para alcançar o próximo sinal ainda aberto. Considere que de início o painel indique uma velocidade de 45 km/h. Alguns segundos depois ela passa para 50 km/h e, finalmente, para 60 km/h. Sabendo que a indicação de 50 km/h no painel demora 8,0 s antes de mudar para 60 km/h, então a distância entre os semáforos é de