The figure illustrates the visual project for making a commemorative medal, with the shape of a right circular cylinder, with diameter 6 cm and thickness 3 mm.
The figure $ABCD$ has the shape of a square and is the base of a prism that crosses the entire medal. The region of the medal external to this prism will be minted in gold. It is intended to mint 100 of these medals.
Consider 3.1 as the approximate value for $\pi$. What is the volume of gold, in cubic centimeter, necessary for the making of these medals?
(A) 288
(B) 297
(C) 567
(D) 990
(E) 1134

The luminance of an object is the quantity that describes the amount of light produced or reflected by its surface. It is defined as the ratio between the luminous intensity, measured in candela (cd), and the square of the distance from the object to the light source, measured in meter (m).
The unit of measurement of the luminance of an object is
(A) $\dfrac{\mathrm{cd}}{\mathrm{m}^2}$
(B) $\dfrac{\mathrm{m}^2}{\mathrm{cd}}$
(C) $\dfrac{\mathrm{cd}}{\mathrm{m}}$
(D) $\dfrac{\mathrm{m}}{\mathrm{cd}}$
(E) $\dfrac{\mathrm{m}}{\mathrm{cd}^2}$

A person has a biofuel car, which runs on natural gas vehicle (NGV) and gasoline. The performance of the car, measured in km/m$^3$, in the case of gas, or measured in km/L, in the case of gasoline, depends, among other factors, on the speed, in km/h, at which the car travels. This relationship is in accordance with the graphs.
During a holiday, this person took a 240 km trip. To obtain an estimate of fuel consumption, assume that throughout the journey a constant speed of $60\,\mathrm{km/h}$ was maintained. Consider that, during half of the journey, only NGV was used and, in the other half, only gasoline. What was paid per cubic meter of NGV and per liter of gasoline corresponded, respectively, to R\$ 2.00 and R\$ 3.00.
What was the difference, in reais, between the total expenses with gasoline and with NGV?
(A) 4
(B) 8
(C) 14
(D) 21
(E) 30

In a country, the first step to obtain a driver's license is the hiring of three products:

• package with 20 theoretical classes;
• package with 10 practical classes;
• vehicle rental for conducting practical classes.

A person who intends to obtain a driver's license researched the value of vehicle rental and the values of each theoretical class and each practical class at three driving schools. The table presents these values.

Driving School	Value of each theoretical class (R\$) & Value of each practical class (R\$)	Vehicle rental (R\$)
I	10	80	400
II	30	50	200
III	20	40	400

She will hire the three products at the same driving school so that the total cost in this first stage is as small as possible. The driving school that will be hired is
(A) I, with a total cost of R\$ 1400.00.
(B) II, with a total cost of R\$ 1300.00.
(C) II, with a total cost of R\$ 1300.00.
(D) III, with a total cost of R\$ 1200.00.
(E) III, with a total cost of R\$ 1200.00.

A flush tank, attached to a toilet, has the shape of a right rectangular parallelepiped whose internal base dimensions are $2.5\,\mathrm{dm}$ and $1.5\,\mathrm{dm}$. In this tank there is a float that interrupts the supply when the height of the water column reaches 2 dm.
Each time the flush is activated, all the volume of water contained in the tank is poured into the toilet. To reduce the volume of water poured each time the flush is activated, a person will place, inside this tank, 300 mL bottles filled with sand and capped, so that they remain submerged when the supply is interrupted.
To guarantee efficient operation, the minimum amount of water poured each time the flush is activated should be 5 L.
The maximum number of bottles that will be placed in this tank, guaranteeing efficient operation, is equal to
(A) 10.
(B) 8.
(C) 4.
(D) 3.
(E) 2.

A pastry chef started producing cakes in the shape of a right circular cylinder, with the radius of the base varying between 12 cm and 16 cm and height of 6 cm. These cakes should be packaged in boxes with the shape of a right prism with a square base, so that it is possible to accommodate the cake inside and still have at least 1 cm of distance remaining between the cake and the internal surfaces of the box, lateral and upper. He originally has boxes in the intended format, whose internal dimensions are 14 cm for the length of the side of the base and 7 cm in height, which do not meet his needs. Therefore, he will buy new boxes, with the same format as the original boxes, but with a larger length of the side of the base, that are suitable for packaging all types of cakes he produces.
The edge of the base of the new boxes should be, at minimum, how many centimeters larger than that of the original boxes?
(A) 4
(B) 12
(C) 16
(D) 18
(E) 20

Suppose $A$ is an $m \times n$ matrix, $V$ an $m \times 1$ matrix, with both $A$ and $V$ having rational entries. If the equation $A X = V$ has a solution in $\mathbb{R}^n$, then the equation has a solution with rational entries. (Here and in Question 5 below of Part $\mathrm{A}$, $\mathbb{R}^n$ is identified with the space of $n \times 1$ real matrices.)

A closed and bounded subset of a complete metric space is compact.

Let $p$ be a prime number. If $P$ is a $p$-Sylow subgroup of some finite group $G$, then for every subgroup $H$ of $G$, $H \cap P$ is a $p$-Sylow subgroup of $H$.

A continuous function on $\mathbb{Q} \cap [0,1]$ can be extended to a continuous function on $[0,1]$.

Let $f : \mathbb{R} \rightarrow \mathbb{R}$ be a differentiable function. Then $f'(x)$ is continuous.

There is a continuous onto function from the unit sphere in $\mathbb{R}^3$ to the complex plane $\mathbb{C}$.

$f : \mathbb{C} \rightarrow \mathbb{C}$ is an entire function such that the function $g(z)$ given by $g(z) = f\left(\frac{1}{z}\right)$ has a pole at $0$. Then $f$ is a surjective map.

Every finite group of order 17 is abelian.

Let $n \geq 2$ be an integer. Given an integer $k$ there exists an $n \times n$ matrix $A$ with integer entries such that $\operatorname{det} A = k$ and the first row of $A$ is $(1, 2, \ldots, n)$.

There is a finite Galois extension of $\mathbb{R}$ whose Galois group is nonabelian.

There is a non-constant continuous function from the open unit disc $$D = \{ z \in \mathbb{C} \mid |z| < 1 \}$$ to $\mathbb{R}$ which takes only irrational values.

There is a field of order 121.

Let $\alpha, \beta$ be two complex numbers with $\beta \neq 0$, and $f(z)$ a polynomial function on $\mathbb{C}$ such that $f(z) = \alpha$ whenever $z^5 = \beta$. What can you say about the degree of the polynomial $f(z)$?

Let $f, g : \mathbb{Z}/5\mathbb{Z} \rightarrow S_5$ be two non-trivial group homomorphisms. Show that there is a $\sigma \in S_5$ such that $f(x) = \sigma g(x) \sigma^{-1}$, for every $x \in \mathbb{Z}/5\mathbb{Z}$.

Suppose $f$ is continuous on $[0, \infty)$, differentiable on $(0, \infty)$ and $f(0) \geq 0$. Suppose $f'(x) \geq f(x)$ for all $x \in (0, \infty)$. Show that $f(x) \geq 0$ for all $x \in (0, \infty)$.

If $f$ and $g$ are continuous functions on $[0,1]$ satisfying $f(x) \geq g(x)$ for every $0 \leq x \leq 1$, and if $\int_0^1 f(x)\, dx = \int_0^1 g(x)\, dx$, then show that $f = g$.

Let $\{a_n\}$ and $\{b_n\}$ be sequences of complex numbers such that each $a_n$ is non-zero, $\lim_{n \rightarrow \infty} a_n = \lim_{n \rightarrow \infty} b_n = 0$, and such that for every natural number $k$, $$\lim_{n \rightarrow \infty} \frac{b_n}{a_n^k} = 0$$ Suppose $f$ is an analytic function on a connected open subset $U$ of $\mathbb{C}$ which contains $0$ and all the $a_n$. Show that if $f(a_n) = b_n$ for every natural number $n$, then $b_n = 0$ for every natural number $n$.

Let $T : \mathbb{R}^3 \rightarrow \mathbb{R}^3$ be an orthogonal transformation such that $\operatorname{det} T = 1$ and $T$ is not the identity linear transformation. Let $S \subset \mathbb{R}^3$ be the unit sphere, i.e., $$S = \left\{ (x, y, z) \mid x^2 + y^2 + z^2 = 1 \right\}$$ Show that $T$ fixes exactly two points on $S$.

Compute $$\int_0^{\infty} \frac{x^{1/3}}{1 + x^2}\, dx$$