The remote control of a toy car comes equipped with a screen that automatically adjusts the scale used in displaying each displacement. The screen displays the image of the displacement, the scale used in generating this image, and the length of this displacement, in centimeters, in accordance with the scale used. The figures represent the remote control screen displaying data from five displacements made by this toy car.
The option that indicates the displacement of greatest length performed by the toy car is
(A) I.
(B) II.
(C) III.
(D) IV.
(E) V.

An artist, who usually makes drawings with sand on the beach, asked a beachgoer to make a small drawing, which would serve as a sketch for a large work of art to be made in the sand. This drawing is represented in the figure.
After completion, the work of art obtained maintained the same proportions as the drawing made by the beachgoer, with the measurements indicated in the figure being enlarged to 30 m.
At what scale does this drawing represent the work of art?
(A) 1 : 1.5
(B) 1 : 2.25
(C) 1 : 10
(D) 1 : 100
(E) 1 : 150

Cortisol is a hormone produced by the adrenal glands and can be considered an important marker of physiological stress. In a study conducted with nurses, it was found that the concentration of salivary cortisol on a work day, denoted by $T$, was, on average, 1.59 times the concentration of salivary cortisol on a day off, denoted by $F$.
In this study, the relationship obtained between $T$ and $F$ was
(A) $T = 1.59 + F$
(B) $F = 1.59 + T$
(C) $\dfrac{T}{F} = 1.59$
(D) $\dfrac{F}{T} = 1.59$
(E) $F \cdot T = 1.59$

A parking lot has 120 spaces for vehicles, and all these spaces are occupied. Each customer pays a monthly fee to use a space, which is calculated based on the parking lot's monthly expenses and the desired profit. The parking lot's monthly expenses are: R\$14,240.00 for maintenance plus R\$36.00 insurance per vehicle. The parking lot's profit is determined by the difference between the amount collected from monthly fees and the expenses incurred. Starting the following month, the insurance value per vehicle will increase by 20\%, and maintenance expenses will remain unchanged. With this, the parking lot owner will adjust the monthly fees to obtain a monthly profit of R\$10,000.00. Despite this adjustment, all spaces will remain occupied.
The value, in reais, of the adjusted monthly fee will be
(A) 185.60.
(B) 226.09.
(C) 245.20.
(D) 268.93.
(E) 285.60.

The owner of an ice cream shop stores ice cream in containers of $20{,}000\,\mathrm{cm}^3$. He serves the ice cream in cups, in portions of 250 mL.
The number of cups he can serve from a full container of ice cream is
(A) 5.
(B) 8.
(C) 50.
(D) 80.
(E) 800.

In a school, all high school students practice one of three sports modalities offered as a physical activity, and each of them practices only one of these activities. The graphs provide some data related to the quantities of students who practice these sports modalities in this school, although some quantities have not been provided.
What is the number of high school students in this school?
(A) 720
(B) 360
(C) 320
(D) 288
(E) 240

The owner of a boat must depart from point $P$ and arrive at point $R$ by means of two linear displacements and navigating at a constant speed. This trip will be made during the night, and since he has only a compass and a clock, he planned his route as follows: $1^{\text{st}}$ - depart from point $P$ in direction 110 and navigate for 4 hours, reaching a point $Q$; $2^{\text{nd}}$ - depart from point $Q$ in direction 90 and navigate for 2 hours, reaching the destination point $R$.
However, when directing the boat for the first displacement, he did so in direction 340, instead of 110. With this, he made the following displacements: $1^{\text{st}}$ - departed from point $P$ in direction 340 and navigated for 4 hours, reaching a point $S$; $2^{\text{nd}}$ - departed from point $S$ in direction 90 and navigated for 2 hours, reaching point $T$.
The boat owner only realized the mistake upon arriving at point $T$. With this, he now needs to define the direction and navigation time that will allow him, departing from point $T$, to reach the destination point $R$ through a straight route.
Consider 0.64 as an approximation for $\cos 50°$. The direction and approximate navigation time that the boat owner should use are, respectively,
(A) 135 and 7 hours and 15 minutes.
(B) 45 and 7 hours and 15 minutes.
(C) 135 and 12 hours.
(D) 135 and 6 hours.
(E) 45 and 6 hours.

A container has a shape such that, when filled with water at a constant flow rate, the distance $D$ from the water surface to the table top, in centimeter, increases in relation to time $T$, in minute, according to a function of the type $$D = k + \operatorname{tg}[p(T + m)],$$ where the parameters $k$, $p$, and $m$ are real numbers, for $T$ varying from 0 to 4 minutes, as illustrated in the figure, in which the vertical asymptotes of the tangent function used in the definition of $D$ are presented.
The algebraic expression that represents the relationship between $D$ and $T$ is
(A) $D = 2.5 + \operatorname{tg}\left[30\left(T - \dfrac{5 - 2\pi}{2}\right)\right]$
(B) $D = 4 + \operatorname{tg}\left[30\left(T + \dfrac{5}{2}\right)\right]$
(C) $D = 4 + \operatorname{tg}\left[2.5\left(T + \dfrac{5 + 2\pi}{2}\right)\right]$
(D) $D = 30 + \operatorname{tg}\left[\dfrac{1}{2}(T - 5)\right]$
(E) $D = 30 + \operatorname{tg}\left[\dfrac{1}{2}\left(T - \dfrac{5}{2}\right)\right]$

The gray squares in the figure represent the city blocks in a part of the neighborhood where João and his friend live. The small square (A), painted in black and located in the upper left corner of a larger square, indicates the house of João's friend. João also lives in a corner house, but at the northeast end of a city block. To reach his friend's house, upon leaving home, João must walk through the block where he lives in the west direction, turn right, walk through three blocks in the north direction, and turn left. His friend's house is in the second block to the west.
The city block where João's house is located is represented by the square with the letter
(A) P.
(B) Q.
(C) R.
(D) S.
(E) T.

A company produced, in a given month, 110 tons of plastic from petroleum derivatives and 80 tons from recycled plastics. The cost to recycle one ton of plastic is R\$ 500.00, which equals 5\% of the cost to produce the same amount of plastic from petroleum derivatives. For the following month, this company's goal is to produce the same amount of plastic that was produced in this month, but with a reduction of at least 50\% in production cost.
For the company to achieve its goal in the following month, the minimum amount of tons of plastic that must be produced from recycling should be
(A) 135.
(B) 140.
(C) 155.
(D) 160.
(E) 175.

In a city, a tunnel will be built that crosses a mountain to facilitate the transit of automobiles and bicycles. Two projects were developed and the schemes with the front views of these projects are presented in the figure.
Project 1 has two tunnels, one exclusive for bicycles and the other for automobiles. Project 2 has a single tunnel, with spaces reserved for exclusive transit of bicycles and automobiles. In both projects, the tunnels have the shape of a straight semicylinder of the same length, with two-way routes for both types of vehicles, separated by walls.
The project to be approved will be the one that presents the smallest cross-sectional area, as it will imply a smaller volume of material removed from the mountain.
Consider 3 as an approximation for $\pi$ and disregard the thicknesses of the walls.
The project to be approved is
(A) 1, as it presents a cross-sectional area measuring $67.5\,\mathrm{m}^2$.
(B) 2, as it presents a cross-sectional area measuring $121.5\,\mathrm{m}^2$.
(C) 1, as it presents a cross-sectional area measuring $135\,\mathrm{m}^2$.
(D) 2, as it presents a cross-sectional area measuring $243\,\mathrm{m}^2$.
(E) either one of the two, as they present cross-sectional areas with equal measurements.

A car that costs 60 thousand reais is sold by a dealership that offers two payment options, both without down payment and without interest:

• option 1: payment in $n$ equal installments;
• option 2: payment in 6 more installments than in option 1 and, with this, the value of each installment becomes R\$ 500.00 less than the value of the installment in option 1. In both payment options, the total value to be paid for the car is the same.

What is the quantity $n$ of installments contained in payment option 1?
(A) 18
(B) 24
(C) 30
(D) 42
(E) 48

The final of a football championship was disputed in 2 regular periods, of 45 minutes each, without added time, with an extension of 30 minutes, also without added time. A player entered at the beginning of the second period, with equipment to measure the distance traveled during his participation in the game. At the end of the second regular period, this player had traveled $4.5\,\mathrm{km}$. He maintained in the extension the same average speed that he had maintained in the second regular period.
The distance traveled by this player during his participation in the match, in kilometers, was
(A) 4.5.
(B) 6.0.
(C) 7.5.
(D) 9.0.
(E) 12.0.

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

Three cubic dice, with faces numbered from 1 to 6, were used in a game. Artur chose two dice, and João got the third. The game consists of both rolling their dice, observing the numbers on the faces facing up, and comparing the largest number obtained by Artur with the number obtained by João. The player who obtains the largest number wins. In case of a tie, the victory goes to João.
The player who has the greatest probability of victory is
(A) Artur, with probability of $\dfrac{2}{3}$
(B) João, with probability of $\dfrac{4}{9}$
(C) Artur, with probability of $\dfrac{91}{216}$
(D) João, with probability of $\dfrac{91}{216}$
(E) Artur, with probability of $\dfrac{125}{216}$

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) $\dfrac{\mathrm{m}}{\mathrm{cd}^2}$

Four friends, each with 100 coins, created a game, in which each one assumes one of four positions, $1, 2, 3$, or $4$, indicated in the figure, and remains there until the end.
The development of the game takes place in rounds and, in all of them, each player transfers and receives a quantity of coins, as follows:

• the player in position 1 transfers 1 coin to the player in position 2;
• the player in position 2 transfers 2 coins to the player in position 3;
• the player in position 3 transfers 3 coins to the player in position 4;
• the player in position 4 transfers 4 coins to the player in position 1, completing the round.

At the end of round $n$, what is the algebraic expression that represents the number of coins of the player in position 1?
(A) $103 + 4n$
(B) $103 + 3n$
(C) $100 + 4n$
(D) $100 + 3n$
(E) $99 + 4n$

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 $[0,1]$.