Four candidates presented themselves to take the exam of a competition. Before starting the exam, the cell phones of the four candidates were collected by the proctor, who stored them, each one, inside a black envelope. At the end of the exam, the proctor returned the four envelopes with the cell phones to the four candidates, in a random manner, since he had not identified the envelopes.
The probability that all candidates received back the envelopes with their respective cell phones is
(A) $\dfrac{1}{2}$
(B) $\dfrac{1}{10}$
(C) $\dfrac{1}{16}$
(D) $\dfrac{1}{24}$
(E) $\dfrac{1}{256}$

In a computer game, a cube is initially positioned as indicated in the figure.
Each displacement made by this cube always occurs in one of the directions defined by the three coordinate axes. When moving from the initial position, this cube moved 3 units closer to the $yz$ plane, moved 5 units away from the $xz$ plane, and moved 4 units closer to the $xy$ plane.
The figure that presents the orthogonal projections of this cube onto the three coordinate planes, after performing the described movements, is
(A), (B), (C), (D), or (E) as indicated in the figures.

A magazine report addressed the use of social networks by Brazilian internet users. Some of the data collected by the report are presented in the infographic.
According to the infographic data, when randomly selecting a Brazilian internet user in the period to which the report refers, the probability that he is a man who accesses some social network is
(A) $\dfrac{30}{90}$
(B) $\dfrac{36}{100}$
(C) $\dfrac{40}{100}$
(D) $\dfrac{40}{90}$
(E) $\dfrac{46}{90}$

A person intends to install a natural gas vehicle (NGV) kit in his car. At the store he chose to make the purchase and installation of this kit, there were five models of cylinders for gas storage, whose capacities, in cubic meters, were, respectively: $10, 14, 17, 21$, and 25. The price of the cylinder is proportional to its capacity. This car will travel 30 km daily, 7 days a week, and the NGV consumption is $1\,\mathrm{m}^3$ for every 13 km traveled. The person will choose the cylinder model with the lowest price and that guarantees only one refueling per week.
Under these conditions, what will be the capacity, in cubic meters, of the cylinder chosen by this person?
(A) 10
(B) 14
(C) 17
(D) 21
(E) 25

In a school cafeteria, there are five foods sold in packages with different quantities of servings.
The nutritional information contained on the labels of these products is indicated in the images.
A student always chooses the food with the lowest total amount of sodium per package.
Which of these products should be chosen by the student?
(A) Potato chips.
(B) Salted sticks.
(C) Multigrain biscuit.
(D) Polvilho biscuit.
(E) Water and salt biscuit.

A factory used a 3D printer to produce the prototype of a part. The prototype has the shape of a convex polyhedron, obtained by the juxtaposition of two distinct solids, one with the shape of a regular hexagonal prism and the other with the shape of a straight hexagonal pyramid frustum. The larger base of the pyramid frustum coincides with one of the bases of the prism.
After printing the prototype, it was sent to the customization sector for painting its surface. The criterion defined for painting considers that congruent faces must be painted with the same color, and non-congruent faces must have different colors. What is the quantity of colors used to paint the prototype?
(A) 9
(B) 8
(C) 6
(D) 4
(E) 3

Research in the area of neurobiology confirms that meditative practice is responsible for considerably reducing respiratory frequency for advanced practitioners, who, after initiating meditation, have their respiratory frequencies reduced until they stabilize at a lower level. The graph presents the relationship of respiratory frequency, in breaths per minute (rpm), in relation to time, in minutes, of an advanced practitioner, in which $(\mathrm{f}_1)$ represents the frequency at instant $\mathrm{t}_1$, when meditative practice begins; and $(\mathrm{f}_2)$, the frequency at instant $t_2$, from which it stabilizes during meditation.
From the instant $\mathrm{t}_1$, when the meditative practice begins, the behavior of respiratory frequency, in relation to time,
(A) remains constant.
(B) is directly proportional to time.
(C) is inversely proportional to time.
(D) decreases until the instant $\mathrm{t}_2$, after which it becomes constant.
(E) decreases proportionally to time, both between $\mathrm{t}_1$ and $\mathrm{t}_2$ and after $t_2$.

Around a circular lagoon, whose radius measures 1 km, there is a bicycle path. Due to frequent bicycle thefts, the city council plans to allocate police officers in strategic positions to patrol this bicycle path, in order to make it fully protected. A point on the bicycle path is considered protected if there is at least one police officer at most 200 m away from that point, positioned on the bicycle path. The figure illustrates a point $P$ on the bicycle path, which will be protected if there is at least one police officer positioned on the dark gray region.
Disregard the width of the bicycle path and use 3 as an approximation for $\pi$.
Under these conditions, the minimum number of police officers to be allocated along this bicycle path to make it protected is
(A) 4.
(B) 8.
(C) 15.
(D) 30.
(E) 60.

In a laboratory, a container holds 10 liters of a solution composed only of substances $\mathrm{S}_1$ and $\mathrm{S}_2$. Of this solution, 99.95\% is $\mathrm{S}_1$. An amount of $\mathrm{S}_1$ will be removed from this solution, maintaining the initial amount of $\mathrm{S}_2$, so that 99.90\% of the new solution is $S_1$.
What is the amount of $\mathrm{S}_1$, in liters, that will be removed?
(A) 0.0050
(B) 0.0100
(C) 0.5000
(D) 4.9775
(E) 5.0000

A fuel distributor owns tanker trucks with a capacity of 30,000 liters each. In any transport carried out by these trucks, the same volume of fuel is discarded because it contains many impurities. This discarded volume is independent of the quantity transported.
A gas station ordered 10,000 liters of gasoline from this distributor, which sent 10,200 liters, considering the volume discarded in transport. Nevertheless, the amount of gasoline delivered to the gas station was 9,900 liters.
In a new order, this gas station requested that exactly double the volume of gasoline ordered in the previous order be delivered.
Using the same truck from the previous delivery, what is the minimum volume of gasoline, in liters, that the distributor should send to guarantee delivery of the ordered quantity in this new order?
(A) 20,100
(B) 20,200
(C) 20,300
(D) 20,400
(E) 20,600

A technology company will standardize the internet connection speed it offers to its customers in ten cities. The company's management decides that its new reference speed standard will be the median of the reference speed values of connections in these ten cities. These values, in megabytes per second (MB/s), are presented in the table.

Cities	Reference speed (MB/s)
C1	390
C2	380
C3	320
C4	390
C5	340
C6	380
C7	390
C8	400
C9	350
C10	360

The reference speed, in megabytes per second, to be adopted by this company is
(A) 360.
(B) 370.
(C) 380.
(D) 390.
(E) 400.

In a digital game, there are three characters: one hero and two villains. The programming is done in such a way that the hero will always be attacked by the villain closest to him. One way to ``confuse'' the villains is to move the hero along trajectories that keep him equidistant from the villains, creating uncertainty between them, and thus preventing him from being attacked.
For the programming of one of the stages of this game, the programmer considered, in the Cartesian plane, the square STUV as the region of movement of the characters, where V and $T$ represent the fixed positions of the villains, and $S$, the initial position of the hero, as shown in the figure.
What is the equation of the trajectory along which the hero can move without being attacked?
(A) $y = -3x + 20$
(B) $y = -3x + 16$
(C) $y = -3x - 20$
(D) $y = 3x + 16$
(E) $y = 3x - 16$

A bookstore sells books of the following literary genres: science fiction, self-help, romance, and biography. The graph presents the inventory of books that this bookstore has, by literary genre and by author's nationality, as well as the demand by literary genre, obtained through a survey conducted with its regular customers.
The bookstore manager will place an order for new copies only of the genre whose quantity in stock is insufficient to meet the demand identified by the survey.
The genre of book from which the manager should order more copies is
(A) science fiction, because it is the one with the highest demand.
(B) biography, because it is the genre with the lowest demand.
(C) self-help, because the quantity in stock is less than the demand.
(D) biography, because it is the genre with the smallest quantity of books in stock.
(E) romance, because it is the one with the smallest inventory of books by Brazilian authors.

A language school offers courses in English, Spanish, French, and German. The graphs present the percentage distribution of enrollments, by language, in 2023, and the distribution of the number of enrollments, by language, in 2024.
To plan the activities for 2025, the school manager estimated that the total number of enrollments will be the same as in 2024, and the percentage distribution of enrollments, by language, will be equal to that recorded in 2023.
According to this estimate, the number of enrollments in the French course for the year 2025 will be
(A) 2.
(B) 12.
(C) 20.
(D) 22.
(E) 40.

In a clinical study, 55 women were randomly distributed into 5 groups of 11 people. To test a new medication, a group will be chosen in which the majority of women are between 20 and 30 years old. The other groups will take placebo or medications already on the market. The table, partially filled, provides some data related to the ages of women in these groups.

Groups	Minimum age	Maximum age	Mean	Median	Mode	Standard deviation
1			25			10
2				25		9
3					25
4			25			1
5	20	35

Even with the incomplete table, it was possible to select one of these groups because, with only the data presented in the table, a group was identified that certainly met the selection criterion.
The group chosen was
(A) 1.
(B) 2.
(C) 3.
(D) 4.
(E) 5.

Pace is a term used by a runner to denote his average rhythm in a race. It represents the average time, in seconds, that this runner takes to cover 1 km.
The diagram presents the time, in seconds, that a runner took to cross the marks that define the first four 1 km sections in a 5 km race, and the time spent to cover each 1 km section.
The best pace that this runner achieved in 5 km races was $281\,\mathrm{s/km}$. For him to repeat his best pace in 5 km races in this race, his time in the $5^{\text{th}}$ section should be how many seconds less than what he spent to cover the $4^{\text{th}}$ section?
(A) 1
(B) 2
(C) 8
(D) 9
(E) 15

Ten couples founded a dance group and decided to establish a board of directors with three positions: president, secretary, and treasurer. For greater representation, it was decided that at most one person per couple could hold a position on this board.
How many different boards can be formed by these 10 couples?
(A) $10 \times 9 \times 8$
(B) $20 \times 18 \times 16$
(C) $20 \times 19 \times 18$
(D) $10 \times 9 \times 8 \times 2$
(E) $20 \times 18 \times 16 \times 2$

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.

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) $D = 30 + \operatorname{tg}\left[\dfrac{1}{2}\left(T - \dfrac{5}{2}\right)\right]$

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.

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

A father bought eight different gifts (among which, a bicycle and a cell phone) to give to his three children. He intends to distribute the gifts so that the oldest and youngest children receive three gifts each, and the middle one receives the two remaining gifts. The oldest will receive, among his gifts, either a bicycle or a cell phone, but not both.
In how many distinct ways can the distribution of gifts be made?
(A) 36
(B) 53
(C) 300
(D) 360
(E) 560

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 $\dfrac{125}{216}$

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) $99 + 4n$

An entrepreneur uses machines whose internal pressure $P$, in atmosphere, depends on the continuous time of use $t$, in hour, and on a positive parameter $K$, which defines the model of the machine, according to the expression: $$P = 4 \cdot \log[-K \cdot (t + 1) \cdot (t - 19)]$$
The manufacturer of these machines recommends to the user that the internal pressure of this type of machine does not exceed 10 atmospheres during its operation.
The entrepreneur intends to buy new machines of this type that should operate, daily, for a continuous period of 10 hours. For this, he needs to define the model of machine to be acquired by choosing the largest possible value of the parameter $K$, in accordance with the manufacturer's recommendation. The largest value to be chosen for $K$ is
(A) $10^{0.5}$
(B) $10^{8}$
(C) $\dfrac{10^{2.5}}{84}$
(D) $\dfrac{10^{2.5}}{99}$
(E) $25 \times 10^{-2}$

The productivity of soybeans in a cultivated area is the average quantity of 50-kilogram sacks that are produced per hectare. The table presents the cultivated area and soybean productivity on a certain property, over five harvests, with periods of one year, from 2011 to 2016.

Harvest	$\mathbf{11\text{-}12}$	$\mathbf{12\text{-}13}$	$\mathbf{13\text{-}14}$	$\mathbf{14\text{-}15}$	$\mathbf{15\text{-}16}$
Cultivated area (hectare)	200	220	250	250	200
Productivity (sacks of $50\,\mathrm{kg}$ per hectare)	40	30	45	45	50

The line graph that represents the soybean production of this property, in tons, in these five harvests is
(A), (B), (C), (D), or (E) as indicated in the figures.