In a factory, a kiln bakes ceramics at a temperature of $1000^{\circ}\mathrm{C}$. At the end of baking, it is turned off and cools down. The temperature of the kiln is expressed in degrees Celsius (${}^{\circ}\mathrm{C}$). The kiln door can be opened safely for the ceramics as soon as its temperature is below $70^{\circ}\mathrm{C}$.
For a natural integer $n$, we denote $T_n$ the temperature in degrees Celsius of the kiln after $n$ hours have elapsed from the moment it was turned off. We therefore have $T_0 = 1000$. The temperature $T_n$ is calculated by the following algorithm:
\begin{verbatim} T←1000 For i going from 1 to n T←0.82 x T+3.6 End For \end{verbatim}

1. Determine the temperature of the kiln, rounded to the nearest unit, after 4 hours of cooling.
2. Prove that, for every natural integer $n$, we have: $T_n = 980 \times 0.82^n + 20$.
3. After how many hours can the kiln be opened safely for the ceramics?

Let $S = 1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \ldots + \frac{1}{100}$. Among the Python scripts below, the one that allows calculating the sum $S$ is:
a. \begin{verbatim} def somme_a() : S = 0 for k in range(100) : S =1/( k+1) return S \end{verbatim}
b. \begin{verbatim} def somme_b() : S = 0 for k in range(100) : S = S + 1/(k + 1) return S \end{verbatim}
c. \begin{verbatim} def somme_c() : k = 0 while S < 100 : S = S+1 /(k+1) return S \end{verbatim}
d. \begin{verbatim} def somme_d() : k = 0 while k < 100: S = S + 1/(k + 1) return S \end{verbatim}

Question 159
A progressão aritmética $(a_n)$ tem primeiro termo $a_1 = 3$ e razão $r = 4$. O valor de $a_{10}$ é
(A) 35 (B) 39 (C) 40 (D) 43 (E) 47

A sequência $(a_n)$ é uma progressão aritmética com $a_1 = 3$ e razão $r = 4$. O valor de $a_{10}$ é
(A) 35 (B) 39 (C) 40 (D) 43 (E) 47

A soma dos termos de uma progressão aritmética finita com 10 termos, primeiro termo igual a 2 e último termo igual a 20, é
(A) 100 (B) 110 (C) 120 (D) 130 (E) 140

The projections for rice production in the period 2012-2021, in a certain producing region, point to a perspective of constant growth in annual production. The table presents the quantity of rice, in tons, that will be produced in the first years of this period, according to this projection.

Year	Production projection (t)
2012	50.25
2013	51.50
2014	52.75
2015	54.00

The total amount of rice, in tonnes, that should be produced in the period from 2012 to 2021 will be
(A) 497.25. (B) 500.85. (C) 502.87. (D) 558.75. (E) 563.25.

The Sun's magnetic activity cycle has a period of 11 years. The beginning of the first recorded cycle occurred at the beginning of 1755 and extended until the end of 1765. Since then, all cycles of the Sun's magnetic activity have been recorded.
In the year 2101, the Sun will be in the magnetic activity cycle number
(A) 32. (B) 34. (C) 33. (D) 35. (E) 31.

QUESTION 160
The sum of the first 10 terms of an arithmetic progression with first term 2 and common difference 3 is
(A) 145
(B) 155
(C) 165
(D) 175
(E) 185

A club has a soccer field with a total area of $8000 \mathrm{~m}^{2}$, corresponding to the grass. Usually, the grass mowing of this field is done by two machines owned by the club for this service. Working at the same pace, the two machines mow together $200 \mathrm{~m}^{2}$ per hour. Due to the urgency of holding a soccer match, the field administrator will need to request machines from the neighboring club equal to his own to do the mowing work in a maximum time of 5 h.
Using the two machines that the club already has, what is the minimum number of machines that the field administrator should request from the neighboring club?
(A) 4
(B) 6
(C) 8
(D) 14
(E) 16

A passion fruit producer uses a water tank with volume $V$ to feed the irrigation system of his orchard. The system draws water through a hole at the bottom of the tank at a constant flow rate. With the water tank full, the system was activated at 7 a.m. on Monday. At 1 p.m. on the same day, it was found that 15\% of the water volume in the tank had already been used. An electronic device interrupts the system's operation when the remaining volume in the tank is 5\% of the total volume, for refilling.
Assuming that the system operates without failures, at what time will the electronic device interrupt the operation?
(A) At 3 p.m. on Monday.
(B) At 11 a.m. on Tuesday.
(C) At 2 p.m. on Tuesday.
(D) At 4 a.m. on Wednesday.
(E) At 9 p.m. on Tuesday.

In a cafeteria, the success of summer sales are juices prepared based on fruit pulp. One of the best-selling juices is strawberry with acerola, which is prepared with $\frac{2}{3}$ of strawberry pulp and $\frac{1}{3}$ of acerola pulp.
For the merchant, the pulps are sold in packages of equal volume. Currently, the strawberry pulp package costs $\mathrm{R}\$ 18.00$ and the acerola one costs $\mathrm{R}\$ 14.70$. However, a price increase is expected for the acerola pulp package next month, rising to $\mathrm{R}\$ 15.30$.
To not increase the price of the juice, the merchant negotiated with the supplier a reduction in the price of the strawberry pulp package.
The reduction, in reais, in the price of the strawberry pulp package should be
(A) 1.20.
(B) 0.90.
(C) 0.60.
(D) 0.40.
(E) 0.30.

In an arithmetic progression, the first term is 3 and the common difference is 4. What is the 10th term of this progression?
(A) 35
(B) 39
(C) 43
(D) 47
(E) 51

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) $99 + 4n$

The sum $$S = 1 + 111 + 11111 + \cdots + \underbrace{11\cdots1}_{2k+1}$$ is equal to . . . . . . .

For an arithmetic sequence $\left\{ a _ { n } \right\}$ $$a _ { 1 } + a _ { 2 } = 10 , \quad a _ { 3 } + a _ { 4 } + a _ { 5 } = 45$$ When this holds, what is the value of $a _ { 10 }$? [2 points]
(1) 47
(2) 45
(3) 43
(4) 41
(5) 39

For an arithmetic sequence $\left\{ a _ { n } \right\}$, $$a _ { 5 } = 4 a _ { 3 } , \quad a _ { 2 } + a _ { 4 } = 4$$ When these conditions hold, what is the value of $a _ { 6 }$? [2 points]
(1) 5
(2) 8
(3) 11
(4) 13
(5) 16

Two sequences $\left\{ a _ { n } \right\} , \left\{ b _ { n } \right\}$ are given by $$\begin{aligned} & a _ { n } = \frac { 1 } { 2 ^ { n - 1 } } \cos \frac { ( n - 1 ) \pi } { 2 } \\ & b _ { n } = \frac { 1 + ( - 1 ) ^ { n - 1 } } { 2 ^ { n } } \end{aligned}$$ Which of the following in are correct? [4 points] 〈Remarks〉 ㄱ. For all natural numbers $k$, $a _ { 3 k } < 0$. ㄴ. For all natural numbers $k$, $a _ { 4 k - 1 } + b _ { 4 k - 1 } = 0$. ㄷ. $\sum _ { n = 1 } ^ { \infty } a _ { n } = \frac { 3 } { 5 } \sum _ { n = 1 } ^ { \infty } b _ { n }$
(1) ㄱ
(2) ㄴ
(3) ㄷ
(4) ㄱ, ㄴ
(5) ㄴ, ㄷ

Two sequences $\left\{ a _ { n } \right\} , \left\{ b _ { n } \right\}$ are given by
$$\begin{aligned} & a _ { n } = \frac { 1 } { 2 ^ { n - 1 } } \cos \frac { ( n - 1 ) \pi } { 2 } \\ & b _ { n } = \frac { 1 + ( - 1 ) ^ { n - 1 } } { 2 ^ { n } } \end{aligned}$$
Which of the following statements in are true? [4 points]

ㄱ. For all natural numbers $k$, $a _ { 3 k } < 0$. ㄴ. For all natural numbers $k$, $a _ { 4 k - 1 } + b _ { 4 k - 1 } = 0$. ㄷ. $\sum _ { n = 1 } ^ { \infty } a _ { n } = \frac { 3 } { 5 } \sum _ { n = 1 } ^ { \infty } b _ { n }$
(1) ㄱ
(2) ㄴ
(3) ㄷ
(4) ㄱ, ㄴ
(5) ㄴ, ㄷ

In the sequence $\left\{ a _ { n } \right\}$, $a _ { 1 } = 1 , a _ { 2 } = 4 , a _ { 3 } = 10$, and the sequence $\left\{ a _ { n + 1 } - a _ { n } \right\}$ is a geometric sequence. Find the value of $a _ { 5 }$. [3 points]

Three numbers $a , 0 , b$ form an arithmetic sequence in this order, and three numbers $2 b , a , - 7$ form a geometric sequence in this order. What is the value of $a$? [3 points]
(1) 10
(2) 12
(3) 14
(4) 16
(5) 18

For an arithmetic sequence $\left\{ a _ { n } \right\}$ with first term 0 and common difference not equal to 0, a sequence $\left\{ b _ { n } \right\}$ satisfies $a _ { n + 1 } b _ { n } = \sum _ { k = 1 } ^ { n } a _ { k }$. Find the value of $b _ { 27 }$. [4 points]

For an arithmetic sequence $\left\{ a _ { n } \right\}$ with $a _ { 2 } = 3 , a _ { 5 } = 24$, find the value of $a _ { 7 }$. [3 points]

Points are marked on the coordinate plane in the following [Steps]. [Step 1] Mark a point at $( 0,1 )$. [Step 2] Mark 3 points at $( 0,3 ) , ( 1,3 ) , ( 2,3 )$ in this order. $\vdots$ [Step $k$ ] Mark $( 2 k - 1 )$ points at $( 0,2 k - 1 ) , ( 1,2 k - 1 ) , ( 2,2 k - 1 ) , \cdots$, $( 2 k - 2,2 k - 1 )$ in this order. (Here, $k$ is a natural number.) $\vdots$ When points are marked in this manner starting from [Step 1], the coordinates of the 100th marked point are $( p , q )$. What is the value of $p + q$? [4 points]
(1) 46
(2) 43
(3) 40
(4) 37
(5) 34

For an arithmetic sequence $\left\{ a _ { n } \right\}$ with common difference 2,
$$a _ { 1 } + a _ { 5 } + a _ { 9 } = 45$$
Find the value of $a _ { 1 } + a _ { 10 }$. [3 points]

An arithmetic sequence $\left\{ a _ { n } \right\}$ satisfies $a _ { 2 } + a _ { 4 } = 8$ and $a _ { 7 } = 52$. Find the common difference. [3 points]