In a country with a constant population equal to 120 million, the inhabitants live either in rural areas or in cities. Population movements can be modeled as follows:

• in 2010, the population consists of 90 million rural inhabitants and 30 million city dwellers;
• each year, $10\%$ of rural inhabitants migrate to the city;
• each year, $5\%$ of city dwellers migrate to rural areas.

For any natural integer $n$, we denote:

• $u_n$ the population in rural areas, in the year $2010 + n$, expressed in millions of inhabitants;
• $v_n$ the population in cities, in the year $2010 + n$, expressed in millions of inhabitants.

We have $u_0 = 90$ and $v_0 = 30$.
Part A

1. Translate the fact that the total population is constant by a relation linking $u_n$ and $v_n$.
2. What formulas can be entered in cells B3 and C3 which, copied downward, allow us to obtain the spreadsheet showing the evolution of $(u_n)$ and $(v_n)$?
3. What conjectures can be made concerning the long-term evolution of this population?

Part B
Deduce from the recurrence relations that, for every natural integer $n$, $R_n = 50 \times 0.85^n + 40$ and determine the expression of $C_n$ as a function of $n$. b. Determine the limit of $R_n$ and of $C_n$ when $n$ tends towards $+\infty$. What can we conclude from this for the population studied? 6. a. Complete the algorithm so that it displays the number of years after which the urban population will exceed the rural population. b. By solving the inequality with unknown $n$, $$50 \times 0.85^n + 40 < 80 - 50 \times 0.85^n,$$ find again the value displayed by the algorithm.

Exercise 5 (5 points) — Candidates who have NOT followed the specialization course
A biologist wishes to study the evolution of the population of an animal species in a reserve. This population is estimated at 12000 individuals in 2016. The constraints of the natural environment mean that the population cannot exceed 60000 individuals.

Part A: a first model

In a first approach, the biologist estimates that the population grows by $5\%$ per year. The annual evolution of the population is thus modelled by a sequence $(v_n)$ where $v_n$ represents the number of individuals, expressed in thousands, in $2016 + n$. We thus have $v_0 = 12$.

1. Determine the nature of the sequence $(v_n)$ and give the expression of $v_n$ as a function of $n$.
2. Does this model meet the constraints of the natural environment?

Part B: a second model

The biologist then models the annual evolution of the population by a sequence $(u_n)$ defined by $u_0 = 12$ and, for every natural integer $n$, $$u_{n+1} = -\frac{1.1}{605}u_n^2 + 1.1\, u_n.$$

1. We consider the function $g$ defined on $\mathbb{R}$ by $$g(x) = -\frac{1.1}{605}x^2 + 1.1\, x$$ a. Justify that $g$ is increasing on $[0;60]$. b. Solve in $\mathbb{R}$ the equation $g(x) = x$.
   
2. We note that $u_{n+1} = g(u_n)$. a. Calculate the value rounded to $10^{-3}$ of $u_1$. Interpret. b. Prove by induction that, for every natural integer $n$, $0 \leqslant u_n \leqslant 55$. c. Prove that the sequence $(u_n)$ is increasing. d. Deduce the convergence of the sequence $(u_n)$. e. It is admitted that the limit $\ell$ of the sequence $(u_n)$ satisfies $g(\ell) = \ell$. Deduce its value and interpret it in the context of the exercise.
   
3. The biologist wishes to determine the number of years after which the population will exceed 50000 individuals with this second model. He uses the following algorithm:
   

   \multirow{2}{*}{Variables}	$n$ a natural integer
   \cline{2-2}	$u$ a real number
   Processing	$n$ takes the value 0
   	$u$ takes the value 12
   	While $\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots$
   	$\quad u$ takes the value $\ldots\ldots\ldots\ldots\ldots$
   $n$ takes the value $\ldots\ldots\ldots\ldots\ldots$
   	End While
   Output	Display $\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots$

   
   Copy and complete this algorithm so that it displays as output the smallest integer $r$ such that $u_r \geqslant 50$.

Exercise 4 (5 points) — Candidates who have followed the specialization course
In a public garden, an artist must install an aquatic artwork. This artwork will consist of two basins A and B as well as a filtering reserve R. Initially, the two basins each contain 100 liters of water. A system of pipes allows the following water transfers to be carried out, every hour and in this order:

• first, half of basin A empties into reserve R;
• then, three quarters of basin B empty into basin A;
• finally, 200 liters of water are added to basin A and 300 liters of water are added to basin B.

The quantities of water in the two basins A and B are modeled using two sequences $(a_n)$ and $(b_n)$: for any natural number $n$, we denote by $a_n$ and $b_n$ the quantities of water in hundreds of liters that will be respectively contained in basins A and B after $n$ hours. For any natural number $n$, we denote by $U_n$ the column matrix $U_n = \binom{a_n}{b_n}$. Thus $U_0 = \binom{1}{1}$.
1. Justify that, for any natural number $n$, $U_{n+1} = MU_n + C$ where $M = \left(\begin{array}{cc} 0.5 & 0.75 \\ 0 & 0.25 \end{array}\right)$ and $C = \binom{2}{3}$.
2. Consider the matrix $P = \left(\begin{array}{cc} 1 & 3 \\ 0 & -1 \end{array}\right)$.
a. Calculate $P^2$. Deduce that the matrix $P$ is invertible and specify its inverse matrix.
b. Show that $PMP$ is a diagonal matrix $D$ that you will specify.
c. Calculate $PDP$.
d. Prove by induction that, for any natural number $n$, $M^n = PD^nP$.
It is admitted henceforth that for any natural number $n$, $M^n = \left(\begin{array}{cc} 0.5^n & 3 \times 0.5^n - 3 \times 0.25^n \\ 0 & 0.25^n \end{array}\right)$.
3. Show that the matrix $X = \binom{10}{4}$ satisfies $X = MX + C$.
4. For any natural number $n$, we define the matrix $V_n$ by $V_n = U_n - X$.
a. Show that for any natural number $n$, $V_{n+1} = MV_n$.
b. It is admitted that, for any non-zero natural number $n$, $V_n = M^n V_0$. Show that for any non-zero natural number $n$, $$U_n = \binom{-18 \times 0.5^n + 9 \times 0.25^n + 10}{-3 \times 0.25^n + 4}.$$
5. a. Show that the sequence $(b_n)$ is increasing and bounded above. Determine its limit.
b. Determine the limit of the sequence $(a_n)$.
c. It is admitted that the sequence $(a_n)$ is increasing. Deduce the capacity of the two basins A and B that must be planned for the feasibility of the project, that is, to avoid any overflow.

A biologist is interested in the evolution of the population of an animal species on an island in the Pacific. At the beginning of 2020, this population had 600 individuals. We consider that the species will be threatened with extinction on this island if its population becomes less than or equal to 20 individuals. The biologist models the number of individuals by the sequence $(u_n)$ defined by:
$$\begin{cases} u_{0} & = 0.6 \\ u_{n+1} & = 0.75 u_{n} \left( 1 - 0.15 u_{n} \right) \end{cases}$$
where for every natural integer $n$, $u_{n}$ denotes the number of individuals, in thousands, at the beginning of the year $2020 + n$.

1. Estimate, according to this model, the number of individuals present on the island at the beginning of 2021 and then at the beginning of 2022.

Let $f$ be the function defined on the interval $[ 0 ; 1 ]$ by
$$f ( x ) = 0.75 x ( 1 - 0.15 x )$$

1. Show that the function $f$ is increasing on the interval $[ 0 ; 1 ]$ and draw up its variation table.
2. Solve in the interval $[ 0 ; 1 ]$ the equation $f ( x ) = x$.

We note for the rest of the exercise that, for every natural integer $n$, $u_{n+1} = f \left( u_{n} \right)$.
4. a. Prove by induction that for every natural integer $n$, $0 \leqslant u_{n+1} \leqslant u_{n} \leqslant 1$. b. Deduce that the sequence $\left( u_{n} \right)$ is convergent. c. Determine the limit $\ell$ of the sequence $(u_{n})$.
5. The biologist has the intuition that the species will sooner or later be threatened with extinction. a. Justify that, according to this model, the biologist is correct. b. The biologist has programmed in Python language the function menace() below:
\begin{verbatim} def menace() : u = 0.6 n = 0 while u>0.02 : u=0.75*u*(1-0.15*u) n = n+1 return n \end{verbatim}
Give the numerical value returned when the function menace() is called. Interpret this result in the context of the exercise.

Exercise 3: Sequences
The population of an endangered species is closely monitored in a nature reserve. Climate conditions as well as poaching cause this population to decrease by $10\%$ each year. To compensate for these losses, 100 individuals are reintroduced into the reserve at the end of each year. We wish to study the evolution of the population size of this species over time. For this, we model the population size of the species by the sequence $(u_n)$ where $u_n$ represents the population size at the beginning of the year $2020 + n$. We admit that for all natural integer $n$, $u_n \geqslant 0$. At the beginning of the year 2020, the studied population has 2000 individuals, thus $u_0 = 2000$.

1. Justify that the sequence $(u_n)$ satisfies the recurrence relation: $$u_{n+1} = 0.9u_n + 100.$$
2. Calculate $u_1$ then $u_2$.
3. Prove by induction that for all natural integer $n$: $1000 < u_{n+1} \leqslant u_n$.
4. Is the sequence $(u_n)$ convergent? Justify your answer.
5. We consider the sequence $(v_n)$ defined for all natural integer $n$ by $v_n = u_n - 1000$. a. Show that the sequence $(v_n)$ is geometric with common ratio 0.9. b. Deduce that, for all natural integer $n$, $u_n = 1000\left(1 + 0.9^n\right)$. c. Determine the limit of the sequence $(u_n)$. Give an interpretation of this in the context of this exercise.
6. We wish to determine the number of years necessary for the population size to fall below a certain threshold $S$ (with $S > 1000$). a. Determine the smallest integer $n$ such that $u_n \leqslant 1020$. Justify your answer by a calculation. b. In the Python program opposite, the variable $n$ denotes the number of years elapsed since 2020, the variable $u$ denotes the population size. Copy and complete this program so that it returns the number of years necessary for the population size to fall below the threshold $S$. \begin{verbatim} def population(S) : n=0 u=2000 while ......: u= ... n = ... return ... \end{verbatim}

At the beginning of 2021, a bird colony had 40 individuals. Observation leads to modeling the population evolution by the sequence $(u_n)$ defined for all natural integers $n$ by: $$\begin{cases} u _ { 0 } & = 40 \\ u _ { n + 1 } & = 0,008 u _ { n } \left( 200 - u _ { n } \right) \end{cases}$$ where $u _ { n }$ denotes the number of individuals at the beginning of the year $( 2021 + n )$.

1. Give an estimate, according to this model, of the number of birds in the colony at the beginning of 2022.

Consider the function $f$ defined on the interval $[ 0 ; 100 ]$ by $f ( x ) = 0,008 x ( 200 - x )$.

1. Solve in the interval $[ 0 ; 100 ]$ the equation $f ( x ) = x$.
2. a. Prove that the function $f$ is increasing on the interval $[ 0 ; 100 ]$ and draw its variation table. b. By noting that, for all natural integers $n , u _ { n + 1 } = f \left( u _ { n } \right)$, prove by induction that, for all natural integers $n$: $$0 \leqslant u _ { n } \leqslant u _ { n + 1 } \leqslant 100$$ c. Deduce that the sequence $(u_n)$ is convergent. d. Determine the limit $\ell$ of the sequence $(u_n)$. Interpret the result in the context of the exercise.
3. Consider the following algorithm: \begin{verbatim} def seuil(p) : n=0 u=40 while u < p: n =n+1 u=0.008*u*(200-u) return(n+2021) \end{verbatim} The execution of seuil(100) returns no value. Explain why using question 3.

The purpose of this part is to study a model of population evolution. In 2022, this population has 3000 individuals. We denote $P _ { n }$ the population size in thousands in the year $2022 + n$. Thus $P _ { 0 } = 3$. According to a model inspired by the Verhulst model, a Belgian mathematician of the XIX${}^{\mathrm{th}}$ century, we consider that, for all natural integer $n$ :
$$P _ { n + 1 } - P _ { n } = P _ { n } \left( 1 - b \times P _ { n } \right) , \text { where } b \text { is a strictly positive real number. }$$
The real number $b$ is a damping factor that allows us to account for the limited nature of the resources in the environment in which these individuals evolve.

1. In this question $b = 0$. a. Justify that the sequence $\left( P _ { n } \right)$ is a geometric sequence and specify its common ratio. b. Determine the limit of $P _ { n }$.
2. In this question $b = 0.2$. a. For all natural integer $n$, we set $v _ { n } = 0.1 \times P _ { n }$. Calculate $v _ { 0 }$ then show that, for all natural integer $n , v _ { n + 1 } = 2 v _ { n } \left( 1 - v _ { n } \right)$. b. In this model, justify that the population will stabilize around a value that you will specify.

Alain owns a swimming pool that contains $50\mathrm{~m}^3$ of water. Recall that $1\mathrm{~m}^3 = 1000\mathrm{~L}$. To disinfect the water, he must add chlorine. The chlorine level in the water, expressed in $\mathrm{mg}\cdot\mathrm{L}^{-1}$, is defined as the mass of chlorine per unit volume of water. Pool specialists recommend a chlorine level between 1 and $3\mathrm{~mg}\cdot\mathrm{L}^{-1}$. Under the action of the ambient environment, particularly ultraviolet rays, chlorine decomposes and gradually disappears. On Wednesday, June 19, he measures a chlorine level of $0.70\mathrm{~mg}\cdot\mathrm{L}^{-1}$.
Part A: study of a discrete model.
To maintain the chlorine level in his pool, Alain decides, starting from Thursday, June 20, to add 15 g of chlorine each day. It is admitted that this chlorine mixes uniformly in the pool water.

1. Justify that this addition of chlorine increases the level by $0.3\mathrm{~mg}\cdot\mathrm{L}^{-1}$.
2. For any natural number $n$, we denote by $v_n$ the chlorine level, in $\mathrm{mg}\cdot\mathrm{L}^{-1}$, obtained with this new protocol $n$ days after Wednesday, June 19. Thus $v_0 = 0.7$. It is admitted that for any natural number $n$, $$v_{n+1} = 0.92 v_n + 0.3.$$ a. Show by induction that for any natural number $n$, $\quad v_n \leqslant v_{n+1} \leqslant 4$. b. Show that the sequence $(v_n)$ is convergent and calculate its limit.
3. In the long term, will the chlorine level comply with the pool specialists' recommendation? Justify your answer.
4. Reproduce and complete the algorithm below written in Python language so that the function \texttt{alerte\_chlore} returns, when it exists, the smallest integer $n$ such that $v_n > s$. \begin{verbatim} def alerte_chlore(s) : n = 0 u=0.7 while...: n = ... u=... return n \end{verbatim}
5. What value is obtained by entering the instruction \texttt{alerte\_chlore(3)}? Interpret this result in the context of the exercise.

Part B: study of a continuous model.
Alain decides to call on a specialized engineering firm. This firm uses a continuous model to describe the chlorine level in the pool. In this model, for a duration $x$ (in days elapsed from Wednesday, June 19), $f(x)$ represents the chlorine level, in $\mathrm{mg}\cdot\mathrm{L}^{-1}$, in the pool. It is admitted that the function $f$ is a solution of the differential equation $(E): y' = -0.08y + \frac{q}{50}$, where $q$ is the quantity of chlorine, in grams, added to the pool each day.

1. Justify that the function $f$ is of the form $f(x) = C\mathrm{e}^{-0.08x} + \frac{q}{4}$ where $C$ is a real constant.
2. a. Express as a function of $q$ the limit of $f$ at $+\infty$. b. Recall that the chlorine level observed on Wednesday, June 19 is equal to $0.7\mathrm{~mg}\cdot\mathrm{L}^{-1}$. It is desired that the chlorine level stabilizes in the long term around $2\mathrm{~mg}\cdot\mathrm{L}^{-1}$. Determine the values of $C$ and $q$ so that these two conditions are satisfied.

A patient must take a dose of 2 ml of a medication every hour. We introduce the sequence $\left( u _ { n } \right)$ such that the term $u _ { n }$ represents the quantity of medication, expressed in ml, present in the body immediately after $n$ doses of medication. We have $u _ { 1 } = 2$ and for every strictly positive natural integer $n$: $u _ { n + 1 } = 2 + 0.8 u _ { n }$.
Part A Using this model, a doctor seeks to determine after how many doses of medication the quantity present in the patient's body is strictly greater than 9 mL.

1. Calculate the value $u _ { 2 }$.
2. Show by induction that: $$u _ { n } = 10 - 8 \times 0.8 ^ { n - 1 } \text { for every strictly positive natural integer } n.$$
3. Determine $\lim _ { n \rightarrow + \infty } u _ { n }$ and give an interpretation of this result in the context of the exercise.
4. Let $N$ be a strictly positive natural integer. Does the inequality $u _ { N } \geqslant 10$ have solutions? Interpret the result of this question in the context of the exercise.
5. Determine after how many doses of medication the quantity of medication present in the patient's body is strictly greater than 9 mL. Justify your approach.

Part B Using the same modeling, the doctor is interested in the average quantity of medication present in the patient's body over time. For this purpose, the sequence ( $S _ { n }$ ) is defined for every strictly positive natural integer $n$ by $$S _ { n } = \frac { u _ { 1 } + u _ { 2 } + \cdots + u _ { n } } { n } .$$ We admit that the sequence ( $S _ { n }$ ) is increasing.

1. Calculate $S _ { 2 }$.
2. Show that for every strictly positive natural integer $n$, $$u _ { 1 } + u _ { 2 } + \cdots + u _ { n } = 10 n - 40 + 40 \times 0.8 ^ { n } .$$
3. Calculate $\lim _ { n \rightarrow + \infty } S _ { n }$.
4. The following mystery function is given, written in Python language: \begin{verbatim} def mystere(k) : n = 1 s =2 while sJustify that this value is strictly greater than 10.

Throughout the exercise, probabilities will be rounded to $10^{-3}$ if necessary.
A binary datum is data that can only take two values: 0 or 1. Data of this type is transmitted successively from one machine to another. Each machine transmits the received data either faithfully, that is, by transmitting the information as it received it (1 becomes 1 and 0 becomes 0), or in the opposite way (1 becomes 0 and 0 becomes 1). The transmission is faithful in $90\%$ of cases, and therefore opposite in $10\%$ of cases. Throughout the exercise, the first machine always receives the value 1.
For any natural integer $n \geqslant 1$, we denote:

• $V_n$ the event: ``the $n$-th machine holds the value 1'';
• $\overline{V_n}$ the event: ``the $n$-th machine holds the value 0''.

Part A

1. a. Copy and complete the probability tree. b. Prove that $P(V_3) = 0{,}82$ and interpret this result in the context of the exercise. c. Given that the third machine received the value 1, calculate the probability that the second machine also received the value 1.
2. For any natural integer $n \geqslant 1$, we denote $p_n = P(V_n)$. The first machine received the value 1, so $p_1 = 1$. a. Prove that for any natural integer $n \geqslant 1$: $$p_{n+1} = 0{,}8\, p_n + 0{,}1.$$ b. Prove by induction that for any natural integer $n \geqslant 1$, $$p_n = 0{,}5 \times 0{,}8^{n-1} + 0{,}5.$$ c. Calculate the limit of $p_n$ as $n$ tends to infinity. Interpret this result in the context of the exercise.

Part B
To model in Python language the transmission of the binary datum described at the beginning of the exercise, we consider the simulation function which takes as a parameter a natural integer $n$ which represents the number of transmissions carried out from one machine to another, and which returns the list of successive values of the binary datum. The incomplete script of this function is given below. We recall that the instruction rand() returns a random number from the interval $[0; 1[$.
\begin{verbatim} def simulation(n): donnee = 1 liste = [donnee] for k in range(n): if rand() <0.1 donnee = 1 - donnee liste.append(donnee) return liste \end{verbatim}
For example, simulation(3) can return $[1, 0, 0, 1]$.

1. Determine the role of the instructions on lines 5 and 6 of the algorithm above.
2. Calculate the probability that simulation(4) returns the list $[1,1,1,1,1]$ and the probability that simulation(6) returns the list $[1,0,1,0,0,1,1]$.

With the objective of working on concentration and movement synchronization of students in one of his classes, a physical education teacher divided this class into three groups (A, B, and C) and stipulated the following activity: students in group A should clap every 2 s, students in group B should clap every 3 s, and students in group C should clap every 4 s.
The teacher reset the stopwatch and the three groups started clapping when it registered 1 s. The movements continued until the stopwatch registered 60 s.
An intern noted on paper the sequence formed by the instants when the three groups clapped simultaneously.
What is the general term of the sequence noted?
(A) $12n$, with $n$ a natural number, such that $1 \leq n \leq 5$.
(B) $24n$, with $n$ a natural number, such that $1 \leq n \leq 2$.
(C) $12(n-1)$, with $n$ a natural number, such that $1 \leq n \leq 6$.
(D) $12(n-1)+1$, with $n$ a natural number, such that $1 \leq n \leq 5$.
(E) $24(n-1)+1$, with $n$ a natural number, such that $1 \leq n \leq 3$.

20. (This question is worth 13 points) Given the sequence $\left\{ a _ { n } \right\}$ satisfying: $a _ { 1 } \in \mathbf { N } ^ { * } , a _ { 1 } \leqslant 36$, and $a _ { n + 1 } = \left\{ \begin{array} { l } 2 a _ { n } , a _ { n } \leqslant 18 , \\ 2 a _ { n } - 36 , a _ { n } > 18 \end{array} ( n = 1,2 , \ldots ) \right.$. Let the set $M = \left\{ a _ { n } \mid n \in \mathbf { N } ^ { * } \right\}$. (I) If $a _ { 1

We set, for all $t \in I$, $$f ( t ) = \sum _ { n = 0 } ^ { + \infty } C _ { n } t ^ { n } \quad \text { and } \quad g ( t ) = 2 t f ( t ) .$$ Deduce that $\mathbb { P } ( T \neq 0 ) = 1 - \sqrt { 1 - 4 p ( 1 - p ) }$. Interpret this result when $p = \frac { 1 } { 2 }$.

Selma designs a toy with identical clips in yellow and blue colors. In step 1, she places one clip on the ground. In each subsequent step, she attaches one clip to each blue part of all the clips she placed in the previous step as shown in the figure, and moves to the next step. Selma completes the first 3 steps of this toy using 7 clips.
Accordingly, after Selma completes step 12, how many more clips has she used in total compared to after completing step 10?
A) $3 \cdot 2^{10}$
B) $3 \cdot 2^{11}$
C) $7 \cdot 2^{9}$
D) $7 \cdot 2^{10}$
E) $2^{11}$