For each of the following statements, indicate whether it is true or false. Each answer must be justified. An unjustified answer earns no points. The five questions of this exercise are independent.

1. We consider the script written in Python language below. \begin{verbatim} def seuil(S) : n=0 u=7 while u < S : n=n+1 u=1.05*u+3 return(n) \end{verbatim} Statement 1: the instruction seuil(100) returns the value 18.
   
2. Let $(S_n)$ be the sequence defined for every natural integer $n$ by $$S_n = 1 + \frac{1}{5} + \frac{1}{5^2} + \ldots + \frac{1}{5^n}.$$ Statement 2: the sequence $(S_n)$ converges to $\frac{5}{4}$.
   
3. Statement 3: in a class composed of 30 students, we can form 870 different pairs of delegates.
   
4. We consider the function $f$ defined on $[1 ; +\infty[$ by $f(x) = x(\ln x)^2$. Statement 4: the equation $f(x) = 1$ admits a unique solution in the interval $[1 ; +\infty[$.
   
5. Statement 5: $$\int_0^1 x\mathrm{e}^{-x}\,\mathrm{d}x = \frac{\mathrm{e} - 2}{\mathrm{e}}.$$

The distance between the points $A = (1, 2)$ and $B = (4, 6)$ is:
(A) 3
(B) 4
(C) 5
(D) 6
(E) 7

The value of $\displaystyle\sum_{k=1}^{5} k^2$ is:
(A) 45
(B) 50
(C) 55
(D) 60
(E) 65

Notice that the quadratic polynomial $p(x) = 1 + x + \frac{1}{2}x(x-1)$ satisfies $p(j) = 2^{j}$ for $j = 0, 1$ and $2$. A polynomial $q(x)$ of degree 7 satisfies $q(j) = 2^{j}$ for $j = 0, 1, 2, 3, 4, 5, 6, 7$. Find the value of $q(10)$.

For a natural number $k$, when $n = 5 ^ { k }$, $f ( n )$ satisfies $$f ( 5 n ) = f ( n ) + 3 , \quad f ( 5 ) = 4$$ Find the value of $\sum _ { k = 1 } ^ { 10 } f \left( 5 ^ { k } \right)$. [4 points]

For a natural number $n \geq 2$, let $C _ { n }$ be the circle obtained by translating the circle $C$ with center at the origin and radius 1 by $\frac { 2 } { n }$ in the $x$-direction. Let $l _ { n }$ be the length of the common chord of circles $C$ and $C _ { n }$. When $\sum _ { n = 2 } ^ { \infty } \frac { 1 } { \left( n l _ { n } \right) ^ { 2 } } = \frac { q } { p }$, find the value of $p + q$. (Here, $p , q$ are coprime natural numbers.) [4 points]

For a natural number $n \geq 2$, consider the set $$\left\{ 3 ^ { 2 k - 1 } \mid k \text{ is a natural number, } 1 \leqq k \leqq n \right\}$$ Let $S$ be the set containing only all possible values obtained by multiplying two distinct elements of this set, and let $f ( n )$ be the number of elements in $S$. For example, $f ( 4 ) = 5$. Find the value of $\sum _ { n = 2 } ^ { 11 } f ( n )$. [4 points]

For a natural number $n \geq 2$, consider the set $$\left\{ 3 ^ { 2 k - 1 } \mid k \text { is a natural number, } 1 \leqq k \leqq n \right\}$$ Let $S$ be the set containing only all possible values obtained by multiplying two distinct elements of this set, and let $f ( n )$ be the number of elements in $S$. For example, $f ( 4 ) = 5$. Find the value of $\sum _ { n = 2 } ^ { 11 } f ( n )$.

For the sequence $\left\{ a _ { n } \right\}$, $$\sum _ { k = 1 } ^ { 10 } \left( a _ { k } + 1 \right) ^ { 2 } = 28 , \sum _ { k = 1 } ^ { 10 } a _ { k } \left( a _ { k } + 1 \right) = 16$$ Find the value of $\sum _ { k = 1 } ^ { 10 } \left( a _ { k } \right) ^ { 2 }$. [4 points]

For two sequences $\left\{ a _ { n } \right\}$ and $\left\{ b _ { n } \right\}$, $$\sum _ { k = 1 } ^ { 5 } a _ { k } = 8 , \quad \sum _ { k = 1 } ^ { 5 } b _ { k } = 9$$ What is the value of $\sum _ { k = 1 } ^ { 5 } \left( 2 a _ { k } - b _ { k } + 4 \right)$? [3 points]
(1) 19
(2) 21
(3) 23
(4) 25
(5) 27

A sequence $\left\{ a _ { n } \right\}$ satisfies $a _ { 1 } = 1$ and for all natural numbers $n$, $$\sum _ { k = 1 } ^ { n } \left( a _ { k } - a _ { k + 1 } \right) = - n ^ { 2 } + n$$ What is the value of $a _ { 11 }$? [3 points]
(1) 88
(2) 91
(3) 94
(4) 97
(5) 100

For a sequence $\left\{ a _ { n } \right\}$, $$\sum _ { k = 1 } ^ { 10 } a _ { k } - \sum _ { k = 1 } ^ { 7 } \frac { a _ { k } } { 2 } = 56 , \quad \sum _ { k = 1 } ^ { 10 } 2 a _ { k } - \sum _ { k = 1 } ^ { 8 } a _ { k } = 100$$ find the value of $a _ { 8 }$. [3 points]

Let $f ( x )$ be an odd function with domain $( - \infty , + \infty )$, satisfying $f ( 1 - x ) = f ( 1 + x )$ and $f ( 1 ) = 2$. Then $f ( 1 ) + f ( 2 ) + f ( 3 ) + \cdots + f ( 50 ) =$
A. $- 50$
B. $0$
C. $2$
D. $50$

Given that $f ( x )$ is an odd function with domain $( - \infty , + \infty )$ satisfying $f ( 1 - x ) = f ( 1 + x )$. If $f ( 1 ) = 2$, then $f ( 1 ) + f ( 2 ) + f ( 3 ) + \cdots + f ( 30 ) =$
A. $- 50$
B. $0$
C. $2$
D. $50$

Given that $f(x), g(x)$ have domain $\mathbf{R}$, and $f(x) + g(2-x) = 5, g(x) - f(x-4) = 7$. If the graph of $y = g(x)$ is symmetric about the line $x = 2$, and $g(2) = 4$, then $\sum_{k=1}^{22} f(k) =$
A. $-21$
B. $-22$
C. $-23$
D. $-24$

We fix $x > 0$ and for all natural integers $n$, we define $F_{n}(x)$ by: $$F_{n}(x) = \ln\left(\frac{n! \, n^{x+1}}{(x+1)(x+2) \ldots (x+n+1)}\right)$$ Deduce that: $$F_{n}(x) = G_{n}(x) - \int_{0}^{n+1} \frac{h(u)}{u+x} du$$ where $$G_{n}(x) = \ln n! + (x+1)\ln n - \left(x+n+\frac{3}{2}\right)\ln(x+n+1) + n+1 + \left(x+\frac{1}{2}\right)\ln x$$

Let $\varepsilon_{1}, \varepsilon_{2}, \varepsilon_{3}, \varepsilon_{4}$ be four strictly positive real numbers pairwise distinct and two strictly positive real numbers $E$ and $N$. Let $\Omega$ be the part, assumed to be non-empty, formed of the quadruplets $x = (x_{1}, x_{2}, x_{3}, x_{4})$ of $\mathbb{R}_{+}^{4}$ satisfying: $$\left\{\begin{array}{l} x_{1} + x_{2} + x_{3} + x_{4} = N \\ \varepsilon_{1} x_{1} + \varepsilon_{2} x_{2} + \varepsilon_{3} x_{3} + \varepsilon_{4} x_{4} = E \end{array}\right.$$
VI.A.1) Let $f$ be a function of class $\mathcal{C}^{1}$ on $\mathbb{R}_{+}^{4}$. Show that $f$ admits a maximum on $\Omega$. We then denote $a = (a_{1}, a_{2}, a_{3}, a_{4}) \in \Omega$ a point at which this maximum is attained.
VI.A.2) Show that if $(x_{1}, x_{2}, x_{3}, x_{4}) \in \Omega$ then $x_{3}$ and $x_{4}$ can be written in the form $$\begin{aligned} & x_{3} = u x_{1} + v x_{2} + w \\ & x_{4} = u^{\prime} x_{1} + v^{\prime} x_{2} + w^{\prime} \end{aligned}$$ where we shall give explicitly $u, v, u^{\prime}, v^{\prime}$ in terms of $\varepsilon_{1}, \varepsilon_{2}, \varepsilon_{3}, \varepsilon_{4}$.
VI.A.3) Assuming that none of the numbers $a_{1}, a_{2}, a_{3}, a_{4}$ is zero, deduce that $$\begin{aligned} & \frac{\partial f}{\partial x_{1}}(a) + u \frac{\partial f}{\partial x_{3}}(a) + u^{\prime} \frac{\partial f}{\partial x_{4}}(a) = 0 \\ & \frac{\partial f}{\partial x_{2}}(a) + v \frac{\partial f}{\partial x_{3}}(a) + v^{\prime} \frac{\partial f}{\partial x_{4}}(a) = 0 \end{aligned}$$
VI.A.4) Show that the vector subspace of $\mathbb{R}^{4}$ spanned by the vectors $(1, 0, u, u^{\prime})$ and $(0, 1, v, v^{\prime})$ admits a supplementary orthogonal subspace spanned by the vectors $(1,1,1,1)$ and $(\varepsilon_{1}, \varepsilon_{2}, \varepsilon_{3}, \varepsilon_{4})$.
VI.A.5) Deduce the existence of two real numbers $\alpha, \beta$ such that for all $i \in \{1,2,3,4\}$ we have $$\frac{\partial f}{\partial x_{i}}(a) = \alpha + \beta \varepsilon_{i}$$

The family $\left(K_p\right)_{p \in \mathbb{N}}$ is the unique family of polynomials such that for all $p \in \mathbb{N}$, the degree of $K_p$ equals $p$ with strictly positive leading coefficient, and for all $N \in \mathbb{N}$, $\left(K_p\right)_{0 \leqslant p \leqslant N}$ is an orthonormal basis of $\mathbb{R}_N[X]$ for $\langle \cdot, \cdot \rangle$.
For all $p \in \llbracket 0; n-1 \rrbracket$, calculate $K_p(1)$.

For $n \in \mathbb{N}^*$ and $(i,j) \in \llbracket 1, n \rrbracket^2$, we denote by $h_{i,j}^{(-1,n)}$ the coefficient at position $(i,j)$ of the matrix $H_n^{-1}$ and we denote by $s_n$ the sum of the coefficients of the matrix $H_n^{-1}$, that is: $$s_n = \sum_{1 \leqslant i,j \leqslant n} h_{i,j}^{(-1,n)}$$
Determine the value of $s_n$.

We define, for every non-zero natural number $n$, the function $k_{n}$ by $$\begin{cases} k_{n}(x) = 1 - \frac{|x|}{n} & \text{if } |x| \leqslant n \\ k_{n}(x) = 0 & \text{otherwise} \end{cases}$$ Express the Fourier transform $\hat{k}_{n}(x)$ using the function defined by $$\varphi(x) = \begin{cases} \left(\frac{\sin x}{x}\right)^{2} & \text{if } x \neq 0 \\ 1 & \text{if } x = 0 \end{cases}$$

In this question, we study the case $\lambda(t) = t^2$ and $f(t) = \dfrac{1}{1+t^2}$ for all $t \in \mathbb{R}^+$.
II.C.1) Determine $E$. What is the value of $Lf(0)$?
II.C.2) Prove that $Lf$ is differentiable.
II.C.3) Show the existence of a constant $A > 0$ such that for all $x > 0$, we have $$Lf(x) - (Lf)^{\prime}(x) = \frac{A}{\sqrt{x}}.$$
II.C.4) We denote $g(x) = e^{-x}Lf(x)$ for $x \geqslant 0$.
Show that for all $x \geqslant 0$, we have $$g(x) = \frac{\pi}{2} - A\int_0^x \frac{e^{-t}}{\sqrt{t}}\,dt.$$
II.C.5) Deduce from this the value of the integral $\displaystyle\int_0^{+\infty} e^{-t^2}\,dt$.

In this part, $\lambda(t) = t$ for all $t \in \mathbb{R}^+$.
In the particular case where $f(t) = e^{-at}t^n$ for all $t \in \mathbb{R}^+$, with $n \in \mathbb{N}$ and $a \in \mathbb{R}$, make explicit $E$, $E^{\prime}$ and calculate $Lf(x)$ for $x \in E^{\prime}$.

In this part, $\lambda(t) = t$ for all $t \in \mathbb{R}^+$ and $f(t) = \dfrac{\sin t}{t}$ for all $t \in \mathbb{R}^{+*}$, $f$ being extended by continuity at 0.
Calculate $(Lf)^{\prime}(x)$ for $x \in E$.

We consider $\psi : x \mapsto \frac { 1 } { ( 1 + x ) ^ { 2 } ( 1 - x ) }$ with power series expansion $\psi ( x ) = \sum _ { n = 0 } ^ { + \infty } v _ { n } x ^ { n }$ for $x \in ] - 1,1 [$. We denote $A_n = \sum_{k=0}^n a_k$ and $\widetilde{a}_n = \frac{A_n}{n+1}$.
Calculate $\widetilde { v } _ { n }$ (arithmetic mean of the numbers $v _ { 0 } , \ldots , v _ { n }$).

Calculate $\sum _ { n = - \infty } ^ { + \infty } \left| \varphi _ { n } ( x ) \right| ^ { 2 }$.