We consider the family of polynomials $$\left\{ \begin{array}{l} H_0 = 1 \\ H_k = \frac{1}{k!} \prod_{j=0}^{k-1} (X - j) \quad \text{for } k \in \llbracket 1, n \rrbracket \end{array} \right.$$
Determine the real sequences $\left(u_k\right)_{k \in \mathbb{N}}$ such that $$u_{k+2} - 2u_{k+1} + u_k = k^3 + 2k^2 + 5k + 7 \quad (k \in \mathbb{N})$$

We assume in this question that $U$ is path-connected. Determine the set of functions $f$ in $\mathcal{H}(U)$ such that $f^2$ also belongs to $\mathcal{H}(U)$.

We assume in this question that $U$ is path-connected. Determine the set of functions $f$ in $\mathcal{H}(U)$ such that $f^2$ also belongs to $\mathcal{H}(U)$.

Let $n \in \mathbb{N}^*$. For $k \in \mathbb{N}$, we denote $\mathcal{B}(n, k)$ the set of maps $\sigma \in \operatorname{MD}(n+1)$ such that $\sigma(2) - \sigma(1) = k+1$. For $k \in \mathbb{N}$ and $s \in \mathbb{N}$, we denote $\mathcal{C}(n, s, k)$ the set of elements $\sigma$ of $\operatorname{MD}(n+2)$ such that $\sigma(2) - \sigma(1) = s+1, \quad n+2-\sigma(2) = k$.
2. Under what condition (necessary and sufficient) on $n$ and $k$ is the set $\mathcal{B}(n, k)$ non-empty? Under what condition (necessary and sufficient) on $n$, $s$ and $k$ is the set $\mathcal{C}(n, s, k)$ non-empty?

Let $A = \operatorname{co}(E)$ where $E$ is the subset of $\mathbb{R}^3$ defined by $$E = \{(0,0,1),(0,0,-1)\} \cup \{(1+\cos(\theta), \sin(\theta), 0), \theta \in [0, 2\pi]\}$$ show that $\operatorname{Ext}(A)$ is non-empty and is not closed.

In this subsection, we assume that $J_n = J_n^{(\mathrm{C})}$, the matrix introduced in subsection A-II.
Show then that $m^+ = 0$ if $\beta \leqslant 1$, and $m^+ > 0$ if $\beta > 1$.

Let $n \geq 1$ be an integer. We say that a matrix $M \in \mathcal{M}_n(\mathbb{R})$ is doubly stochastic if for all $i, j \in \{1, \ldots, n\}$ we have $$M_{ij} \geq 0 \quad \text{and} \quad \sum_{k=1}^n M_{ik} = \sum_{k=1}^n M_{kj} = 1.$$ We denote by $B_n$ the set of doubly stochastic matrices in $\mathcal{M}_n(\mathbb{R})$.
Show that $B_n$ is a polytope and determine its dimension.

Let $a , b , c$ and $d$ be four non-negative real numbers where $a + b + c + d = 1$. The number of different ways one can choose these numbers such that $a ^ { 2 } + b ^ { 2 } + c ^ { 2 } + d ^ { 2 } = \max \{ a , b , c , d \}$ is
(A) 1 .
(B) 5 .
(C) 11 .
(D) 15 .

Let $f : \mathbb { Z } \rightarrow \mathbb { Z }$ be a function satisfying $f ( 0 ) \neq 0 = f ( 1 )$. Assume also that $f$ satisfies equations $( \mathbf { A } )$ and $( \mathbf { B } )$ below.
$$\begin{aligned} f ( x y ) & = f ( x ) + f ( y ) - f ( x ) f ( y ) \\ f ( x - y ) f ( x ) f ( y ) & = f ( 0 ) f ( x ) f ( y ) \end{aligned}$$
for all integers $x , y$.
(i) Determine explicitly the set $\{ f ( a ) : a \in \mathbb { Z } \}$.
(ii) Assuming that there is a non-zero integer $a$ such that $f ( a ) \neq 0$, prove that the set $\{ b : f ( b ) \neq 0 \}$ is infinite.

Suppose $n \geq 2$. Consider the polynomial
$$Q _ { n } ( x ) = 1 - x ^ { n } - ( 1 - x ) ^ { n } .$$
Show that the equation $Q _ { n } ( x ) = 0$ has only two real roots, namely 0 and 1.

Consider four natural numbers $a , b , c$ and $d$ satisfying $1 < a < b < c < d$. Suppose that two sets using these numbers, $A = \{ a , b , c , d \}$ and $B = \left\{ a ^ { 2 } , b ^ { 2 } , c ^ { 2 } , d ^ { 2 } \right\}$, satisfy the following two conditions:
(i) Just two elements belong to the intersection $A \cap B$, and the sum of these two elements is greater than or equal to 15, and less than or equal to 25.
(ii) The sum of all the elements belonging to the union $A \cup B$, is less than or equal to 300.
We are to find the values of $a , b , c$ and $d$.
First, set $A \cap B = \{ x , y \}$, where $x < y$. Since $x \in B$ and $y \in B$, it follows from (i) that $y = \mathbf { A B }$ and that $x$ is either $\mathbf{C}$ or $\mathbf { D }$. (Write the answers in the order $\mathbf { C } < \mathbf { D }$.) Here, when we consider (ii), we see that $x = \mathbf { E }$. Hence $A$ includes the elements $\mathbf { F }$, $\mathbf{F}$ and $\mathbf{F}$.
Furthermore, when we denote the remaining element of $A$ by $z$, from (ii) we see that $z$ satisfies
$$z ^ { 2 } + z \leqq \mathbf { G H } .$$
Hence we have $z = \mathbf { I }$. From the above we obtain
$$a = \mathbf { J } , \quad b = \mathbf { K } , \quad c = \mathbf { L } \text { and } d = \mathbf { M N } .$$