For the rest of this problem, we assume that $\varphi$ is a non-degenerate symmetric bilinear form on $E$, and we denote by $q$ its quadratic form.
Let $F$ be a singular vector subspace of $E$. We assume that $(e_1, \ldots, e_s)$ is a basis of $F \cap F^\perp$. We denote by $G$ a supplementary subspace of $F \cap F^\perp$ in $F$.
a) Show that $G$ is non-singular.
b) Demonstrate by induction on the dimension of $F \cap F^\perp$ (starting with $\operatorname{dim}(F \cap F^\perp) = 1$, then $\operatorname{dim}(F \cap F^\perp) > 1$) that there exist $s$ planes $P_1, \ldots, P_s$ of $E$ such that the following three properties are verified:

1. For all $i \in \{1,\ldots,s\}$, $(P_i, q_{/P_i})$ is an artinian plane containing $e_i$.
2. For all $(i,j) \in \{1,\ldots,s\}^2$ with $i \neq j$, $P_i$ is orthogonal to $P_j$.
3. For all $i \in \{1,\ldots,s\}$, $P_i$ is orthogonal to $G$.

Let $n$ be a non-zero natural number. Show by induction $$\forall k \in \llbracket 0, 2^n - 1 \rrbracket, \quad k \in \operatorname{Im} \Phi_n$$ where $\Phi_n : \{0,1\}^n \rightarrow \llbracket 0, 2^n - 1 \rrbracket$, $(x_j)_{j \in \llbracket 1,n \rrbracket} \mapsto \sum_{j=1}^{n} x_j 2^{n-j}$.

Using the recurrence relation $2\beta_{n+1} = \sum_{k=0}^{n} \binom{n}{k} \beta_k \beta_{n-k}$ and the analogous relation $2\alpha_{n+1} = \sum_{k=0}^{n} \binom{n}{k} \alpha_k \alpha_{n-k}$ with $\alpha_0 = \beta_0 = 1$, deduce that $\beta_n = \alpha_n$ for every $n \in \mathbb{N}$.

Let $n$ be a non-zero natural number. We set $$\Phi_n : \left|\, \begin{aligned} \{0,1\}^n &\rightarrow \llbracket 0, 2^n - 1 \rrbracket \\ (x_j)_{j \in \llbracket 1,n \rrbracket} &\mapsto \sum_{j=1}^{n} x_j 2^{n-j} \end{aligned} \right.$$
Show by induction $$\forall k \in \llbracket 0, 2^n - 1 \rrbracket, \quad k \in \operatorname{Im} \Phi_n.$$

Show that, for all $n \in \mathbb{N}$ and $\theta \in \mathbb{R}$, $T_n(\cos\theta) = \cos(n\theta)$.
The sequence of polynomials $\left(T_n\right)_{n \in \mathbb{N}}$ is defined by $T_0 = 1, T_1 = X$ and $\forall n \in \mathbb{N}, T_{n+2} = 2X T_{n+1} - T_n$.

Show that, for all $n \in \mathbb{N}$ and $\theta \in \mathbb{R}$, $T_n(\cos\theta) = \cos(n\theta)$.
The sequence of polynomials $\left(T_n\right)_{n \in \mathbb{N}}$ is defined by $T_0 = 1, T_1 = X$ and $\forall n \in \mathbb{N}, T_{n+2} = 2X T_{n+1} - T_n$.

Let $\mu_1$ and $\mu_2$ be two probabilities on $\mathbb{N}^*$. We assume that $\forall r \in \mathbb{N}^*, \mu_1(\mathbb{N}^* r) = \mu_2(\mathbb{N}^* r)$.
We recall that we denote by $(p_i)_{i \in \mathbb{N}^*}$ the sequence of prime numbers, ordered in increasing order.
Show that for all $r \in \mathbb{N}^*$ and all integer $n \geqslant 1$: $$\mu_1\left(\mathbb{N}^* r \backslash \bigcup_{i=1}^{n} \mathbb{N}^* r p_i\right) = \mu_2\left(\mathbb{N}^* r \backslash \bigcup_{i=1}^{n} \mathbb{N}^* r p_i\right).$$

Let $\mu_1$ and $\mu_2$ be two probabilities on $\mathbb{N}^*$. We suppose that $\forall r \in \mathbb{N}^*, \mu_1(\mathbb{N}^* r) = \mu_2(\mathbb{N}^* r)$, where $\mathbb{N}^* r$ denotes the set of strictly positive multiples of $r$. Using the result of 17a, show that for all $r \in \mathbb{N}^*$ and all integer $n \geqslant 1$: $$\mu_1\left(\mathbb{N}^* r \backslash \bigcup_{i=1}^{n} \mathbb{N}^* r p_i\right) = \mu_2\left(\mathbb{N}^* r \backslash \bigcup_{i=1}^{n} \mathbb{N}^* r p_i\right)$$

Let $c > 0$, and let $\left(a_n\right)_{n \in \mathbb{N}}$ be a sequence of positive real numbers such that $a_{n+1} \leq a_n - c(a_n)^2$ for all $n \in \mathbb{N}$. Show $a_n \leq a_0/(1 + nca_0)$ for all $n \in \mathbb{N}$. Hint: adapt the reasoning from question 10.c)