For any polynomial $f ( x )$ with real coefficients and of degree 2011 , there is a real number $b$ such that $f ( b ) = f ^ { \prime } ( b )$.

A differentiable function $f : \mathbb { R } \rightarrow \mathbb { R }$ satisfies $f ( 1 ) = 2 , f ( 2 ) = 3$ and $f ( 3 ) = 1$. Show that $f ^ { \prime } ( x ) = 0$ for some $x$.

For $n > 1$, a configuration consists of $2n$ distinct points in a plane, $n$ of them red, the remaining $n$ blue, with no three points collinear. A pairing consists of $n$ line segments, each with one blue and one red endpoint, such that each of the given $2n$ points is an endpoint of exactly one segment. Prove the following. a) For any configuration, there is a pairing in which no two of the $n$ segments intersect. (Hint: consider total length of segments.) b) Given $n$ red points (no three collinear), we can place $n$ blue points such that any pairing in the resulting configuration will have two segments that do not intersect. (Hint: First consider the case $n = 2$.)

Show that there is no differentiable function $f : \mathbb{R} \longrightarrow \mathbb{R}$ such that $f(0) = 1$ and $f^{\prime}(x) \geq (f(x))^{2}$ for every $x \in \mathbb{R}$.

[12 points] Throughout this problem we are interested in real valued functions $f$ satisfying two conditions: at each $x$ in its domain, $f$ is continuous and $f(x^{2}) = f(x)^{2}$. Prove the following independent statements about such functions. The hints below may be useful.
(i) There is a unique such function $f$ with domain $[0,1]$ and $f(0) \neq 0$.
(ii) If the domain of such $f$ is $(0, \infty)$, then ($f(x) = 0$ for every $x$) OR ($f(x) \neq 0$ for every $x$).
(iii) There are infinitely many such $f$ with domain $(0, \infty)$ such that $\int_{0}^{\infty} f(x)\, dx < 1$.
Hints: (1) Suppose a number $a$ and a sequence $x_{n}$ are in the domain of a continuous function $f$ and $x_{n}$ converges to $a$. Then $f(x_{n})$ must converge to $f(a)$. For example $f(0.5^{n}) \rightarrow f(0)$ and $f(2^{\frac{1}{n}}) \rightarrow f(1)$ if all the mentioned points are in the domain of $f$. In parts (i) and (ii) suitable sequences may be useful. (2) Notice that $f(x) = x^{r}$ satisfies $f(x^{2}) = f(x)^{2}$.

Show that there is no polynomial $p ( x )$ for which $\cos ( \theta ) = p ( \sin \theta )$ for all angles $\theta$ in some nonempty interval.
Hint: Note that $x$ and $| x |$ are different functions but their values are equal on an interval (as $x = | x |$ for all $x \geq 0$). You may want to show as a first step that this cannot happen for two polynomials, i.e., if polynomials $f$ and $g$ satisfy $f ( x ) = g ( x )$ for all $x$ in some interval, then $f$ and $g$ must be equal as polynomials, i.e., in each degree they must have the same coefficient.

[15 points] Two distinct real numbers $r$ and $s$ are said to form a good pair $(r, s)$ if
$$r^3 + s^2 = s^3 + r^2$$
(i) Find a good pair $(a, \ell)$ with the largest possible value of $\ell$. Find a good pair $(s, b)$ with the smallest possible value $s$. For every good pair $(c, d)$ other than the two you found, show that there is a third real number $e$ such that $(d, e)$ and $(c, e)$ are also good pairs.
(ii) Show that there are infinitely many good pairs of rational numbers.
Hints (use these or your own method): The function $f(x) = x^3 - x^2$ may be useful. If $(r, s)$ is a good pair, can you express $s$ in terms of $r$? You may use that there are infinitely many right triangles with integer sides such that no two of these triangles are similar to each other.

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 $x \in E$ such that $q(x) = 0$ and such that $x \neq 0$.
We propose to demonstrate that there exists a plane $\Pi \subset E$ containing $x$ such that $(\Pi, q_{/\Pi})$ is an artinian plane (where $q_{/\Pi}$ denotes the restriction of the application $q$ to the plane $\Pi$).
a) Demonstrate that there exists $z \in E$ such that $\varphi(x,z) = 1$.
b) We set $y = z - \frac{q(z)}{2}x$. Compute $q(y)$.
c) Conclude.

Let $\mathcal{A}$ be an open bounded non-empty subset of $\mathbb{R}^{2}$. We denote by $C(\mathcal{A})$ the set of continuous functions $f$ from $\mathbb{R}^{2}$ to $[0,1]$ such that $\forall (x,y) \in \mathbb{R}^{2} \setminus \mathcal{A},\, f(x,y) = 0$ (in other words $f$ is zero outside $\mathcal{A}$). Show that the supremum $$\sup_{f \in C(\mathcal{A})} \iint_{\mathbb{R}^{2}} f(x,y)\,dx\,dy$$ exists and defines a real number denoted $V(\mathcal{A})$.

Prove that, for any continuous application $f : C(0,1) \rightarrow \mathbb{R}$, the set $\mathcal{D}_f$ admits exactly one element.

Let $m$ be an integer greater than or equal to 2. We consider a polynomial $P \in \mathcal{P}_m$.
Deduce that there exists a polynomial $T \in \mathcal{P}_{m-2}$ such that $P + (1 - x^2 - y^2) T$ is a harmonic polynomial.

We denote by $\mathcal{S}_{n}^{\dagger}(\mathbb{R})$ the set of $n \times n$ symmetric matrices whose eigenvalues are all simple. Let $M \in \mathcal{S}_{n}^{\dagger}(\mathbb{R})$. Determine a real $r > 0$ such that the open ball of $\mathcal{S}_{n}(\mathbb{R})$ centered at $M$ with radius $r$ is included in $\mathcal{S}_{n}^{\dagger}(\mathbb{R})$. Deduce that $\mathcal{S}_{n}^{\dagger}(\mathbb{R})$ is an open set of $\mathcal{S}_{n}(\mathbb{R})$.

Let $V \geqslant 0$ be a real number.
8a. Give an example of an integer simplex in $\mathbb{R}^2$ with volume greater than or equal to $V$ and having no interior integer points.
8b. Give an example of an integer simplex in $\mathbb{R}^3$ with volume greater than or equal to $V$ whose only integer points are the vertices.

Let $\mathcal{S}$ be a simplex of $\mathbb{R}^n$ and $k$ an integer such that $\operatorname{Vol}(\mathcal{S}) > k$.
15a. Show that there exist $x \in [0,1[^n$ and $(k+1)$ elements of $\mathbb{Z}^n$ $u_0, \ldots, u_k$ such that $x \in \mathcal{S} - u_i$ for $i = 0, \ldots, k$. One may study the sets $(u + [0,1[^n) \cap \mathcal{S}$ when $u$ ranges over $\mathbb{Z}^n$; and admit — outside the CPGE curriculum — that the volume of a simplex is its Lebesgue measure, which is sub-additive.
15b. Deduce from this the existence of the $(k+1)$ points $v_0, \ldots, v_k$ that satisfy the conditions of Theorem 1.
15c. Prove Theorem 1, that is, here we assume only that $\operatorname{Vol}(\mathcal{S}) \geqslant k$.

Let $t_1, \ldots, t_n$ be strictly positive real numbers such that $\sum_{i=1}^n t_i = 1$ and let $N \geqslant n$ be an integer. We wish to show that there exist non-negative integers $p_1, \ldots, p_n$ and $q$ such that
i) $1 \leqslant q \leqslant N^{n-1}$,
ii) $\sum_{i=1}^n p_i = q$,
iii) $\left|qt_1 - p_1\right| \leqslant \frac{n}{N}$,
iv) for all $i = 2, \ldots, n$, $\left|qt_i - p_i\right| \leqslant \frac{1}{N}$.
16a. By considering the vectors with coordinates $\left(\{kt_2\}, \ldots, \{kt_n\}\right) \in [0,1[^{n-1}$ when $k$ ranges over $\{0, \ldots, N^{n-1}\}$, show that there exist integers $p_2, \ldots, p_n, q \geqslant 0$ satisfying conditions i) and iv).
16b. Conclude.

The purpose of this question is to show that for any strictly positive integers $n$ and $k$, there exists a constant $\alpha(k,n) \in ]0,1[$ such that, if $t_1, \ldots, t_n$ are strictly positive real numbers satisfying $1 > \sum_{i=1}^n t_i > 1 - \alpha(k,n)$, then there exist non-negative integers $p_1, \ldots, p_n \geqslant 0$ and $q$ such that $$\sum_{i=1}^n p_i = q > 0, \quad \text{and for all } i = 1, \ldots, n, \quad (kq+1)t_i > kp_i.$$ We proceed by induction on $n$.
17a. Handle the case $n = 1$ by showing that the constant $\alpha(k,1) = \frac{1}{k+1}$ works.
We assume the statement is true up to rank $n-1 \geqslant 1$. In particular, $\alpha(k,n-1) > 0$ is defined for all $k \geqslant 1$. We set for $k \geqslant 1$ $$\alpha(k,n) = \frac{1}{4kN^{n-1}} \quad \text{where} \quad N = 1 + \max\left(\frac{4k}{\alpha(k,n-1)}, 2kn(n+1)\right).$$ We are given $t_1 \geqslant t_2 \geqslant \cdots \geqslant t_n > 0$, and we assume that $\sum_{i=1}^n t_i = 1 - \alpha$ with $0 < \alpha < \alpha(k,n)$.
17b. If $t_n < \alpha(k,n-1) - \alpha$, establish the statement at rank $n$.
17c. If $t_n \geqslant \alpha(k,n-1) - \alpha$, apply the result of question 16 to the $\frac{t_i}{1-\alpha}$, $i = 1, \ldots, n$. With its notation, show that $$\alpha(k,n) < \min\left(\frac{1}{n+1}, \frac{1}{2}\alpha(k,n-1)\right) \quad \text{and} \quad 1 - qk\frac{\alpha}{1-\alpha} \geqslant \frac{1}{2}.$$ Conclude by distinguishing the cases $i \geqslant 2$ and $i = 1$.

Let $f \in \mathbb{R}^{N}$ and $J_{f} : \Sigma_{N} \rightarrow \mathbb{R}$ defined by $J_{f}(p) = H_{N}(p) + \sum_{i=1}^{N} p_{i} f_{i}$. We denote $$J_{f,*} = \sup\{J_{f}(p) \mid p \in \Sigma_{N}\}$$ the supremum of $J_{f}$ on $\Sigma_{N}$ and $\Sigma_{N}(f) = \{p \in \Sigma_{N} \mid J_{f}(p) = J_{f,*}\}$ the set of $p$ in $\Sigma_{N}$ for which the supremum is attained.
Show that $\Sigma_{N}(f)$ is non-empty.

Let $f \in \mathbb{R}^{N}$ and $J_{f} : \Sigma_{N} \rightarrow \mathbb{R}$ defined by $J_{f}(p) = H_{N}(p) + \sum_{i=1}^{N} p_{i} f_{i}$. We denote $J_{f,*} = \sup\{J_{f}(p) \mid p \in \Sigma_{N}\}$ and $\Sigma_{N}(f) = \{p \in \Sigma_{N} \mid J_{f}(p) = J_{f,*}\}$.
Let $p \in \Sigma_{N}$.
(a) Suppose that $p_{1} = 0$ and $p_{2} > 0$. Show that there exists $p'$ in $\Sigma_{N}$ such that $J_{f}(p') > J_{f}(p)$ (you may look for $p'$ close to $p$).
(b) Deduce that if $p \in \Sigma_{N}(f)$, then $p_{i} > 0$ for all $i \in \{1, \ldots, N\}$.

We now assume that $S _ { + } = S _ { - }$ and we assume that $\left( x _ { + } \mid q \right) = 0$. We denote by $z = \binom { x _ { + } } { 0 }$, $R ^ { + } = \left( \begin{array} { c c } S _ { + } & 0 \\ 0 & + 1 \end{array} \right) , R ^ { - } = \left( \begin{array} { c c } S _ { + } & 0 \\ 0 & - 1 \end{array} \right)$.
(a) Show that $O z = R^{+} z = R^{-} z$.
(b) We now write $$O = \left( \begin{array} { c c } \alpha ^ { \prime } & { } ^ { t } q ^ { \prime } \\ r ^ { \prime } & P ^ { \prime } \end{array} \right)$$ where $P ^ { \prime } \in M _ { n - 1 } ( \mathbb { R } )$. Construct then $z ^ { \prime } = \binom { \eta ^ { \prime } } { x ^ { \prime } } \in \mathbb { R } ^ { n }$ with $x ^ { \prime } \in \mathbb { R } ^ { n - 1 }$ strictly positive and $\eta ^ { \prime } \geq 0$ such that there exists a sign diagonal matrix $R ^ { \prime }$ satisfying $O z ^ { \prime } = R ^ { \prime } z ^ { \prime }$.
(c) In the case where $\eta ^ { \prime } = 0$, and using question 1(c), show that there exists a sign diagonal matrix $S$ such that $O \left( z + z ^ { \prime } \right) = S \left( z + z ^ { \prime } \right)$ and conclude.

Let $\Lambda$ be a non-empty subset of $\mathbb{R}_*^+$ closed under addition. We define $$\Gamma = \left\{z \in \mathbb{R}_+^* \mid \exists (x, y) \in \Lambda,\, z = y - x\right\}, \quad \text{and} \quad r(\Lambda) = \inf \Gamma.$$ Give two examples of such sets $\Lambda$, one for which $r(\Lambda) > 0$ and another for which $r(\Lambda) = 0$.

Let $\Lambda$ be a non-empty subset of $\mathbb{R}_*^+$ closed under addition, with $$\Gamma = \left\{z \in \mathbb{R}_+^* \mid \exists (x, y) \in \Lambda,\, z = y - x\right\}, \quad r(\Lambda) = \inf \Gamma.$$ We assume that $r(\Lambda) > 0$. Show that there exist $(a, b) \in \Lambda^2$ such that $b - a \in [r(\Lambda), 2r(\Lambda)[$.

Let $\Lambda$ be a non-empty subset of $\mathbb{R}_*^+$ closed under addition, with $r(\Lambda) > 0$. Let $(a, b) \in \Lambda^2$ such that $b - a \in [r(\Lambda), 2r(\Lambda)[$ and denote $d = b - a$. Show that there exists $n_0 \in \mathbb{N}$ such that $n_0 a + n_0 d > (n_0 + 1)a$, then that there exists $k \in \mathbb{N}$ such that $a = kd$.

Let $\Lambda$ be a non-empty subset of $\mathbb{R}_*^+$ closed under addition, with $r(\Lambda) = 0$. Let $\eta > 0$. Show that there exists $A \geqslant 0$ such that for all $x > A$, $$\Lambda \cap [x, x + \eta] \neq \varnothing$$

We assume that for all $d \geqslant 0$, $\mathbb{P}(X \in d\mathbb{Z}) < 1$, and that $g$ is of class $\mathscr{C}^1$, with support in $[0, K]$ with $K > 0$. With $c := \lim_{x \rightarrow +\infty} \sup_{t \geqslant x} f'(t)$, show that there exists a sequence $y_n \rightarrow +\infty$ such that $f'\left(y_n\right) \rightarrow c$ when $n \rightarrow +\infty$.

Let $m \geq 2$ be a natural integer and $E$ an $\mathbb{R}$-vector space of dimension $2m+1$ equipped with a scalar product $(.|.)$. Let $T, M$ be two endomorphisms of $E$ satisfying (H1)–(H4). We set $F^+ = \operatorname{ker}(M - \operatorname{Id}_E)$, $F^- = \operatorname{ker}(M + \operatorname{Id}_E)$. Let $G$ be the set of elements $u \in E$ satisfying (a) $u \in \operatorname{Im}(T)$ and (b) $\forall v \in E, S(u,v) = 0$.
We now say that a pair $(w_1, w_2) \in E \times E$ is a characterizing pair of $G$ if $w_1$ and $w_2$ satisfy the three properties:
(A) $w_1 \in F^+$, $T(w_1) \in G^\perp$ and $T(w_1) \neq 0_E$,
(B) $w_2 \in F^-$, $T(w_2) \in G^\perp$ and $T(w_2) \neq 0_E$,
(C) $w_i \in \operatorname{Im}(T^2)^\perp$ for $i = 1$ and $i = 2$.
Deduce from the previous questions the existence of a characterizing pair of $G$.