A standard parabola is the graph of a quadratic polynomial $y = x^2 + ax + b$ with leading coefficient 1. Three standard parabolas with vertices $V_1$, $V_2$, $V_3$ intersect pairwise at points $A_1$, $A_2$, $A_3$. Let $A \mapsto s(A)$ be the reflection of the plane with respect to the $x$ axis.

Prove that standard parabolas with vertices $s(A_1)$, $s(A_2)$, $s(A_3)$ intersect pairwise at the points $s(V_1)$, $s(V_2)$, $s(V_3)$.

Does there exist a continuously differentiable function $f : \mathbb{R} \to \mathbb{R}$ such that for every $x \in \mathbb{R}$ we have $f(x) > 0$ and $f'(x) = f(f(x))$?

Let $n$ be a positive integer and let $$a_k = \frac{1}{\binom{n}{k}}, \qquad b_k = 2^{k-n}, \qquad \text{for } k = 1, 2, \dots, n.$$ Show that $$\tag{1} \frac{a_1 - b_1}{1} + \frac{a_2 - b_2}{2} + \cdots + \frac{a_n - b_n}{n} = 0.$$

Let $f : [a,b] \to [a,b]$ be a continuous function and let $p \in [a,b]$. Define $p_0 = p$ and $p_{n+1} = f(p_n)$ for $n = 0, 1, 2, \dots$. Suppose that the set $T_p = \{ p_n : n = 0, 1, 2, \dots \}$ is closed, i.e., if $x \notin T_p$ then there is a $\delta > 0$ such that for all $x' \in T_p$ we have $|x' - x| \ge \delta$. Show that $T_p$ has finitely many elements.

Prove or disprove the following statements:

(a) There exists a monotone function $f : [0,1] \to [0,1]$ such that for each $y \in [0,1]$ the equation $f(x) = y$ has uncountably many solutions $x$.

(b) There exists a continuously differentiable function $f : [0,1] \to [0,1]$ such that for each $y \in [0,1]$ the equation $f(x) = y$ has uncountably many solutions $x$.

For an $n \times n$ matrix $M$ with real entries let $\|M\| = \sup\limits_{x \in \mathbb{R}^n \setminus \{0\}} \dfrac{\|Mx\|_2}{\|x\|_2}$, where $\|\cdot\|_2$ denotes the Euclidean norm on $\mathbb{R}^n$. Assume that an $n \times n$ matrix $A$ with real entries satisfies $\|A^k - A^{k-1}\| \le \dfrac{1}{2002 k}$ for all positive integers $k$. Prove that $\|A^k\| \le 2002$ for all positive integers $k$.

Compute the determinant of the $n \times n$ matrix $A = [a_{ij}]$, $$a_{ij} = \begin{cases} (-1)^{|i-j|}, & \text{if } i \ne j, \\ 2, & \text{if } i = j. \end{cases}$$

Two hundred students participated in a mathematical contest. They had 6 problems to solve. It is known that each problem was correctly solved by at least 120 participants. Prove that there must be two participants such that every problem was solved by at least one of these two students.

For each $n \ge 1$ let $$a_n = \sum_{k=0}^{\infty} \frac{k^n}{k!}, \qquad b_n = \sum_{k=0}^{\infty} (-1)^k \frac{k^n}{k!}.$$ Show that $a_n \cdot b_n$ is an integer.

In the tetrahedron $OABC$, let $\angle BOC = \alpha$, $\angle COA = \beta$ and $\angle AOB = \gamma$. Let $\sigma$ be the angle between the faces $OAB$ and $OAC$, and let $\tau$ be the angle between the faces $OBA$ and $OBC$. Prove that $$\gamma > \beta \cdot \cos\sigma + \alpha \cdot \cos\tau.$$

Let $A$ be an $n \times n$ matrix with complex entries and suppose that $n > 1$. Prove that $$A \overline{A} = I_n \iff \exists S \in GL_n(\mathbb{C}) \text{ such that } A = S \overline{S}^{-1}.$$ (If $A = [a_{ij}]$ then $\overline{A} = [\overline{a_{ij}}]$, where $\overline{a_{ij}}$ is the complex conjugate of $a_{ij}$; $GL_n(\mathbb{C})$ denotes the set of all $n \times n$ invertible matrices with complex entries, and $I_n$ is the identity matrix.)

Let $f : \mathbb{R}^n \to \mathbb{R}$ be a convex function whose gradient $\nabla f = \left( \frac{\partial f}{\partial x_1}, \dots, \frac{\partial f}{\partial x_n} \right)$ exists at every point of $\mathbb{R}^n$ and satisfies the condition $$\exists L > 0\ \forall x_1, x_2 \in \mathbb{R}^n\quad \|\nabla f(x_1) - \nabla f(x_2)\| \le L \|x_1 - x_2\|.$$ Prove that $$\tag{1} \forall x_1, x_2 \in \mathbb{R}^n\quad \|\nabla f(x_1) - \nabla f(x_2)\|^2 \le L \langle \nabla f(x_1) - \nabla f(x_2),\ x_1 - x_2 \rangle.$$ In this formula $\langle a, b \rangle$ denotes the scalar product of the vectors $a$ and $b$.