Školjka

%V0 Let $\mathbb{Q}_{>0}$ be the set of positive rational numbers. Let $f : \mathbb{Q}_{>0} \to \mathbb{R}$ be a function satisfying the conditions $$ f(x) f(y) \geq f(xy) \quad \text{and} \quad f(x + y) \geq f(x) + f(y) $$ for all $x, y \in \mathbb{Q}_{>0}$. Given that $f(a) = a$ for some rational $a > 1$, prove that $f(x) = x$ for all $x \in \mathbb{Q}_{>0}$.

%V0 In the plane, $2013$ red points and $2014$ blue points are marked so that no three of the marked points are collinear. One needs to draw $k$ lines not passing through the marked points and dividing the plane into several regions. The goal is to do it in such a way that no region contains points of both colors. Find the minimal value of $k$ such that the goal is attainable for every possible configuration of $4027$ points.

%V0 Let $n \geq 2$ be an integer. Consider all circular arrangements of the numbers $0, 1, \ldots, n$; the $n + 1$ rotations of an arrangement are considered to be equal. A circular arrangement is called [i]beautiful[/i] if, for any four distinct numbers $0 \leq a, b, c, d \leq n$ with $a + c = b + d$, the chord joining numbers $a$ and $c$ does not intersect the chord joining numbers $b$ and $d$. Let $M$ be the number of beautiful arrangements of $0, 1, \ldots, n$. Let $N$ be the number of pairs $(x, y)$ of positive integers such that $x + y \leq n$ and $\operatorname{gcd}(x, y) = 1$. Prove that $$ M = N + 1 \text{.} $$

%V0 Let $ABC$ be an acute-angled triangle with orthocenter $H$, and let $W$ be a point on side $BC$. Denote by $M$ and $N$ the feet of the altitudes from $B$ and $C$, respectively. Denote by $\omega_1$ the circumcircle of $BWN$, and let $X$ be the point on $\omega_1$ which is diametrically opposite to $W$. Analogously, denote by $\omega_2$ the circumcircle of $CWM$, and let $Y$ be the point on $\omega_2$ which is diametrically opposite to $W$. Prove that $X$, $Y$ and $H$ are collinear.

%V0 Let the excircle of the triangle $ABC$ lying opposite to $A$ touch its side $BC$ at the point $A_1$. Define the points $B_1$ and $C_1$ analogously. Suppose that the circumcentre of the triangle $A_1B_1C_1$ lies on the circumcircle of the triangle $ABC$. Prove that the triangle $ABC$ is right-angled.

%V0 Prove that for any pair of positive integers $k$ and $n$ there exist $k$ positive integers $m_1, m_2, \ldots, m_k$ such that $$ 1 + \frac{2^k - 1}{n} = \left( 1 + \frac{1}{m_1} \right) \left( 1 + \frac{1}{m_2} \right) \cdots \left( 1 + \frac{1}{m_k} \right) \text{.} $$