Determine all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ which satisfy conditions: $f(x) + f(y) \ge xy$ for all real $x, y$ and for each real $x$ exists real $y$, such that $f(x) + f(y) = xy$.

Let $n$ be a positive integer, which gives remainder $4$ of dividing by $8$. Numbers $1 = k_1 < k_2 < ... < k_m = n$ are all positive diivisors of $n$. Show that if $i \in \{ 1, 2, ..., m - 1 \}$ isn't divisible by $3$, then $k_{i + 1} \le 2k_{i}$.

Bisector of side $BC$ intersects circumcircle of triangle $ABC$ in points $P$ and $Q$. Points $A$ and $P$ lie on the same side of line $BC$. Point $R$ is an orthogonal projection of point $P$ on line $AC$. Point $S$ is middle of line segment $AQ$. Show that points $A, B, R, S$ lie on one circle.

Let $ABCD$ be a trapezoid with bases $AB$ and $CD$. Circle of diameter $BC$ is tangent to line $AD$. Prove, that circle of diameter $AD$ is tangent to line $BC$.

Let $A_1, A_2, ..., A_k$ be $5$-element subsets of set $\{1, 2, ..., 23\}$ such that, for all $1 \le i < j \le k$ set $A_i \cap A_j$ has at most three elements. Show that $k \le 2018$.

Let $k$ be a positive integer and $a_1, a_2, ...$ be a sequence of terms from set $\{ 0, 1, ..., k \}$. Let $b_n = \sqrt[n] {a_1^n + a_2^n + ... + a_n^n}$ for all positive integers $n$. Prove, that if in sequence $b_1, b_2, b_3, ...$ are infinitely many integers, then all terms of this series are integers.