PL-OM 2018

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.