In triangle $ABC$ the angle $\angle A$ is the smallest. Points $D, E$ lie on sides $AB, AC$ so that $\angle CBE=\angle DCB=\angle BAC$. Prove that the midpoints of $AB, AC, BE, CD$ lie on one circle.

Let $P$ be a polynomial with real coefficients. Prove that if for some integer $k$ $P(k)$ isn't integral, then there exist infinitely many integers $m$, for which $P(m)$ isn't integral.

Find the biggest natural number $m$ that has the following property: among any five 500-element subsets of $\{ 1,2,\dots, 1000\}$ there exist two sets, whose intersection contains at least $m$ numbers.

Solve the system $$\begin{cases} x+y+z=1\\ x^5+y^5+z^5=1\end{cases}$$in real numbers.

Prove that diagonals of a convex quadrilateral are perpendicular if and only if inside of the quadrilateral there is a point, whose orthogonal projections on sides of the quadrilateral are vertices of a rectangle.

Prove that for each positive integer $a$ there exists such an integer $b>a$, for which $1+2^a+3^a$ divides $1+2^b+3^b$.