CZ-SK-MO 2022

We say that a positive integer $k$ is fair if the number of $2021$ -digit palindromes that are a multiple of $k$ is the same as the number of $2022$ -digit palindromes that are a multiple of $k$ . Does the set $M = \{1, 2,..,35\}$ contain more numbers that are fair or those that are not fair?

(A palindrome is an integer that reads the same forward and backward.)

(David Hruska)

Given a scalene acute triangle $ABC$ , let M be the midpoints of its side $BC$ and $N$ the midpoint of the arc $BAC$ of its circumcircle. Let $\omega$ be the circle with diameter $BC$ and $D$ , $E$ its intersections with the bisector of angle $\angle BAC$ . Points $D'$ , $E'$ lie on $\omega$ such that $DED'E'$ is a rectangle. Prove that $D'$ , $E'$ , $M$ , $N$ lie on a single circle.

(Patrik Bak)

Let $ABCD$ be a convex quadrilateral with $AB = BC = CD$ and $P$ its intersection of diagonals. Denote by $O_1$ , $O_2$ the circumcenters of triangles $ABP$ , $CDP$ , respectively. Prove that $O_1BCO_2$ is a parallelogram.

(Patrik Bak)

Find all integers $n$ such that $2^n + n^2$ is a square of an integer.

(Tomas Jurik )