In a sequence of $71$ nonzero real numbers, each number (apart from the fitrst one and the last one) is one less than the product of its two neighbors. Prove that the first and the last number are equal. (Josef Tkadlec)

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 )

Consider any graph with $50$ vertices and $225$ edges. We say that a triplet of its (mutually distinct) vertices is connected if the three vertices determine at least two edges. Determine the smallest and the largest possible number of connected triples. (Jan Mazak, Josef Tkadlec)