CZ-SK-MO 2024

Let the interior point $P$ of the convex quadrilateral $ABCD$ be such that $$|\angle PAD| = |\angle ADP| = |\angle CBP| = |\angle PCB| = |\angle CPD|.$$ Let $O$ be the center of the circumcircle of the triangle $CPD$ . Prove that $|OA| = |OB|$ .

There were $10$ boys and $10$ girls at the party. Every boy likes a different 'positive' number of girls. Every girl likes a different positive number of boys. Define the largest non-negative integer $n$ such that it is always possible to form at least $n$ disjoint pairs in which both like the other.

Let $(a_k)^{\infty}_{k=0}$ be a sequence of real numbers such that if $k$ is a non-negative integer, then $$a_{k+1} = 3a_k - \lfloor 2a_k \rfloor - \lfloor a_k \rfloor.$$ Definitely all positive integers $n$ such that if $a_0 = 1/n$ , then this sequence is constant after a certain term.