KURSCHAK 2022

A square has been divided into $2022$ rectangles with no two of them having a common interior point. What is the maximal number of distinct lines that can be determined by the sides of these rectangles?

Let $p$ and $q$ be prime numbers of the form $4k+3$ . Suppose that there exist integers $x$ and $y$ such that $x^2-pqy^2=1$ . Prove that there exist positive integers $a$ and $b$ such that $|pa^2-qb^2|=1$ .

Let $a_{i,j}\enspace(\forall\enspace 1\leq i\leq n, 1\leq j\leq n)$ be $n^2$ real numbers such that $a_{i,j}+a_{j,i}=0\enspace\forall i, j$ (in particular, $a_{i,i}=0\enspace\forall i$ ). Prove that $${1\over n}\sum_{i=1}^{n}\left(\sum_{j=1}^{n} a_{i,j}\right)^2\leq{1\over2}\sum_{i=1}^{n}\sum_{j=1}^{n} a_{i,j}^2.$$