NO-ABEL 2019

You have an $n \times n$ grid of empty squares. You place a cross in all the squares, one at a time. When you place a cross in an empty square, you receive $i+j$ points if there were $i$ crosses in the same row and $j$ crosses in the same column before you placed the new cross. Which are the possible total scores you can get?

$find$ all non negative integers $m$ , $n$ such that $mn-1$ divides $n^3-1$

The diagonals of a convex quadrilateral $ABCD$ intersect at $E$ . The triangles $ABE, BCE, CDE$ and $DAE$ have centroids $K,L,M$ and $N$ , and orthocentres $Q,R,S$ and $T$ . Show that the quadrilaterals $QRST$ and $LMNK$ are similar.