DE-BUNDESWETTBEWERB 2020

Show that there are infinitely many perfect squares of the form $50^m-50^n$ , but no perfect square of the form $2020^m+2020^n$ , where $m$ and $n$ are positive integers.

Konstantin moves a knight on a $n \times n$ - chess board from the lower left corner to the lower right corner with the minimal number of moves.

Then Isabelle takes the knight and moves it from the lower left corner to the upper right corner with the minimal number of moves.

For which values of $n$ do they need the same number of moves?

Let $AB$ be the diameter of a circle $k$ and let $E$ be a point in the interior of $k$ . The line $AE$ intersects $k$ a second time in $C \ne A$ and the line $BE$ intersects $k$ a second time in $D \ne B$ .

Show that the value of $AC \cdot AE+BD\cdot BE$ is independent of the choice of $E$ .