SE-SKOLORNAS 2020

How many of the numbers $1\cdot 2\cdot 3$ , $2\cdot 3\cdot 4$ ,..., $2020 \cdot 2021 \cdot 2022$ are divisible by $2020$ ?

The medians of the sides $AC$ and $BC$ in the triangle $ABC$ are perpendicular to each other. Prove that $\frac12 <\frac{|AC|}{|BC|}<2$ .

Determine all bounded functions $f: R \to R$ , such that $f (f (x) + y) = f (x) + f (y)$ , for all real $x, y$ .

Which is the least positive integer $n$ for which it is possible to find a (non-degenerate) $n$ -gon with sidelengths $1, 2,. . . , n$ , and where all vertices have integer coordinates?

Find all integers $a$ such that there is a prime number of $p\ge 5$ that divides ${p-1 \choose 2}$ $+ {p-1 \choose 3} a$ $+{p-1 \choose 4} a^2$ + ...+ ${p-1 \choose p-3} a^{p-5} .$

A finite set of axis parallelcubes in space has the property of each point of the room is located in a maximum of M different cubes. Show that you can divide the amount of cubes in $8 (M - 1) + 1$ subsets (or less) with the property that the cubes in each subset lacks common points. (An axis parallel cube is a cube whose edges are parallel to the coordinate axes.)