Find all functions $f: R \to R$, such that equality $f (xf (y) - yf (x)) = f (xy) - xy$ holds for all $x, y \in R$.

There are $100$ cities in Matland. Every road in Matland connects two cities, does not pass through any other city and does not form crossroads with other roads (although roads can go through tunnels one after the other). Driving in Matlandia by road, it is possible to get from any city to any other. Prove that that it is possible to repair some of the roads of Matlandia so that from an odd number of repaired roads would go in each city.

The tangents of the circumcircle $\Omega$ of the triangle $ABC$ at points $B$ and $C$ intersect at point $P$. The perpendiculars drawn from point $P$ to lines $AB$ and $AC$ intersect at points$ D$ and $E$ respectively. Prove that the altitudes of the triangle $ADE$ intersect at the midpoint of the segment $BC$.

We shall call an integer n cute if it can be written in the form $n = a^2 + b^3 + c^3 + d^5$, where $a, b, c$ and $d$ are integers. a) Determine if the number $2020$ is cute. b) Find all cute integers