Let $n{}$ be a positive integer. What is the smallest sum of digits that $5^n + 6^n + 2022^n$ can take?

Let $ABC$ be a triangle with $AB > AC$ with incenter $I{}$. The internal bisector of the angle $BAC$ intersects the $BC$ at the point $D{}$. Let $M{}$ the midpoint of the segment $AD{}$, and let $F{}$ be the second intersection point of $MB$ with the circumcircle of the triangle $BIC$. Prove that $AF$ is perpendicular to $FC$.

Find all functions $f : \mathbb{R} \to\mathbb{R}$ such that $f(0)\neq 0$ and \[f(f(x)) + f(f(y)) = f(x + y)f(xy),\]for all $x, y \in\mathbb{R}$.

Find the largest positive integer $k{}$ for which there exists a convex polyhedron $\mathcal{P}$ with 2022 edges, which satisfies the following properties: The degrees of the vertices of $\mathcal{P}$ don’t differ by more than one, and It is possible to colour the edges of $\mathcal{P}$ with $k{}$ colours such that for every colour $c{}$, and every pair of vertices $(v_1, v_2)$ of $\mathcal{P}$, there is a monochromatic path between $v_1$ and $v_2$ in the colour $c{}$. Viktor Simjanoski, Macedonia