Find all reals $A,B,C$ such that there exists a real function $f$ satisfying $f(x+f(y))= Ax+By+C$ for all reals $x,y$.

Assume that $n\ge 3$ people with different names sit around a round table. We call any unordered pair of them, say $M,N$, dominating if 1) they do not sit in adjacent seats 2) on one or both arcs connecting $M,N$ along the table, all people have names coming alphabetically after $M,N$. Determine the minimal number of dominating pairs.

Let $ABC$ be a triangle and $D,E$ be points on $BC,CA$ such that $AD,BE$ are angle bisectors of $\triangle ABC$. Let $F,G$ be points on the circumcircle of $\triangle ABC$ such that $AF||DE$ and $FG||BC$. Prove that $\frac{AG}{BG}= \frac{AB+AC}{AB+BC}$.

The difference between the cubes of two consecutive positive integers is equal to $n^2$ for a positive integer $n$. Show that $n$ is the sum of two squares.