NMC 2008

Determine all real numbers $A, B$ and $C$ such that there exists a real function $f$ that satisfies

$$f(x+f(y))=A x+B y+C$$

for all real $x$ and $y$.

Assume that $n \geq 3$ people with different names sit around a round table. We call any unordered pair of them, say $M$ and $N$, dominating, if

(i) $M$ and $N$ do not sit on adjacent seats, and

(ii) on one (or both) of the arcs connecting $M$ and $N$ along the table edge, all people have names that come alphabetically after the names of $M$ and $N$.

Determine the minimal number of dominating pairs.

Let $A B C$ be a triangle and let $D$ and $E$ be points on $B C$ and $C A$, respectively, such that $A D$ and $B E$ are angle bisectors of $A B C$. Let $F$ and $G$ be points on the circumcircle of $A B C$ such that $A F$ and $D E$ are parallel and $F G$ and $B C$ are parallel. Show that

$$\frac{A G}{B G}=\frac{A C+B C}{A B+C B}$$

The difference between the cubes of two consecutive positive integers is a square $n^{2}$, where $n$ is a positive integer. Show that $n$ is the sum of two squares.