AT-OEMO-FED1 2012

Determine all functions $f: \mathbb{Z}\to\mathbb{Z}$ satisfying the following property: For each pair of integers $m$ and $n$ (not necessarily distinct), $\mathrm{gcd}(m, n)$ divides $f(m) + f(n)$ .

Note: If $n\in\mathbb{Z}$ , $\mathrm{gcd}(m, n)=\mathrm{gcd}(|m|, |n|)$ and $\mathrm{gcd}(n, 0)=n$ .

Determine all solutions $(n, k)$ of the equation $n!+An = n^k$ with $n, k \in\mathbb{N}$ for $A = 7$ and for $A = 2012$.

Consider a stripe of $n$ fieds, numbered from left to right with the integers $1$ to $n$ in ascending order. Each of the fields is colored with one of the colors $1$ , $2$ or $3$ . Even-numbered fields can be colored with any color. Odd-numbered fields are only allowed to be colored with the odd colors $1$ and $3$ . How many such colorings are there such that any two neighboring fields have different colors?

Let $ABC$ be a scalene (i.e. non-isosceles) triangle. Let $U$ be the center of the circumcircle of this triangle and $I$ the center of the incircle. Assume that the second point of intersection different from $C$ of the angle bisector of $\gamma = \angle ACB$ with the circumcircle of $ABC$ lies on the perpendicular bisector of $UI$ . Show that $\gamma$ is the second-largest angle in the triangle $ABC$ .