SILK-ROAD 2013

Circle with center $I$ , inscribed in a triangle $ABC$ , touches the sides $BC$ and $AC$ at points $A_1$ and $B_1$ respectively. On rays $A_1I$ and $B_1I$ , respectively, let be the points $A_2$ and $B_2$ such that $IA_2=IB_2=R$ , where $R$ is the radius of the circumscribed circle of the triangle $ABC$ . Prove that: a) $AA_2 = BB_2 = OI$ where $O$ is the center of the circumscribed circle of the triangle $ABC$ , b) lines $AA_2$ and $BB_2$ intersect on the circumcircle of the triangle $ABC$ .

Find all non-decreasing functions $f\,:\,\mathbb{N}\to\mathbb{N}$ , such that $f(f(m)f(n)+m)=f(mf(n))+f(m)$

In the film there is $n$ roles. For each $i$ ( $1 \le i \le n$ ), the role of number $i$ can play $a_i$ a person, and one person can play only one role. Every day a casting is held, in which participate people for $n$ roles, and from each role only one person. Let $p$ be a prime number such that $p \ge a_1, \ldots, a_n, n$ . Prove that you can have $p^k$ castings such that if we take any $k$ people who are tried in different roles, they together participated in some casting ( $k$ is a natural number not exceeding $n$ ).