Determine all pairs of positive integers $m, n,$ satisfying the equality $(2^{m}+1;2^n+1)=2^{(m;n)}+1$ , where $(a;b)$ is the greatest common divisor

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 $n$ ).