BG-MO 1995

Find the number of integers $n > 1$ which divide $a^{25} - a$ for every integer $a$ .

Let triangle ABC has semiperimeter $p$ . E,F are located on AB such that $CE=CF=p$ . Prove that the C-excircle of triangle ABC touches the circumcircle (EFC).

Two players $A$ and $B$ take stones one after the other from a heap with $n \ge 2$ stones. $A$ begins the game and takes at least one stone, but no more than $n -1$ stones. Thereafter, a player on turn takes at least one, but no more than the other player has taken before him. The player who takes the last stone wins. Who of the players has a winning strategy?

Points $A_1,B_1,C_1$ are selected on the sides $BC$ , $CA$ , $AB$ respectively of an equilateral triangle $ABC$ in such a way that the inradii of the triangles $C_1AB_1$ , $A_1BC_1$ , $B_1CA_1$ and $A_1B_1C_1$ are equal. Prove that $A_1,B_1,C_1$ are the midpoints of the corresponding sides.

Let $A = \{1,2,...,m + n\}$ , where $m,n$ are positive integers, and let the function f : $A \to A$ be defined by: $f(m) = 1$ , $f(m+n) = m+1$ and $f(i) = i+1$ for all the other $i$ . (a) Prove that if $m$ and $n$ are odd, then there exists a function $g : A \to A$ such that $g(g(a)) = f(a)$ for all $a \in A$ . (b) Prove that if $m$ is even, then there is a function $g : A\to A$ such that $g(g(a))=f(a)$ for all $a \in A$ is and only if $n = m$ .