BALKAN 1990

The sequence $(a_{n})_{n\geq 1}$ is defined by $a_{1} = 1, a_{2} = 3$ , and $a_{n + 2} = (n + 3)a_{n + 1} - (n + 2)a_{n}, \forall n \in \mathbb{N}$ . Find all values of $n$ for which $a_{n}$ is divisible by $11$ .

The polynomial $P(X)$ is defined by $P(X)=(X+2X^{2}+\ldots +nX^{n})^{2}=a_{0}+a_{1}X+\ldots +a_{2n}X^{2n}$ . Prove that $a_{n+1}+a_{n+2}+\ldots +a_{2n}=\frac{n(n+1)(5n^{2}+5n+2)}{24}$ .

Let $ABC$ be an acute triangle and let $A_{1}, B_{1}, C_{1}$ be the feet of its altitudes. The incircle of the triangle $A_{1}B_{1}C_{1}$ touches its sides at the points $A_{2}, B_{2}, C_{2}$ . Prove that the Euler lines of triangles $ABC$ and $A_{2}B_{2}C_{2}$ coincide.

Find the least number of elements of a finite set $A$ such that there exists a function $f : \left\{1,2,3,\ldots \right\}\rightarrow A$ with the property: if $i$ and $j$ are positive integers and $i-j$ is a prime number, then $f(i)$ and $f(j)$ are distinct elements of $A$ .