Triangle $ABC$ is right angled at $C$. Lines $AM$ and $BN$ are internal angle bisectors. $AM$ and $BN$ intersect altitude $CH$ at points $P$ and $Q$ respectively. Prove that the line which passes through the midpoints of segments $QN$ and $PM$ is parallel to $AB$.

The sequence $a_n$ is defined by $a_0=a_1=1$ and $a_{n+1}=14a_n-a_{n-1}-4$,for all positive integers $n$. Prove that all terms of this sequence are perfect squares.

Prove that in the set consisting of $\binom{2n}{n}$ people we can find a group of $n+1$ people in which everyone knows everyone or noone knows noone.

Find all primes $p,q$ such that $p$ divides $30q-1$ and $q$ divides $30p-1$.

Let $x_1,x_2,\ldots,x_n$ be nonnegative real numbers of sum equal to $1$. Let $F_n=x_1^{2}+x_2^{2}+\cdots +x_n^{2}-2(x_1x_2+x_2x_3+\cdots +x_nx_1)$. Find: a) $\min F_3$; b) $\min F_4$; c) $\min F_5$.

In triangle $ABC$, $I$ is the incenter. We have chosen points $P,Q,R$ on segments $IA,IB,IC$ respectively such that $IP\cdot IA=IQ \cdot IB=IR\cdot IC$. Prove that the points $I$ and $O$ belong to Euler line of triangle $PQR$ where $O$ is circumcenter of $ABC$.