PL-OM 2010

$ABCD$ is a parallelogram in which angle $DAB$ is acute. Points $A, P, B, D$ lie on one circle in exactly this order. Lines $AP$ and $CD$ intersect in $Q$ . Point $O$ is the circumcenter of the triangle $CPQ$ . Prove that if $D \neq O$ then the lines $AD$ and $DO$ are perpendicular.

On the side $BC$ of the triangle $ABC$ there are two points $D$ and $E$ such that $BD < BE$ . Denote by $p_1$ and $p_2$ the perimeters of triangles $ABC$ and $ADE$ respectively. Prove that $$p_1 > p_2 + 2\cdot \min\{BD, EC\}.$$

Prime number $p>3$ is congruent to $2$ modulo $3$ . Let $a_k = k^2 + k +1$ for $k=1, 2, \ldots, p-1$ . Prove that product $a_1a_2\ldots a_{p-1}$ is congruent to $3$ modulo $p$ .

Real number $C > 1$ is given. Sequence of positive real numbers $a_1, a_2, a_3, \ldots$ , in which $a_1=1$ and $a_2=2$ , satisfy the conditions $$a_{mn}=a_ma_n,$$ $$a_{m+n} \leq C(a_m + a_n),$$ for $m, n = 1, 2, 3, \ldots$ . Prove that $a_n = n$ for $n=1, 2, 3, \ldots$ .