OIM 2016

Find all prime numbers $p,q,r,k$ such that $pq+qr+rp = 12k+1$

Let $ABC$ be an acute triangle and $\Gamma$ its circumcircle. The lines tangent to $\Gamma$ through $B$ and $C$ meet at $P$ . Let $M$ be a point on the arc $AC$ that does not contain $B$ such that $M \neq A$ and $M \neq C$ , and $K$ be the point where the lines $BC$ and $AM$ meet. Let $R$ be the point symmetrical to $P$ with respect to the line $AM$ and $Q$ the point of intersection of lines $RA$ and $PM$ . Let $J$ be the midpoint of $BC$ and $L$ be the intersection point of the line $PJ$ and the line through $A$ parallel to $PR$ . Prove that $L, J, A, Q,$ and $K$ all lie on a circle.

Determine the maximum number of bishops that we can place in a $8 \times 8$ chessboard such that there are not two bishops in the same cell, and each bishop is threatened by at most one bishop.

Note: A bishop threatens another one, if both are placed in different cells, in the same diagonal. A board has as diagonals the $2$ main diagonals and the ones parallel to those ones.

The circumferences $C_1$ and $C_2$ cut each other at different points $A$ and $K$ . The common tangent to $C_1$ and $C_2$ nearer to $K$ touches $C_1$ at $B$ and $C_2$ at $C$ . Let $P$ be the foot of the perpendicular from $B$ to $AC$ , and let $Q$ be the foot of the perpendicular from $C$ to $AB$ . If $E$ and $F$ are the symmetric points of $K$ with respect to the lines $PQ$ and $BC$ , respectively, prove that $A, E$ and $F$ are collinear.

Let $k$ be a positive integer and $a_1, a_2,$ $\cdot \cdot \cdot$ $, a_k$ digits. Prove that there exists a positive integer $n$ such that the last $2k$ digits of $2^n$ are, in the following order, $a_1, a_2,$ $\cdot \cdot \cdot$ $, a_k , b_1, b_2,$ $\cdot \cdot \cdot$ $, b_k$ , for certain digits $b_1, b_2,$ $\cdot \cdot \cdot$ $, b_k$