TUYMAADA 2018

A circle touches the side $AB$ of the triangle $ABC$ at $A$ , touches the side $BC$ at $P$ and intersects the side $AC$ at $Q$ . The line symmetrical to $PQ$ with respect to $AC$ meets the line $AP$ at $X$ . Prove that $PC=CX$ .

Proposed by S. Berlov

Prove that for every positive integer $d > 1$ and $m$ the sequence $a_n=2^{2^n}+d$ contains two terms $a_k$ and $a_l$ ( $k \neq l$ ) such that their greatest common divisor is greater than $m$ .

Proposed by T. Hakobyan

The numbers $1, 2, 3, \dots, 1024$ are written on a blackboard. They are divided into pairs. Then each pair is wiped off the board and non-negative difference of its numbers is written on the board instead. $512$ numbers obtained in this way are divided into pairs and so on. One number remains on the blackboard after ten such operations. Determine all its possible values.

Proposed by A. Golovanov

Quadrilateral $ABCD$ with perpendicular diagonals is inscribed in a circle with centre $O$ . The tangents to this circle at $A$ and $C$ together with line $BD$ form the triangle $\Delta$ . Prove that the circumcircles of $BOD$ and $\Delta$ are tangent.

Proposed by A. Kuznetsov