CAUCASUS 2022

Let $S$ be the set of all $5^6$ positive integers, whose decimal representation consists of exactly 6 odd digits. Find the number of solutions $(x,y,z)$ of the equation $x+y=10z$ , where $x\in S$ , $y\in S$ , $z\in S$ .

Point $P$ is chosen on the leg $CB$ of right triangle $ABC$ ( $\angle ACB = 90^\circ$ ). The line $AP$ intersects the circumcircle of $ABC$ at point $Q$ . Let $L$ be the midpoint of $PB$ . Prove that $QL$ is tangent to a fixed circle independent of the choice of point $P$ .