PL-OM 2014

A tetrahedron $ABCD$ with acute-angled faces is inscribed in a sphere with center $O$ . A line passing through $O$ perpendicular to plane $ABC$ crosses the sphere at point $D'$ that lies on the opposide side of plane $ABC$ than point $D$ . Line $DD'$ crosses plane $ABC$ in point $P$ that lies inside the triangle $ABC$ . Prove, that if $\angle APB=2\angle ACB$ , then $\angle ADD'=\angle BDD'$ .

Denote the set of positive rational numbers by $\mathbb{Q}_{+}$ . Find all functions $f: \mathbb{Q}_{+}\rightarrow \mathbb{Q}_{+}$ that satisfy $$\underbrace{f(f(f(\dots f(f}_{n}(q))\dots )))=f(nq)$$ for all integers $n\ge 1$ and rational numbers $q>0$ .

Find all pairs $(x,y)$ of positive integers that satisfy $$2^x+17=y^4$$ .

In an acute triangle $ABC$ point $D$ is the point of intersection of altitude $h_a$ and side $BC$ , and points $M, N$ are orthogonal projections of point $D$ on sides $AB$ and $AC$ . Lines $MN$ and $AD$ cross the circumcircle of triangle $ABC$ at points $P, Q$ and $A, R$ . Prove that point $D$ is the center of the incircle of $PQR$ .