PL-OM 2002

Find all the natural numbers $a,b,c$ such that:

1. $a^2+1$ and $b^2+1$ are primes
2. $(a^2+1)(b^2+1)=(c^2+1)$

On sides $AC$ and $BC$ of acute-angled triangle $ABC$ rectangles with equal areas $ACPQ$ and $BKLC$ were built exterior. Prove that midpoint of $PL$ , point $C$ and center of circumcircle are collinear.

There is given a triangle $ABC$ in a space. A sphere does not intersect the plane of $ABC$ . There are $4$ points $K, L, M, P$ on the sphere such that $AK, BL, CM$ are tangent to the sphere and $\frac{AK}{AP} = \frac{BL}{BP} = \frac{CM}{CP}$ . Show that the sphere touches the circumsphere of $ABCP$ .