NL-NWO 1982

Which is greater: $17091982!^2$ or $17091982^{17091982}$ ?

In a triangle $ABC$ , $M$ is the midpoint of $AB$ and $P$ an arbitrary point on side $AC$ . Using only a straight edge, construct point $Q$ on $BC$ such that $P$ and $Q$ are at equal distance from $CM$ .

Five marbles are distributed at a random among seven urns. What is the expected number of urns with exactly one marble?

Determine $\gcd (n^2+2,n^3+1)$ for $n=9^{753}$ .