DE-MO 2003

There are four circles $k_1 , k_2 , k_3$ and $k_4$ of equal radius inside the triangle $ABC$ . The circle $k_1$ touches the sides $AB, CA$ and the circle $k_4$ , $k_2$ touches the sides $AB,BC$ and $k_4$ , and $k_3$ touches the sides $AC, BC$ and $k_4.$ Prove that the center of $k_4$ lies on the line connecting the incenter and circumcenter of $ABC.$

$n$ is a positive integer. Let $a(n)$ be the smallest number for which $n\mid a(n)!$ Find all solutions of: $$\frac{a(n)}{n}=\frac{2}{3}$$

Prove that there are infinitely many coprime, positive integers $a,b$ such that $a$ divides $b^2 -5$ and $b$ divides $a^2 -5.$