NO-ABEL 2006 Problem 3

(a) Let $a$ and $b$ be rational numbers such that line $y = ax + b$ intersects the circle $x^2 + y^2 = 5$ at two different points. Show that if one of the intersections has two rational coordinates, so does the other intersection. (b) Show that there are infinitely many triples ( $k, n, m$ ) that are such that $k^2 + n^2 = 5m^2$ , where $k, n$ and $m$ are integers, and not all three have any in common prime factor.