IMO 1988 LL HKG31

Origin: HKG

Problem

The circle x2 + y2 = r2 meets the coordinate axes at A = (r, 0), B = (−r, 0), C = (0, r), and D = (0, −r). Let P = (u, v) and Q = (−u, v) be two points on the circumference of the circle. Let N be the point of intersection of PQ and the y-axis, and let M be the foot of the perpendicular drawn from P to the x-axis. If r2 is odd, u = pm > qn = v, where p and q are prime numbers, and m and n are natural numbers, show that |AM| = 1, |BM| = 9, |DN| = 8, |PQ| = 8.