Russian problems Problem 69 A spy-plane circles point A at a distance 10 km with speed 1000 km/h. A missile is fired towards the plane from A at the same speed and moves so that it is always on the line between A and the plane. How long does it take to hit? Solution
: 18pi sec. Let C be the position of the spy-plane at the moment the missile is fired. Let B be the point a quarter of the way around the circle from C (in the direction the spy-plane is moving). Then the missile moves along the semi-circle on diameter AB and hits the plane at B. To see this take a point P on the quarter circle and let the line AX meet the semi-circle at Q. Let O be the center of the semicircle. The angle BOQ is twice the angle BAQ, so the arc BP is the same length as the arc BQ. Hence also the arc AQ is the same length as the arc CP. Russian 61-70 (C) John Scholes jscholes@kalva.demon.co.uk 30 Sep 1998