IMO 1971 LL NET32

Origin: NET

Problem

Two half-lines a and b, with the common endpoint O, make an acute angle \alpha. Let A on a and B on b be points such that OA = OB, and let b′ be the line through A parallel to b. Let \beta be the circle with center B and radius BO. We construct a sequence of half-lines c1, c2, c3, . . . , all lying inside the angle \alpha, in the following manner: (i) c1 is given arbitrarily; (ii) for every natural number k, the circle \beta intercepts on ck a segment that is of the same length as the segment cut on b′ by a and ck+1. Prove that the angle determined by the lines ck and b has a limit as k tends to infinity and find that limit.