48th Putnam 1987 Problem B4 Define the sequences x i and y i as follows. Let (x 1 , y 1 ) = (0.8, 0.6) and let (x n+1 , y n+1 ) = (x n cos y n - y n sin y n , x n sin y n + y n cos y n ) for n ≥ 1. Find lim n→∞ x n and lim n→∞ y n .
Put y 0 = sin -1 0.6. Then the relations give immediately x n+1 = cos(y 0 + y 1 + ... + y n ), y n+1 = sin(y 0 + y 1 + ... + y n ). Since sin x  0, we have sin x < π - x for 0 < x < π. Hence y 0 + y 1 + ... + y n < π implies y 0 + y 1 + ... + y n+1 < π. So all y n are positive. Hence ∑ y n is monotone increasing and bounded above. So it converges. Hence y n → 0. But that implies that ∑ y n → π, and hence x n → -1. 48th Putnam 1987 © John Scholes jscholes@kalva.demon.co.uk 7 Jan 2001