OP1, OP2, ... , OP2n+1 are unit vectors in a plane. P1, P2, ... , P2n+1 all lie on the same side of a line through O. Prove that |OP1 + ... + OP2n+1| ≥ 1.
Solution
We proceed by induction on n. It is clearly true for n = 1. Assume it is true for 2n-1. Given OPi for 2n+1, reorder them so that all OPi lie between OP2n and OP2n+1. Then u = OP2n + OP2n+1 lies along the angle bisector of angle P2nOP2n+1 and hence makes an angle less than 90o with v = OP1 + OP2 + ... + OP2n-1 (which must lie between OP1 and OP2n-1 and hence between OP2n and OP2n+1. By induction |v| ≥ 1. But |u + v| ≥ |v| (use the cosine formula). Hence the result is true for 2n+1.
It is clearly best possible: take OP1 = ... = OPn = -OPn+1 = ... = -OP2n, and OP2n+1 in an arbitrary direction.
Solutions are also available in: Samuel L Greitzer, International Mathematical Olympiads 1959-1977, MAA 1978, and in István Reiman, International Mathematical Olympiad 1959-1999, ISBN 189-8855-48-X.
© John Scholes
jscholes@kalva.demon.co.uk
10 Oct 1998