IMO 1978 LL SWE35

Origin: SWE

Problem

A sequence (an)N 0 of real numbers is called concave if 2an \geq an−1 + an+1 for all integers n, 1 \leqn \leqN −1. (a) Prove that there exists a constant C > 0 such that N  n=0 an \geqC(N −1) N  n=0 a2 n (1) for all concave positive sequences (an)N 0 . (b) Prove that (1) holds with C = 3/4 and that this constant is best possible.