IMO 1988 LL BUL2

Origin: BUL

Problem

Let an =  (n + 1)2 + n2  , n = 1, 2, . . . , where [x] denotes the integer part of x. Prove that (a) there are infinitely many positive integers m such that am+1−am > 1; (b) there are infinitely many positive integers m such that am+1−am = 1.