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there are infinitely many positive integers m

Source: IMO LongList 1988, Bulgaria 2, Problem 2 of ILL

October 22, 2005
floor functionalgebra unsolvedalgebra

Problem Statement

Let [(n+1)2+n2],n=1,2,…,\left[\sqrt{(n+1)^2 + n^2} \right], n = 1,2, \ldots, where [x][x] denotes the integer part of x.x. Prove that i.) there are infinitely many positive integers mm such that am+1−am>1;a_{m+1} - a_m > 1; ii.) there are infinitely many positive integers mm such that am+1−am=1.a_{m+1} - a_m = 1.
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