IMO 2016 Shortlist A5
(a) Prove that for every positive integer n, there exists a fraction a b where a and b are integers satisfying 0 < b ⩽ √...
Category: Algebra
Problem
(a) Prove that for every positive integer n, there exists a fraction a b where a and b are integers satisfying 0 < b ⩽ √ n + 1 and √ n ⩽ a b ⩽ √ n + 1. (b) Prove that there are infinitely many positive integers n such that there is no fraction a b where a and b are integers satisfying 0 < b ⩽ √ n and √ n ⩽ a b ⩽ √ n + 1. 4 IMO 2016 Hong Kong