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Inequalities in floor function

Source: Indonesian MO (INAMO) 2009, Day 1, Problem 2

August 8, 2009
inequalitiesfunctionfloor functionnumber theory unsolvednumber theory

Problem Statement

For any real x x, let ⌊x⌋ \lfloor x\rfloor be the largest integer that is not more than x x. Given a sequence of positive integers a1,a2,a3,… a_1,a_2,a_3,\ldots such that a1>1 a_1>1 and \left\lfloor\frac{a_1\plus{}1}{a_2}\right\rfloor\equal{}\left\lfloor\frac{a_2\plus{}1}{a_3}\right\rfloor\equal{}\left\lfloor\frac{a_3\plus{}1}{a_4}\right\rfloor\equal{}\cdots Prove that \left\lfloor\frac{a_n\plus{}1}{a_{n\plus{}1}}\right\rfloor\leq1 holds for every positive integer n n.
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