MathDB
IMO Shortlist 2010 - Problem N4

Source:

July 17, 2011
modular arithmeticDivisibilitynumber theoryIMO Shortlist

Problem Statement

Let a,ba, b be integers, and let P(x)=ax3+bx.P(x) = ax^3+bx. For any positive integer nn we say that the pair (a,b)(a,b) is nn-good if n∣P(m)−P(k)n | P(m)-P(k) implies n∣m−kn | m - k for all integers m,k.m, k. We say that (a,b)(a,b) is very goodvery \ good if (a,b)(a,b) is nn-good for infinitely many positive integers n.n. * Find a pair (a,b)(a,b) which is 51-good, but not very good. * Show that all 2010-good pairs are very good.
Proposed by Okan Tekman, Turkey
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