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difference is not divided by n

Source: Vietnam TST 1992 for the 33nd IMO, problem 1

June 25, 2005
floor functionnumber theory unsolvednumber theory

Problem Statement

Let two natural number n>1n > 1 and mm be given. Find the least positive integer kk which has the following property: Among kk arbitrary integers a1,a2,…,aka_1, a_2, \ldots, a_k satisfying the condition ai−aja_i - a_j ( 1≤i<j≤k1 \leq i < j \leq k) is not divided by nn, there exist two numbers ap,asa_p, a_s (p≠sp \neq s) such that m+ap−asm + a_p - a_s is divided by nn.
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