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Find the smallest positive integer m[Iran Second Round 1993]

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November 25, 2010
pigeonhole principlenumber theory proposednumber theory

Problem Statement

Let n,rn, r be positive integers. Find the smallest positive integer mm satisfying the following condition. For each partition of the set {1,2,…,m}\{1, 2, \ldots ,m \} into rr subsets A1,A2,…,ArA_1,A_2, \ldots ,A_r, there exist two numbers aa and bb in some Ai,1≤i≤rA_i, 1 \leq i \leq r, such that 1<ab<1+1n. 1 < \frac ab < 1 +\frac 1n.
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