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China Mathematics Olympiads (National Round) 2007 Problem 6

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November 28, 2010
pigeonhole principlenumber theory unsolvednumber theory

Problem Statement

Find a number n≥9n \geq 9 such that for any nn numbers, not necessarily distinct, a1,a2,…,ana_1,a_2, \ldots , a_n, there exists 9 numbers ai1,ai2,…,ai9,(1≤i1<i2<…<i9≤n)a_{i_1}, a_{i_2}, \ldots , a_{i_9}, (1 \leq i_1 < i_2 < \ldots < i_9 \leq n) and bi∈4,7,i=1,2,…,9b_i \in {4,7}, i =1, 2, \ldots , 9 such that b1ai1+b2ai2+…+b9ai9b_1a_{i_1} + b_2a_{i_2} + \ldots + b_9a_{i_9} is a multiple of 9.
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