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Gulf Mathematical Olympiad 2013 - Problem 1

Source: Gulf Mathematical Olympiad 2013

April 5, 2013
pigeonhole principlealgebra unsolvedalgebra

Problem Statement

Let a1,a2,…,a2na_1,a_2,\ldots,a_{2n} be positive real numbers such that ajan+j=1a_ja_{n+j}=1 for the values j=1,2,…,nj=1,2,\ldots,n.
a. Prove that either the average of the numbers a1,a2,…,ana_1,a_2,\ldots,a_n is at least 1 or the average of the numbers an+1,an+2,…,a2na_{n+1},a_{n+2},\ldots,a_{2n} is at least 1.
b. Assuming that n≥2n\ge2, prove that there exist two distinct numbers j,kj,k in the set {1,2,…,2n}\{1,2,\ldots,2n\} such that ∣aj−ak∣<1n−1.|a_j-a_k|<\frac{1}{n-1}.
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