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Polish MO Finals 2014, Problem 2

Source: Polish MO Finals 2014

July 27, 2016
contestsalgebra

Problem Statement

Let k≥2k\ge 2, n≥1n\ge 1, a1,a2,…,aka_1, a_2,\dots, a_k and b1,b2,…,bnb_1, b_2, \dots, b_n be integers such that 1<a1<a2<⋯<ak<b1<b2<⋯<bn1<a_1<a_2<\dots <a_k<b_1<b_2<\dots <b_n. Prove that if a1+a2+⋯+ak>b1+b2+⋯+bna_1+a_2+\dots +a_k>b_1+b_2+\dots + b_n, then a1⋅a2⋅…⋅ak>b1⋅b2⋅…⋅bna_1\cdot a_2\cdot \ldots \cdot a_k>b_1\cdot b_2 \cdot \ldots \cdot b_n.
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