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Two every other sequences with sum not exceeding 1

Source: 2022 China TST, Test 4 P5

April 30, 2022
algebraInequalitycombinatorics

Problem Statement

Let nn be a positive integer, x1,x2,…,x2nx_1,x_2,\ldots,x_{2n} be non-negative real numbers with sum 44. Prove that there exist integer pp and qq, with 0≤q≤n−10 \le q \le n-1, such that \sum_{i=1}^q x_{p+2i-1} \le 1 \mbox{ and } \sum_{i=q+1}^{n-1} x_{p+2i} \le 1, where the indices are take modulo 2n2n.
Note: If q=0q=0, then ∑i=1qxp+2i−1=0\sum_{i=1}^q x_{p+2i-1}=0; if q=n−1q=n-1, then ∑i=q+1n−1xp+2i=0\sum_{i=q+1}^{n-1} x_{p+2i}=0.
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