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inequality

Source: 2021 Korea Winter Program Practice Test

February 8, 2021
algebrainequalities

Problem Statement

n≥2n\ge2 is a given positive integer. i≤ai≤ni\leq a_i \leq n satisfies for all 1≤i≤n1\leq i\leq n, and SiS_i is defined as a1+a2+...+ai(S0=0)a_1+a_2+...+a_i(S_0=0). Show that there exists such 1≤k≤n1\leq k\leq n that satisfies ak2+Sn−k<2Sn−n(n+1)2a_k^2+S_{n-k}<2S_n-\frac{n(n+1)}{2}.
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