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sum \frac{a_i}{ s-a_i}+\frac{ka_n}{s-a_n} >= \frac{n-1}{n-2

Source: China Northern MO 2015 grade 10 p8 CNMO

May 5, 2024
algebrainequalities

Problem Statement

Given a positive integer n≥3n \ge 3. Find the smallest real number kk such that for any positive real number except a1,a2,..,ana_1, a_2,..,a_n, ∑i=1n−1ais−ai+kans−an≥n−1n−2\sum_{i=1}^{n-1}\frac{a_i}{ s-a_i}+\frac{ka_n}{s-a_n} \ge \frac{n-1}{n-2} where, s=a1+a2+..+ans=a_1+a_2+..+a_n
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