Problem

Source: China Northern MO 2015 grade 10 p8 CNMO

Tags: algebra, inequalities



Given a positive integer $n \ge 3$. Find the smallest real number $k$ such that for any positive real number except $a_1, a_2,..,a_n$, $$\sum_{i=1}^{n-1}\frac{a_i}{ s-a_i}+\frac{ka_n}{s-a_n} \ge \frac{n-1}{n-2}$$where, $s=a_1+a_2+..+a_n$