Problem

Source: China Mathematical Olympiad 1992 problem2

Tags: inequalities, quadratics, inequalities unsolved



Given nonnegative real numbers $x_1,x_2,\dots ,x_n$, let $a=min\{x_1, x_2,\dots ,x_n\}$. Prove that the following inequality holds: \[ \sum^{n}_{i=1}\dfrac{1+x_i}{1+x_{i+1}}\le n+\dfrac{1}{(1+a)^2}\sum^{n}_{i=1}(x_i-a)^2 \quad\quad (x_{n+1}=x_1),\] and equality occurs if and only if $x_1=x_2=\dots =x_n$.