Problem

Source: Spain MO 2024 P2

Tags: algebra, inequalities, Spain



Let $n$ be a positive integer. Let $x_1, x_2, \dots, x_n > 1$ be real numbers whose product is $n+1$. Prove that \[\left(\frac{1}{1^2(x_1-1)}+1\right)\left(\frac{1}{2^2(x_2-1)}+1\right)\cdots\left(\frac{1}{n^2(x_n-1)}+1\right)\geq n+1\]and find for which values equality holds.