Question 32
Let $n \ge 2$ and $a_1, a_2, \dots, a_n, b_1, b_2, \dots, b_n$ be non-negative integers. Prove that
$$
\left(\frac{n}{n-1}\right)^{n-1}\left(\frac{1}{n}\sum_{i=1}^{n} a_i^2\right)+\left(\frac{1}{n}\sum_{i=1}^{n} b_i\right)^2 \geqslant \prod_{i=1}^{n}\left(a_i^2+b_i^2\right)^{\frac{1}{n}}.
$$
$$
\left(\frac{n}{n-1}\right)^{n-1}\left(\frac{1}{n}\sum_{i=1}^{n} a_i^2\right)+\left(\frac{1}{n}\sum_{i=1}^{n} b_i\right)^2 \geqslant \prod_{i=1}^{n}\left(a_i^2+b_i^2\right)^{\frac{1}{n}}.
$$