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2022 VJIMC, 3

Let $x_1,\ldots,x_n$ be given real numbers with $0<m\le x_i\le M$ for each $i\in\{1,\ldots,n\}$. Let $X$ be the discrete random variable uniformly distributed on $\{x_1,\ldots,x_n\}$. The mean $\mu$ and the variance $\sigma^2$ of $X$ are defined as $$\mu(X)=\frac{x_1+\ldots+x_n}n\text{ and }\sigma^2(X)=\frac{(x_1-\mu(X))^2+\ldots+(x_n-\mu(X))^2}n.$$ By $X^2$ denote the discrete random variable uniformly distributed on $\{x_1^2,\ldots,x_n^2\}$. Prove that $$\sigma^2(X)\ge\left(\frac m{2M^2}\right)^2\sigma^2(X^2).$$

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