Question 1
Consider $n$ inequalities
$$4x^2-4a_ix+(a_i-1)^2\le0,$$
where $a_i\in[\frac12,5]$ for $i=1,2,\ldots,n$. Let $x_i$ be any solution of the inequality with parameter $a_i$. Prove that
$$\frac{\sum_{k=1}^n x_k^2}{n}\le\frac{\sum_{k=1}^n x_k}{n}+1.$$
$$4x^2-4a_ix+(a_i-1)^2\le0,$$
where $a_i\in[\frac12,5]$ for $i=1,2,\ldots,n$. Let $x_i$ be any solution of the inequality with parameter $a_i$. Prove that
$$\frac{\sum_{k=1}^n x_k^2}{n}\le\frac{\sum_{k=1}^n x_k}{n}+1.$$