Question 1
Let $x_1\ge x_2\ge\cdots\ge x_n$ and $y_1\ge y_2\ge\cdots\ge y_n$ be two $n$-tuples of numbers. Prove that
$$\sum_{i=1}^n(x_i-y_i)^2\le\sum_{i=1}^n(x_i-z_i)^2$$
is true when $z_1,z_2,\ldots,z_n$ denote $y_1,y_2,\ldots,y_n$ taken in another order.
$$\sum_{i=1}^n(x_i-y_i)^2\le\sum_{i=1}^n(x_i-z_i)^2$$
is true when $z_1,z_2,\ldots,z_n$ denote $y_1,y_2,\ldots,y_n$ taken in another order.