MD-MO 2008 Problem 4

Let $n$ be a positive integer. Find all $x_1,x_2,\ldots,x_n$ that satisfy the relation: $$\sqrt{x_1-1}+2\cdot \sqrt{x_2-4}+3\cdot \sqrt{x_3-9}+\cdots+n\cdot\sqrt{x_n-n^2}=\frac{1}{2}(x_1+x_2+x_3+\cdots+x_n).$$