50th Mathematical Olympiad in Ukraine, Fourth Round (March 24, 2010)

Real numbers $a_1,a_2,\dots ,a_n$ satisfy the following conditions $a_1+a_2+\cdots +a_n=n$ and $a_1\ge a_2\ge \cdots \ge a_n\ge 0$. Prove the inequality: $$n{a}_1\ge a_1^2+a_2^2+\cdots +a_n^2.$$