$$a_1, a_2, \ldots, a_{2007}$$ are $$1, 2, \ldots, 2007$$ in some order. If $$x$$ is the greatest of $$1 \cdot a_1, 2 \cdot a_2, \ldots, 2007 \cdot a_{2007}$$, prove that $$x \ge (1004)^2$$.

PLACEHOLDER ANSWER - this is a proof question. The 1004 numbers $$a_{1004}, a_{1005}, \ldots, a_{2007}$$ are distinct values from 1 to 2007, but only 1003 values are smaller than 1004. So some $$a_k$$ with $$k \ge 1004$$ satisfies $$a_k \ge 1004$$. Then $$x \ge k \cdot a_k \ge 1004 \times 1004 = (1004)^2$$.