USAMO 1998 Problem 3

Let $a_0,a_1,\cdots ,a_n$ be numbers from the interval $(0,\pi/2)$ such that $$\tan (a_0-\frac{\pi}{4})+ \tan (a_1-\frac{\pi}{4})+\cdots +\tan (a_n-\frac{\pi}{4})\geq n-1.$$ Prove that $$\tan a_0\tan a_1 \cdots \tan a_n\geq n^{n+1}.$$