MK-MO 2016 Problem 5

Let $n\ge3$ and $a_1,a_2,...,a_n \in \mathbb{R^{+}}$ , such that $\frac{1}{1+a_1^4} + \frac{1}{1+a_2^4} + ... + \frac{1}{1+a_n^4} = 1$ . Prove that: $$a_1a_2...a_n \ge (n-1)^{\frac n4}$$