Putnam 2003 A2

Let $a_1, a_2, \ldots, a_n$ and $b_1, b_2, \ldots, b_n$ be nonnegative real numbers. Show that $$(a_1a_2\cdots a_n)^{1/n} + (b_1b_2\cdots b_n)^{1/n} \leq \left[(a_1 + b_1)(a_2 + b_2)\cdots(a_n + b_n)\right]^{1/n}.$$