IrMO 1992 Problem 10

If, for $k=1,2,\dots ,n$ , $a_k$ and $b_k$ are positive real numbers, prove that $$\sqrt[n]{a_1a_2\cdots a_n}+\sqrt[n]{b_1b_2\cdots b_n}\le \sqrt[n]{(a_1+b_1)(a_2+b_2)\cdots (a_n+b_n)};$$ and that equality holds if, and only if, $$\frac{a_1}{b_1}=\frac{a_2}{b_2}=\cdots =\frac{a_n}{b_n}.$$