RO-NMO 2017 Problem 93

Let be two natural numbers $n$ and $a.$ a) Prove that there exists an $n\text{-tuplet}$ of natural numbers $\left( a_1,a_2,\ldots ,a_n\right)$ that satisfy the following equality. $$1+\frac{1}{a} =\prod_{i=1}^n \left( 1+\frac{1}{a_i} \right)$$ b) Show that there exist only finitely such $n\text{-tuplets} .$