Problem

Source: Romanian National Olympiad 2017, grade ix, p.3

Tags: number theory, combinatorics



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} . $