EGMO 2018 Problem 2

Consider the set

$$A=\left\{1+\frac{1}{k}: k=1,2,3, \ldots\right\}$$

(a) Prove that every integer $x \geq 2$ can be written as the product of one or more elements of $A$, which are not necessarily different. (b) For every integer $x \geq 2$, let $f(x)$ denote the minimum integer such that $x$ can be written as the product of $f(x)$ elements of $A$, which are not necessarily different. Prove that there exist infinitely many pairs $(x, y)$ of integers with $x \geq 2, y \geq 2$, and

$$f(x y)<f(x)+f(y) .$$

(Pairs $\left(x_{1}, y_{1}\right)$ and $\left(x_{2}, y_{2}\right)$ are different if $x_{1} \neq x_{2}$ or $\left.y_{1} \neq y_{2}\right)$. (Mihail Baluna, Romania)