Problem

Source: Indian IMOTC 2013, Team Selection Test 3, Problem 3

Tags: induction, algebra proposed, algebra



Let $h \ge 3$ be an integer and $X$ the set of all positive integers that are greater than or equal to $2h$. Let $S$ be a nonempty subset of $X$ such that the following two conditions hold: if $a + b \in S$ with $a \ge h, b \ge h$, then $ab \in S$; if $ab \in S$ with $a \ge h, b \ge h$, then $a + b \in S$. Prove that $S = X$.