Problem

Source: Iran National Olympiad - 2014 Second Round - D2P2

Tags: combinatorics unsolved, combinatorics



A subset $S$ of positive real numbers is called powerful if for any two distinct elements $a, b$ of $S$, at least one of $a^{b}$ or $b^{a}$ is also an element of $S$. a) Give an example of a four elements powerful set. b) Prove that every finite powerful set has at most four elements.