Putnam 1995, Problem A-1

Let \( S \) be a set of real numbers which is closed under multiplication (that is, if \( a \) and \( b \) are in \( S \), then so is \( ab \)). Let \( T \) and \( U \) be disjoint subsets of \( S \) whose union is \( S \). Given that the product of any three (not necessarily distinct) elements of \( T \) is in \( T \) and that the product of any three elements of \( U \) is in \( U \), show that at least one of the two subsets \( T, U \) is closed under multiplication.

Solution:
Suppose on the contrary that there exist \( t_1, t_2 \in T \) with \( t_1 t_2 \in U \) and \( u_1, u_2 \in U \) with \( u_1 u_2 \in T \). Then \( (t_1 t_2) u_1 u_2 \in U \) while \( t_1 t_2 (u_1 u_2) \in T \), contradiction.