Putnam 2001, Problem A-1

Consider a set $S$ and a binary operation $*$, i.e., for each $a, b \in S$, $a * b \in S$. Assume $(a * b) * a = b$ for all $a, b \in S$. Prove that $a * (b * a) = b$ for all $a, b \in S$.

Solution:
The hypothesis implies $((b*a)*b)*(b*a) = b$ for all $a,b \in S$ (by replacing $a$ by $b*a$), and hence $a*(b*a) = b$ for all $a,b \in S$ (using $(b*a)*b = a$).