Putnam 1972 
Topic: Algebra

Problem: Let $A$ and $B$ be two elements in a group such that $ABA = BA^2B$, $A^3 = 1$ and $B^{2n-1} = 1$ for some positive integer $n$. Find the value of $B$.

Solution: From $ABA = BA^2B = BA^{-1}B$, we have \[ AB^2 = ABA \cdot A^{-1}B = BA^{-1}BA^{-1}B = BA^{-1} \cdot ABA = B^2A. \] By induction, \[ AB^{2r} = B^{2r}A \] so that \[ AB = AB^{2n} = B^{2n}A = BA. \] Since $A$ and $B$ commute, $ABA = BA^2B$ implies \[ A^2B = A^2B^2, \] or \[ B = B^2, \] or \[ B = \boxed{1}. \]