Putnam Puzzler 2010
Question:
Consider a set S and a binary operation *, i.e. for
each a, b in S, a*b is also in S.
Assume that (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 that ((b*a)*b)*(b*a) = b
for all a, b in S by replacing a
with b*a. Hence, a*(b*a) = b for all
a,b in S using (b*a)*b = a.