Putnam 2001 - Problem A-1 (On a binary operation)

Let $*$ be a binary operation acting on a non-empty set $S$, such that $$(a*b)*a=b,$$ for all $a,b\in S$.

Prove that $$a*(b*a)=b,$$ for all $a,b \in S$.

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The solution in mathlinks says $((b*a)*b)(b*a)=b$ out of the blue. Why? – Carry on Smiling Mar 21 '14 at 23:18
Let $a=(b*a),b=b.$ – coffeemath Mar 21 '14 at 23:22
what do you mean let $a=b*a$ – Carry on Smiling Mar 21 '14 at 23:33
@coffeemath can we just go ahead and say $a=(b*a)$ is that legit? – Carry on Smiling Mar 21 '14 at 23:43
user4140 Yes it's legit. To se it rewrite (ab)*a=b with new letters like (xy)*x=y, then let this x be (a*b) and this y be b. – coffeemath Mar 22 '14 at 0:10

We have that $$a*(b*a)=\big((b*a)*b\big)*(b*a)=b$$ Note that the second "=" holds since $$(c*b)*c=b,$$ where $c=b*a$.

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yes, the only part I don't get is the second equivalence – Carry on Smiling Mar 21 '14 at 23:33
Why is $((b*a)*b)*(b*a)=b$??? – Carry on Smiling Mar 21 '14 at 23:35
I really don't understand it. Could you elaborate please? – Carry on Smiling Mar 21 '14 at 23:41
@user4140: See update. – Yiorgos S. Smyrlis Mar 21 '14 at 23:47
Ohhhhhh, I finally get it. Thanks. – Carry on Smiling Mar 22 '14 at 0:00

Because $b*a,b\in S$, so by the formula $((b*a)*b)*(b*a)=b$. Then $a*(b*a)=b$.

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By what formula? – Carry on Smiling Mar 21 '14 at 23:43
@user4140 By the formula $a∗(b∗a)=b$, just replace $a$ by $a*b$. – user2345215 Mar 21 '14 at 23:47