AMC 10 Mock Exam Question 23
A binary operation $\diamondsuit$ has the properties that $a\,\diamondsuit\, (b\,\diamondsuit \,c) = (a\,\diamondsuit \,b)\cdot c$ and that $a\,\diamondsuit \,a=1$ for all nonzero real numbers $a, b,$ and $c$. (Here $\cdot$ represents multiplication). The solution to the equation $2016 \,\diamondsuit\, (6\,\diamondsuit\, x)=100$ can be written as $\tfrac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. What is $p+q?$
$\displaystyle \textbf{(A) } 109\qquad\textbf{(B) } 201\qquad\textbf{(C) } 301\qquad\textbf{(D) } 3,049\qquad\textbf{(E) } 33,601$
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