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AMC 10 Mock Exam Discussion For 9/22: Question 23

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AMC 10 Mock Exam Discussion For 9/22: Question 23
by John Lensmire - Friday, September 22, 2023, 3:00 PM

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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Note: The question above is a past AMC problem. Solutions submitted must be written by students. Copied solutions will be disqualified.