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Problem 418

AMC 12 late, AIME early

Algebra Difficulty 4.8 Find the answer

23. For any real numbers $a$ , $b$ , $c$ , define an operation ※ with the following properties:
(1) $a ※(b ※ c)=(a ※ b) \cdot c$ ,
(2) $a ※ a=1$ ,

where “ $\cdot$ ” denotes multiplication.
If the solution to the equation $2016 \times(6 ※ x)=100$ is $x=\frac{p}{q}(p, q$ are positive integers, $(p, q)=1)$ , then the value of $p+q$ is $(\quad)$ .
(A) 109
(B) 201
(C) 301
(D) 3049
(E) 33601

Your answer

Official solution

23. A.
$\begin{array}{l} \text { Given } 2016 ※(6 ※ x)=(2016 ※ 6) \cdot x \\ \Rightarrow(2016 ※ 6) \cdot 6=\frac{600}{x} \\ \Rightarrow 2016 ※(6 ※ 6)=\frac{600}{x} . \end{array}$

Since $a ※ a=1$ , we have,
$\begin{array}{l} 6 ※ 6=2016 ※ 2016=1 . \\ \text { Therefore } 2016 ※(2016 ※ 2016)=\frac{600}{x} \\ \Rightarrow(2016 ※ 2016) \cdot 2016=\frac{600}{x} \\ \Rightarrow 1 \cdot 2016=\frac{600}{x} \Rightarrow x=\frac{25}{84} \\ \Rightarrow p+q=25+84=109 . \end{array}$

Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic, difficulty and ordering added by this site.