Question

A binary operation $\diamond$ has the properties that $a\diamond(b\diamond c)=(a\diamond b)\cdot c$ and that $a\diamond a=1$ for all nonzero real numbers $a,$ $b,$ and $c.$ (Here the dot $\cdot$ represents the usual multiplication operation.) The solution to the equation $2016\diamond(6\diamond x)=100$ can be written as $\dfrac{p}{q},$ where $p$ and $q$ are relatively prime positive integers. What is $p+q?$

$109$

$201$

$301$

$3049$

$33{,}601$

Answer: A

Concepts:custom operation functional equation

Difficulty rating: 1910

Solution:

Setting $b=c=a$ gives $a\diamond 1=a\diamond(a\diamond a)$ $=(a\diamond a)\cdot a=a.$ Then setting $c=b$ gives $a=a\diamond 1$ $=a\diamond(b\diamond b)=(a\diamond b)\cdot b,$ so $a\diamond b=\dfrac{a}{b}.$

Therefore $\begin{gathered} 2016\diamond(6\diamond x)\\ =2016\diamond\dfrac{6}{x}\\ =\dfrac{2016}{6/x}\\ =336x=100, \end{gathered}$ so $x=\dfrac{100}{336}=\dfrac{25}{84}$ and $p+q=25+84=109.$

Thus, the correct answer is A.

2016 AMC 12A Problem 20

Answer: A

Difficulty rating: 1910

Solution:

Problem 20 in Other Years