2016 AMC 12A Problem 20

Problem 20 of 25HarderAlgebra

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?

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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,

a\diamond b=\dfrac{a}{b}.

Therefore

\begin{gathered} 2016\diamond(6\diamond x)\\ =2016\diamond\dfrac{6}{x}\\ =\dfrac{2016}{\frac{6}{x}}\\ =336x=100, \end{gathered}

x=\dfrac{100}{336}=\dfrac{25}{84}

and

p+q=25+84=109.

Thus, the correct answer is A.

Tagged: custom operation · functional equation