2004 AMC 10A Problem 2

Problem 2 of 25EasierAlgebra

For any three real numbers

a,

b,

and

c,

with

b \neq c,

the operation

\diamond

is defined by

\diamond(a, b, c) = \dfrac{a}{b - c}.

What is

\diamond(\diamond(1, 2, 3), \diamond(2, 3, 1), \diamond(3, 1, 2))?

Show solution

Solution

The inner values are

\begin{aligned} \diamond(1,2,3) &= \dfrac{1}{2-3} = -1, \\ \diamond(2,3,1) &= \dfrac{2}{3-1} = 1, \\ \diamond(3,1,2) &= \dfrac{3}{1-2} = -3. \end{aligned}

Therefore

\begin{aligned} \diamond(-1, 1, -3) &= \dfrac{-1}{1 - (-3)} \\ &= -\dfrac{1}{4}. \end{aligned}

Thus, the correct answer is B.