Question

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

$-\dfrac{1}{2}$

$-\dfrac{1}{4}$

$0$

$\dfrac{1}{4}$

$\dfrac{1}{2}$

Answer: B

Concepts:custom operation fraction

Difficulty rating: 980

Small Hint:

Evaluate the three inner expressions before combining them

Big Hint:

$\diamond(1,2,3) = -1,$ $\diamond(2,3,1) = 1,$ and $\diamond(3,1,2) = -3$

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.

2004 AMC 10A Problem 2

Answer: B

Difficulty rating: 980

Small Hint:

Big Hint:

Solution:

Problem 2 in Other Years