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2004 AMC 10A Problem 2

2004 AMC 10A Problem 2AMC 10 Step by Step

Name: Date: September 13, 2026

Functions · difficulty 1 of 5 · about 2 min · position P1-10

For any three real numbers $a$ , $b$ , and $c$ , with $b\neq c$ , the operation $\otimes$ is defined by: $\otimes(a,b,c)=\frac{a}{b-c}$ What is $\otimes(\otimes(1,2,3),\otimes(2,3,1),\otimes(3,1,2))$ ?

A)
$-\frac{1}{2}$
B)
$-\frac{1}{4}$
C)
$0$
D)
$\frac{1}{4}$
E)
$\frac{1}{2}$

Answer

Key insight

Evaluate the three inner operations first to get -1, 1, -3, then apply the rule once more to those three numbers.

Solution

Work from the inside out. The rule divides the first argument by the difference of the other two:

$\otimes(1,2,3) = \dfrac{1}{2-3} = -1$ ,
$\otimes(2,3,1) = \dfrac{2}{3-1} = 1$ ,
$\otimes(3,1,2) = \dfrac{3}{1-2} = -3$ .

These three values become the new arguments, in that order:
$\otimes(-1,\,1,\,-3) = \frac{-1}{1-(-3)} = \frac{-1}{4}.$

The answer is $\boxed{\textbf{(B)}\ -\frac{1}{4}}$ .

Why this works

A made-up operation is just a function with a recipe; nested applications are evaluated innermost first, exactly like nested parentheses. Writing each intermediate value down before combining them keeps the signs under control.

The trap

A sign slip in a denominator such as 1 - (-3), or mixing up which argument goes on top.

Common mistakes

A sign slip in a denominator such as 1 - (-3), or mixing up which argument goes on top.
Feeding the inner results into the rule in the wrong order (for instance dividing by $-1 - 1$ ).

Techniques

Set up the equation/formula and compute; no special trick needed