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2004 AMC 10A Problem 2

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2004 AMC 10A Problem 2AMC 10 Step by Step
Name:                             Date: October 28, 2026
Functions · difficulty 1 of 5 · about 2 min · position P1-10

For any three real numbers aa , bb , and cc , with b≠cb\neq c , the operation ⊗\otimes is defined by: ⊗(a,b,c)=ab−c\otimes(a,b,c)=\frac{a}{b-c} What is ⊗(⊗(1,2,3),⊗(2,3,1),⊗(3,1,2))\otimes(\otimes(1,2,3),\otimes(2,3,1),\otimes(3,1,2)) ?

  1. A)−12-\frac{1}{2}
  2. B)−14-\frac{1}{4}
  3. C)00
  4. D)14\frac{1}{4}
  5. E)12\frac{1}{2}

Answer

B

Key insight

Work from the inside out. The three inner values are -1, 1 and -3; then use the same rule once more on those three numbers.

Solution

The rule says: take the first number and divide it by (second number minus third number). The inner calculations must be done first, just like parentheses.

1. First inner value: ⊗(1,2,3)=12−3=1−1=−1\otimes(1,2,3) = \dfrac{1}{2-3} = \dfrac{1}{-1} = -1.
2. Second inner value: ⊗(2,3,1)=23−1=22=1\otimes(2,3,1) = \dfrac{2}{3-1} = \dfrac{2}{2} = 1.
3. Third inner value: ⊗(3,1,2)=31−2=3−1=−3\otimes(3,1,2) = \dfrac{3}{1-2} = \dfrac{3}{-1} = -3.
4. Now put these three results, in the same order, into the rule:
⊗(−1, 1, −3)=−11−(−3)=−11+3=−14.\otimes(-1,\,1,\,-3) = \frac{-1}{1-(-3)} = \frac{-1}{1+3} = -\frac{1}{4}.

The answer is (B) −14\boxed{\textbf{(B)}\ -\frac{1}{4}}.

Why this works

A made-up operation is just a recipe. When one recipe is used inside another, finish the inner ones first, the same way you handle nested parentheses. Writing each middle result on its own line makes the minus signs much easier to keep straight.

The trap

A sign slip in a bottom such as 1 - (-3), which should be 4, or putting the wrong number on top.

Concise solution

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

  • ⊗(1,2,3)=12−3=−1\otimes(1,2,3) = \dfrac{1}{2-3} = -1,
  • ⊗(2,3,1)=23−1=1\otimes(2,3,1) = \dfrac{2}{3-1} = 1,
  • ⊗(3,1,2)=31−2=−3\otimes(3,1,2) = \dfrac{3}{1-2} = -3.

These three values become the new arguments, in that order:
⊗(−1, 1, −3)=−11−(−3)=−14.\otimes(-1,\,1,\,-3) = \frac{-1}{1-(-3)} = \frac{-1}{4}.

The answer is (B) −14\boxed{\textbf{(B)}\ -\frac{1}{4}}.

Review in the AoPS books

Techniques

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