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2010 AMC 10A Problem 6

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2010 AMC 10A Problem 6AMC 10 Step by Step
Name:                             Date: October 28, 2026
Functions · difficulty 1 of 5 · about 1 min · position P1-10

For positive numbers xx and yy the operation x♠yx\spadesuit y is defined as

x♠y=x−1yx\spadesuit y = x -\dfrac{1}{y}

What is 2♠(2♠2)2\spadesuit (2\spadesuit 2) ?

  1. A)23\dfrac{2}{3}
  2. B)11
  3. C)43\dfrac{4}{3}
  4. D)53\dfrac{5}{3}
  5. E)22

Answer

C

Key insight

Do the inside first: 2 spade 2 = 3/2. Then 2 minus 1 divided by 3/2 is 2 - 2/3 = 4/3.

Solution

The rule says: take the first number and subtract 11 divided by the second number. Work from the inside of the brackets outward.

1. Inside bracket. 2♠22\spadesuit 2 means x=2x = 2 and y=2y = 2:
2−12=32.2 - \frac{1}{2} = \frac{3}{2}.
2. Outside. Now the problem is 2♠322\spadesuit \tfrac32, so x=2x = 2 and y=32y = \tfrac32:
2−13/2.2 - \frac{1}{3/2}.
3. Dividing 11 by 32\tfrac32 flips the fraction: 13/2=23\dfrac{1}{3/2} = \dfrac{2}{3}.
4. So the value is 2−23=63−23=432 - \dfrac23 = \dfrac63 - \dfrac23 = \dfrac43.

The answer is (C) 43\boxed{\textbf{(C)}\ \dfrac{4}{3}}.

Why this works

A made-up symbol is just a rule for combining two numbers. The brackets tell you which part to do first, and that result becomes the second number in the next step. The only tricky part here is 11 divided by a fraction, which is that fraction turned upside down.

The trap

Thinking 1 divided by 3/2 is 3/2 instead of 2/3, which gives 1/2, not a choice, and leads to guessing.

Concise solution

Work from the inside out. The inner expression is
2♠2=2−12=32.2\spadesuit 2 = 2 - \frac{1}{2} = \frac{3}{2}.

Now substitute that as the second input:
2♠32=2−13/2=2−23=43.2\spadesuit \frac{3}{2} = 2 - \frac{1}{3/2} = 2 - \frac{2}{3} = \frac{4}{3}.

The answer is (C) 43\boxed{\textbf{(C)}\ \dfrac{4}{3}}.

Review in the AoPS books

Techniques

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