Problem 2 · 2000 AMC 8
Easy
Arithmetic & Operations
number-systems
Which of these numbers is less than its reciprocal?
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Answer: A — −2.
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Hint 1 of 2
A reciprocal flips a number toward 1. For a positive number bigger than 1, flipping makes it smaller — so the answer must be where flipping makes it *bigger*. Which sign of number does that?
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Hint 2 of 2
Flipping (taking 1/x) shrinks numbers far from 0 toward ±1. So a number 'less than its reciprocal' has to be a negative number to the LEFT of −1 — that's the family to look in.
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Approach: think about which way the flip moves each candidate
- Knock out the trick cases instantly: 0 has no reciprocal, and 1 and −1 are their own reciprocals (so they tie, not 'less than').
- That leaves 2 and −2. Flipping 2 gives ½, which is smaller — so 2 is *more* than its reciprocal. Flipping −2 gives −½, and −2 < −½, so −2 IS less than its reciprocal. Answer −2.
- The intuition: on the number line, taking a reciprocal pulls everything toward ±1. A negative number left of −1 gets pulled rightward (bigger), so it ends up below its reciprocal — that's the only way the inequality can hold.
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