Problem 4 · 2024 AMC 8
Medium
Number Theory
digit-sumperfect-squarework-backward
When Yunji added all the integers from 1 to 9, she mistakenly left out a number. Her incorrect sum turned out to be a square number. What number did Yunji leave out?
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Answer: E — She left out 9.
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Hint 1 of 2
Don't test all nine cases. Find the FULL total 1+2+…+9 first — the answer is that total minus one small number, so it lands just below it. Which perfect square is nearby?
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Hint 2 of 2
Technique: full sum is 45 (pair 1+9, 2+8, …). Removing x gives 45 − x, somewhere in 36–44. Which value lands exactly on a perfect square?
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Approach: the full sum minus the missing number is a perfect square
- The insight: start from the FULL total and only then take one number away. 1 + 2 + … + 9 = 45 (it pairs up: 1+9, 2+8, 3+7, 4+6, and a lonely 5 — four 10s plus 5).
- Dropping a number from 1 to 9 leaves a sum between 45−9 = 36 and 45−1 = 44. So we just need a perfect square in the window 36–44.
- Only 36 = 62 lives there (the next square, 49, is too big). So the sum became 36.
- Removed amount = 45 − 36 = 9. Why this transfers: when something is "almost" a known total, compute the whole thing first, then the small correction — far easier than testing every number.
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