Problem 14 · 2012 Math Kangaroo
Medium
Algebra & Patterns
arithmetic-series
For a ski race consecutive starting numbers are handed out. One number was accidentally given out twice. The sum of all the numbers handed out is 857. Which number was given out twice?
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Answer: D — 37
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Hint 1 of 2
First add up 1+2+…+n and see which n lands just below 857.
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Hint 2 of 2
The leftover above that triangular number is the repeated starting number.
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Approach: match a triangular number, then read the extra
- If the numbers run \(1\) to \(n\), their sum is \(\frac{n(n+1)}{2}\) and the repeated number adds a little extra.
- \(1+\cdots+40 = 820\), and \(857 - 820 = 37\), which is a valid number between 1 and 40 (while \(1+\cdots+41 = 861\) is already too big).
- So the repeated number is 37, choice D.
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