Problem 22 · 2010 Math Kangaroo
Stretch
Logic & Word Problems
sum-constraintwork-backward
100 people take part in a race where no one can tie. Everybody is questioned after the race as to which place they have achieved and all answer with a number between 1 and 100. The sum of all answers is 4000. What is the minimum number of people who have lied about their result?
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Answer: D — 12
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Hint 1 of 2
If everyone told the truth the answers would sum to 1+2+...+100.
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Hint 2 of 2
Each liar can only pull the total down by so much; find the most one liar removes.
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Approach: compare the true sum with the stated sum
- Truthful answers sum to 5050, but the stated total is 4000, short by 1050.
- A truthful person states their real rank; a liar can drop their claim, removing at most (rank − 1).
- The 11 highest ranks lying remove at most 99+98+...+89 = 1034, not enough.
- So at least 12 people must have lied.
Mark:
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