Problem 13 · 2013 Math Kangaroo
Hard
Logic & Word Problems
sum-constraintcasework
36 children each voted once for one of five students in their class. The winner received 12 votes and the student placed last received just 4 votes. If every student received a different number of votes, how many votes did the second-placed student receive?
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Answer: B — 8 or 9
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Hint 1 of 3
The five vote counts are all different and add up to 36, with 12 on top and 4 at the bottom.
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Hint 2 of 3
Subtract the known 12 and 4 to find what the middle three must total.
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Hint 3 of 3
The second-place student has the biggest of those three middle counts, so test which biggest values can work.
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Approach: pin the middle three to sum 20 with distinct values
- The top has 12 and the bottom has 4, so the other three students share \(36 - 12 - 4 = 20\) votes.
- Those three are all different and each is strictly between 4 and 12, and the second-place count is the largest of the three.
- If second place got 9, the others could be 5 and 6 (\(9+6+5=20\)); if second got 8, the others could be 5 and 7 (\(8+7+5=20\)); but 10 forces the other two to sum 10 with distinct values above 4, which is impossible.
- So second place got 8 or 9, which is choice B.
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