Problem 18 · 2023 Math Kangaroo
Medium
Logic & Word Problems
Number Theory
work-backwardsum-constraint
Max and two of his friends are standing in a line. The number of people in the line is a multiple of 3. He notices that there are the same number of people in front of him as there are behind him. Both of his friends are behind him: one is in position 19, the other in position 28 of the line. In which position of the line is Max?
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Answer: D — 17
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Hint 1 of 2
Equal numbers in front and behind means Max is exactly in the middle, so the line length is odd.
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Hint 2 of 2
The length is a multiple of 3 and at least 28; the smallest odd multiple of 3 that is ≥ 28 fixes everything.
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Approach: use the middle position and the multiple-of-3, length-≥28 conditions
- Equal people in front and behind put Max in the middle, so the total number is odd.
- The total is a multiple of 3 and must be at least 28 (a friend stands at position 28), so the smallest such odd value is 33.
- Max's middle position in a line of 33 is position (33+1)/2 = 17.
- So the answer is 17 (D).
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