2015 AMC 8 Problem 23
Attempt Problem 23 of the 2015 AMC 8 below, then check your answer against the video solution and professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2015 AMC 8 solutions, or check the answer key.
All problems are used with official legal permission of the Mathematical Association of America (MAA).
23.
Tom has twelve slips of paper which he wants to put into five cups labeled
He wants the sum of the numbers on the slips in each cup to be an integer. Furthermore, he wants the five integers to be consecutive and increasing from to The numbers on the papers are If a slip with goes into cup and a slip with goes into cup then the slip with must go into what cup?
Answer: D
Small hint:
The five cup sums must be consecutive integers with total
Big hint:
After filling cup test where the slip can still leave possible sums.
Video solution:
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Written solution:
The sum of all the slips is so the five consecutive integer cup sums must average Therefore cups and must have sums and respectively.
Cup already contains a and must sum to so it must contain another Cup already contains a so the other slips in must sum to
The slip cannot go in because cup would need another It cannot go in which is already full. It cannot go in or because either would then need another and no remaining slips can make that total. Cup works, for example with
Thus, D is the correct answer.
Problem 23 in other years
- 1985 AMC 8
- 1986 AMC 8
- 1987 AMC 8
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- 1989 AMC 8
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- 1991 AMC 8
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