A person takes at least one aspirin a day for 30 days. If he takes 45 aspirin altogether, in some sequence of consecutive days he takes exactly 14 aspirin.

Solution


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Copyright © 1996-2017 Alexander Bogomolny

A person takes at least one aspirin a day for 30 days. If he takes 45 aspirin altogether, in some sequence of consecutive days he takes exactly 14 aspirin.

Let ai be the total number of aspirin consumed up to and including the ith day, for i = 1, ..., 30. Combine these with the numbers a1 + 14, ..., a30 + 14, providing 60 numbers, all positive and less or equal 45 + 14 = 59. Hence two of these 60 numbers are identical. Since all ai's and, hence, (ai + 14)'s are distinct (at least one aspirin a day consumed), then aj = ai + 14, for some i<j. Thus, on days i + 1 to j, the person consumes exactly 14 aspirin.

In [Engel, p. 60] we find the following variant: A chessmaster has 77 days to prepare for a tournament. He wants to play at least one game per day, but no more than 132 games. Prove that there is a sequence of successive days on which he plays exactly 21 games.

References

  1. A. Engel, Problem-Solving Strategies, Springer Verlag, 1998

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