1976 AHSME Problems/Problem 12
Problem 12
A supermarket has crates of apples. Each crate contains at least apples and at most apples. What is the largest integer such that there must be at least crates containing the same number of apples?
Solution
To find the largest number of "repeated" crates necessary, we must account for all the possibilities of the number of apples in each crate. Since each crate contains a minimum of apples and a maximum of apples, there are different amounts possible for the number of apples per crate.
Now, we have to count for the worst case scenario: the amounts are repeated as many times as possible.
can go into exactly times because , which is less than . This leaves a remainder of crates.
The worst case scenario would be that these crates have a different number of apples each. It doesn't actually matter how many apples are in these crates because any of the values would be repeated again anyway. So, the answer is ~jiang147369
See Also
1976 AHSME (Problems • Answer Key • Resources) | ||
Preceded by Problem 11 |
Followed by Problem 13 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30 | ||
All AHSME Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.