2002 AMC 12P Problems/Problem 13
- The following problem is from both the 2002 AMC 12P #13 and 2002 AMC 10P #24, so both problems redirect to this page.
Problem
What is the maximum value of for which there is a set of distinct positive integers for which
Solution
Note that
When , .
When , .
Therefore, we know .
Now we must show that works. We replace some integer within the set with an integer to account for the amount under , which is .
Essentially, this boils down to writing as a difference of squares. Assume there exist positive integers and where and such that .
We can rewrite this as . Since , either and or and . We analyze each case separately.
Case 1: and
Solving this system of equations gives and . However, , so this case does not yield a solution.
Case 2: and
Solving this system of equations gives and . This satisfies all the requirements of the problem.
The list has terms whose sum of squares equals . Since is impossible, the answer is .
See also
2002 AMC 10P (Problems • Answer Key • Resources) | ||
Preceded by Problem 23 |
Followed by Problem 25 | |
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 | ||
All AMC 10 Problems and Solutions |
2002 AMC 12P (Problems • Answer Key • Resources) | |
Preceded by Problem 12 |
Followed by Problem 14 |
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 | |
All AMC 12 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.