Difference between revisions of "2010 AIME I Problems/Problem 5"
Tempaccount (talk | contribs) (Adding problem section) |
Tempaccount (talk | contribs) (Remove extra problem section) |
||
Line 1: | Line 1: | ||
− | |||
− | |||
__TOC__ | __TOC__ | ||
== Problem == | == Problem == |
Revision as of 16:02, 9 August 2018
Contents
[hide]Problem
Positive integers , , , and satisfy , , and . Find the number of possible values of .
Solution
Solution 1
Using the difference of squares, , where equality must hold so and . Then we see is maximal and is minimal, so the answer is .
Solution 2
Since must be greater than , it follows that the only possible value for is (otherwise the quantity would be greater than ). Therefore the only possible ordered pairs for are , , ... , , so has possible values.
See Also
2010 AIME I (Problems • Answer Key • Resources) | ||
Preceded by Problem 4 |
Followed by Problem 6 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.