Difference between revisions of "2014 AIME II Problems/Problem 12"
Futurewriter (talk | contribs) (→Solution 2) |
m (→Solution 1) |
||
Line 21: | Line 21: | ||
<math>\framebox{399}</math> | <math>\framebox{399}</math> | ||
− | <math>\color{red}{\text{Note: This solution forgot the case of dividing by 0}}</math> | + | <math>\color{red}{\text{Note: This solution forgot the case of dividing by 0}} \leftarrow \text{Don't we all love trolls?}</math> |
==Solution 2== | ==Solution 2== |
Revision as of 22:18, 25 February 2017
Contents
Problem
Suppose that the angles of satisfy Two sides of the triangle have lengths 10 and 13. There is a positive integer so that the maximum possible length for the remaining side of is Find
Solution 1
Note that . Thus, our expression is of the form . Let and .
Using the fact that , we get , or .
Squaring both sides, we get . Cancelling factors, .
Expanding, .
Simplification leads to and .
Therefore, . So could be or . We eliminate and use law of cosines to get our answer:
Solution 2
As above, we can see that
Expanding, we get
Note that , or
Thus , or .
Now we know that , so we can just use the Law of Cosines to get
See also
2014 AIME II (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 | ||
All AIME Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.