Problem:
For how many integers does a triangle with side lengths , and have all its angles acute?
Answer Choices:
A.
B.
C.
D.
E. more than
Solution:
In any triangle with sides , the angle opposite is acute iff . This follows from the Law of Cosines. Applying this fact in turn to the angle opposite and , we find
The first line tells us that . The second tells us that (since is an integer). The third is satisfied for every . Thus there are integer values which meet all the conditions: