Mock AIME 1 2006-2007 Problems/Problem 1
has positive integer side lengths of ,, and . The angle bisector of hits at . If , and the maximum value of where and are relatively prime positive intgers, find . (Note denotes the area of ).
Assume without loss of generality that . Then the hypotenuse of right triangle either has length 17, in which case , or has length , in which case , by the Pythagorean Theorem.
In the first case, you can either know your Pythagorean triples or do a bit of casework to find that the only solution is . In the second case, we have , a factorization as a product of two different positive integers, so we must have and from which we get the solution .
Now, note that the area and , and since is an angle bisector we have so .
In our first case, this value may be either or . In the second, it may be either or . Of these four values, the last is clearly the greatest. 17 and 145 are relatively prime, so our answer is .