2006 iTest Problems/Problem U7
Problem
Triangle has integer side lengths, including , and a right angle, . Let and denote the inradius and semiperimeter of respectively. Find the perimeter of the triangle ABC which minimizes .
Solutions
Solution 1 (credit to NikoIsLife)
Let and . By the Pythagorean Theorem, , and applying difference of squares yields . Because and have the same parity (due to being integers), both and are even.
Let abd ; then . Additionally,
Therefore,
Note that because , we must have . We can do some optimization by using the derivative -- if we let , then
which equals if . Since if and if , we can confirm that results in the absolute minimum of . However, the case where does not happen if are integers, and since is a factor of , we need to test the largest factor of less than and the smallest factor of greater than .
The largest factor of less than is (which does not work), and the smallest factor of greater than is . Therefore, , which means that , , and . Our wanted perimeter is .
Solution 2
As before, label the other leg and the hypotenuse . Let the opposite angle to be , and let ; let the area be and the semiperimeter . Then we have . This means that . By calculus, we know that this function is minimized at , which corresponds to and ; by geometry, we know that this function, expressed in terms of , is symmetric around this point.
Then we proceed as before, searching for Diophantine solutions of with closest to , and we find that is the closest. (We can do so by noting that we would want .) Then the perimeter is as before, and we are done.
~duck_master
Alternate solutions are always welcome. If you have a different, elegant solution to this problem, please add it to this page.
See Also
2006 iTest (Problems) | ||
Preceded by: Problem U6 |
Followed by: Problem U8 | |
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