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 .

Solution

First, label the other leg and the hypotenuse . To minimize , must be minimized and must be maximized. Through logic, it becomes clear that the triangle must be as close to equilateral as possible to maximize and minimize (Think about stretching one vertice of an equilateral triangle. The perimeter increases faster than the inradius).