2006 iTest Problems/Problem U7

Revision as of 21:42, 17 November 2019 by Someonenumber011 (talk | contribs) (Solution)

Problem

Triangle $ABC$ has integer side lengths, including $BC  =  696$, and a right angle, $\angle ABC$. Let $r$ and $s$ denote the inradius and semiperimeter of $ABC$ respectively. Find the perimeter of the triangle ABC which minimizes $\frac{s}{r}$.

Solution

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

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