Difference between revisions of "2006 iTest Problems/Problem U7"

(Problem)
(Solution)
Line 4: Line 4:
  
 
==Solution==
 
==Solution==
First, label the other leg <math>x</math> and the hypotenuse <math>y</math>. To minimize
+
 
 +
First, label the other leg <math>x</math> and the hypotenuse <math>y</math>. To minimize <math>\frac{s}{r}</math>, <math>r</math> must be minimized and <math>s</math> must be maximized. Through logic, it becomes clear that the triangle must be as close to equilateral as possible to maximize <math>r</math> and minimize <math>s</math>(Think about stretching one vertice of an equilateral triangle. The perimeter increases faster than the inradius).

Revision as of 21:42, 17 November 2019

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).

Invalid username
Login to AoPS