Difference between revisions of "2006 iTest Problems/Problem U7"
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Revision as of 17:04, 19 March 2020
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 maximize and
must be minimized. 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).
From the Pythagorean theorem, , applying difference of squares yields
. Since the question states
and
must be integers, we can find possible values of
and
by finding the prime factorization of
, which is
. The two values of
and
that are closest to each other are the values that satisfy
, and
. Solving the system yields
and
. Thus, the perimeter is
~Someonenumber011