2006 iTest Problems/Problem U7
Contents
[hide]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, Answer Key) | ||
Preceded by: Problem U6 |
Followed by: Problem U8 | |
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