Difference between revisions of "2003 Indonesia MO Problems/Problem 8"
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==Problem== | ==Problem== | ||
| − | Given a triangle <math>ABC</math> with <math>C</math> as the right angle, and the sidelengths of the triangle are integers. Determine the | + | Given a triangle <math>ABC</math> with <math>C</math> as the right angle, and the sidelengths of the triangle are integers. Determine the possible side lengths of the triangle if the product of the legs of the right triangle equals to three times the perimeter of the triangle. |
==Solution== | ==Solution== | ||
Latest revision as of 01:19, 19 May 2024
Problem
Given a triangle 
with 
as the right angle, and the sidelengths of the triangle are integers. Determine the possible side lengths of the triangle if the product of the legs of the right triangle equals to three times the perimeter of the triangle.
Solution
Let 
and 
be the legs of the triangle. We can express the comparison as an equation.
![\[ab = 3(a+b+\sqrt{a^2 + b^2})\]](http://latex.artofproblemsolving.com/a/a/9/aa99fa0563c57a9bdb71f6c2d6cb2996df1e7438.png)
We can manipulate the equation to find a relationship between and 
Since 
we can divide both sides by 
without affecting the solution.
![\[ab-6a-6b+18 = 0\]](http://latex.artofproblemsolving.com/0/c/9/0c98b9d794e4422738e7c3f5cabec1e9e3c4561b.png)
We can use Simon's Favorite Factoring Trick to find integral solutions.
![\[(a-6)(b-6) =18\]](http://latex.artofproblemsolving.com/3/c/f/3cf182afacea4bb5e8dd04e38533a9d9a9cc39de.png)
From the equation, we find that the three possible triangles can have the following side lengths: 
See Also
| 2003 Indonesia MO (Problems) | ||
| Preceded by Problem 7 |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 | Followed by Last Problem |
| All Indonesia MO Problems and Solutions | ||
