Difference between revisions of "2003 Indonesia MO Problems/Problem 8"
Rockmanex3 (talk | contribs) (Solution to Problem 8 — Pythagorean Triples and number theory) |
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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 02:19, 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.
We can manipulate the equation to find a relationship between
and
Since
we can divide both sides by
without affecting the solution.
We can use Simon's Favorite Factoring Trick to find integral solutions.
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 |