Mathematical Food for Thought

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Pythagorean x3. Topic: Geometry/NT. Level: AMC. February 22nd, 2007

Problem: (2007 AMC12B – #23) How many non-congruent right triangles with positive integer leg lengths have areas that are numerically equal to times their perimeters?

Solution: Well, basically, you should know the Pythagorean triple generating formula, i.e. , , . Substitute accordingly and we have to solve the diophantine equation

which conveniently simplifies to

.

Obviously then so look at these cases:

: We can take to get the triples .

: We can take to get the triples both of which are already counted.

: We can take to get the triples .

: We can take to get the triple , which is already counted.

So we have triangles. QED.

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Comment: Not too hard if you knew the generating formula for Pythagorean triples. It was a little annoying having to check for repeated triples, but at least there weren’t that many.

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Practice Problem: (2007 AMC 12B – #24) How many pairs of positive integers are there such that and

is an integer?

2 Responses to “Pythagorean x3. Topic: Geometry/NT. Level: AMC.”

1. t0rajir0u Says:

http://www.artofproblemsolving.com/Forum/weblog_entry.php?t=135187