Difference between revisions of "2005 Canadian MO Problems/Problem 2"
Line 14: | Line 14: | ||
*[[2005 Canadian MO Problems/Problem 1|Previous problem]] | *[[2005 Canadian MO Problems/Problem 1|Previous problem]] | ||
*[[2005 Canadian MO]] | *[[2005 Canadian MO]] | ||
+ | |||
+ | |||
+ | [[Category:Olympiad Number Theory Problems]] |
Revision as of 12:54, 4 September 2006
Problem
Let be a Pythagorean triple, i.e., a triplet of positive integers with
.
- Prove that
.
- Prove that there does not exist any integer
for which we can find a Pythagorean triple
satisfying
.
Solution
First part:
. By AM-GM we have
if
is a positive real number other than 1. If
then
so
and
and
and thus
.