2005 Canadian MO Problems/Problem 2
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
We have
By AM-GM, we have
where is a positive real number not equal to one. If
, then
. Thus
and
. Therefore,