2020 CAMO Problems/Problem 1
Contents
[hide]Problem 1
Let (meaning takes positive real numbers to positive real numbers) be a nonconstant function such that for any positive real numbers and , Prove that there is a constant such that for all positive real numbers .
Solution
Because , we can find that It's obvious that if there exists two real numbers and , which satisfies and
Then, for , ,
Then,
The fraction is also satisfies for
Then, we can solve this problem using mathematical induction
~~Andy666
Solution (2)
Let denote a substitution of for and be the inverse of when it exists.
By we get so the domain of (x) must be in the interval
(*) from here,
Taking
so let for some real constant .
by substitution into (*); we know that so so
so
where
-Shushninja
See also
2020 CAMO (Problems • Resources) | ||
Preceded by First problem |
Followed by Problem 2 | |
1 • 2 • 3 • 4 • 5 • 6 | ||
All CAMO Problems and Solutions |
2020 CJMO (Problems • Resources) | ||
Preceded by Problem 1 |
Followed by Problem 3 | |
1 • 2 • 3 • 4 • 5 • 6 | ||
All CJMO Problems and Solutions |
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