2008 IMO Problems/Problem 4
Problem
Find all functions (so
is a function from the positive real numbers) such that
for all positive real numbes satisfying
Solution
Considering and
which satisfy the constraint
we get the following equation:
\[\frac{(f(1))^2 + (f(x))^2}{f(x) + f(x)} = \frac{1+x^2}{x+x} \Leftrightarrow x((f(1))^2 + (f(x))^2}) = (1+x^2)f(x)\] (Error compiling LaTeX. Unknown error_msg)
At once considering we get
and kowing that
the only possible solution is
since
is impossible.
So we get the quadratic equation:
\[x(f(x))^2} - (1+x^2)f(x) + x = 0\] (Error compiling LaTeX. Unknown error_msg)
Solving for as a function of
we get:
At once we see that for one value of ,
can only take one of 2 possible values:
.
Take into consideration that but
verifies the quadratic equation and thus so far we can't say that $f(x)=x \, \forall_{x \x \in \mathbb{R}^+}$ (Error compiling LaTeX. Unknown error_msg) or alternatively $f(x)=\frac{1}{x} \, \forall_{x \x \in \mathbb{R}^+\\}$ (Error compiling LaTeX. Unknown error_msg). This is indeed the case but we haven't proved it yet.
To prove the previous assertion consider 2 values such that
while having
Consider now the original functional equation with which verifies the constraint. Substituting we have:
Now either or
. (notice that
by hypothesis)
If then we have
and since
the only solution is
.
If then we have
and since
the only solution is
.
So the only solutions are or
in which case both alternatives imply
. Thus we conclude that solutions to the functional equation are a subset of $\left\{f(x)=x \ \forall_{x \x \in \mathbb{R}^+},\ f(x)=\frac{1}{x}\ \forall_{x \x \in \mathbb{R}^+} \right\}$ (Error compiling LaTeX. Unknown error_msg).
Finally plug each of these 2 functions into the functional equation and verify that they indeed are solutions.
This is trivial since is an obvious solution and for
we have:
provided that
which is the original constraint.
So the functional equation has 2 solutions:
\[f(x) = x\ \forall_{x \x \in \mathbb{R}^+}\ \vee\ f(x)=\frac{1}{x}\ \forall_{x \x \in \mathbb{R}^+}\] (Error compiling LaTeX. Unknown error_msg)