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I was trying to solve the following problem:

Find all functions $f:(0,\infty)\to\Bbb R$ satisfying $$f(x)f(y)+f\left(\frac5x\right)f\left(\frac5y\right)=2f(xy)$$ for every $x,y\in(0,\infty)$ and $f(5)=1$.

It's easy to show that $f(1)=1$ (take $ x=y=1$). If we take $y=\frac5x$, we obtain $f(x)f(y)=f(xy)$ and $f(x)=f\left(\frac5x\right)$. Therefore $f(x)^2=1$ for every $x>0$.

Here is my problem: The solution should be $f(x)\equiv 1$ and I can't show that (I tried contradiction).

Help!

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  • $\begingroup$ I got $f(x)f(5/x)=1$ how did you get $f(x)=f(5/x)$ ? $\endgroup$ – Deepesh Meena Aug 22 '17 at 4:37
  • $\begingroup$ Taking y=1, f(x)+f(5/x)=2f(x), then f(5/x)=f(x) $\endgroup$ – Ángela Flores Aug 22 '17 at 4:40
  • $\begingroup$ For the record, where did you get this problem from? $\endgroup$ – Parcly Taxel Aug 22 '17 at 6:10
  • $\begingroup$ I get it from a test for local high school students. I'll try to find the original source. $\endgroup$ – Ángela Flores Aug 22 '17 at 6:14
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$f(x)=1$ is not the only function satisfying the given relation.

Define an auxiliary function $g(x)=f(5^x)$, then the given relation is $$g(x)g(y)+g(1-x)g(1-y)=2g(x+y)$$ and the derived equations are $$g(x)g(y)=g(x+y)\tag1$$ $$g(x)=g(1-x)$$ $$g(0)=g(1)=1\tag2$$ From the last two equations above we can get $$g(x)=g(-x)\tag3$$ (corresponding to $f(x)=f(1/x)$). From equations $(1,2,3)$ we can derive the original relation and conditions, so we lose nothing by using only them.

It is clear that $g(x)=1$ for all integer $x$, but we can set $g(x)=-1$ for all non-integer $x$ and still satisfy all relations. In general, we can have for an arbitrary $n\in\Bbb N^+$: $$g(x)=\begin{cases}1&x\in\frac1n\Bbb Z\\-1&\text{otherwise}\end{cases}$$ and the relations would still be satisfied. Because $\frac1n\Bbb Z$ (the set of numbers whose product with $n$ is an integer) is closed under addition and subtraction, $g(x)=1$ is only fixed for those $x$ that are members of that set, leaving us free to map non-members to 1. Of course, $g(x)=1$ for all $x$ is also a solution.

Translated back, $f(x)$ is either constant 1 (the given solution) or $$\begin{cases}1&\log_5x\in\frac1n\Bbb Z\\-1&\text{otherwise}\end{cases}$$ where $n\in\Bbb N^+$.

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