2020 CAMO Problems/Problem 1

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Problem 1

Let $f:\mathbb R_{>0}\to\mathbb R_{>0}$ (meaning $f$ takes positive real numbers to positive real numbers) be a nonconstant function such that for any positive real numbers $x$ and $y$, \[f(x)f(y)f(x+y)=f(x)+f(y)-f(x+y).\]Prove that there is a constant $a>1$ such that \[f(x)=\frac{a^x-1}{a^x+1}\]for all positive real numbers $x$.

Solution

Because $f(x)f(y)f(x+y)=f(x)+f(y)-f(x+y)$, we can find that \[f(x+y)=\frac{f(x)+f(y)}{1+f(x)*f(y)}\] It's obvious that if there exists two real numbers $x$ and $y$, which satisfies \[f(x)=\frac{a^x-1}{a^x+1}\] and \[f(y)=\frac{a^y-1}{a^y+1}\]

Then, for $f(x+y)$, \[f(x+y)=\frac{f(x)+f(y)}{1+f(x)*f(y)}\], \[f(x+y)=\frac{2*a^x*a^y-2}{2*a^x*a^y+2}\]

Then, \[f(x+y)=\frac{a^x*a^y-1}{a^x*a^y+1}\]

The fraction is also satisfies for $f(x+y)$

Then, we can solve this problem using mathematical induction

~~Andy666

See also

2020 CAMO (ProblemsResources)
Preceded by
First problem
Followed by
Problem 2
1 2 3 4 5 6
All CAMO Problems and Solutions
2020 CJMO (ProblemsResources)
Preceded by
Problem 1
Followed by
Problem 3
1 2 3 4 5 6
All CJMO Problems and Solutions

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