Difference between revisions of "2020 CAMO Problems/Problem 1"
(→Solution) |
(→Solution) |
||
| Line 8: | Line 8: | ||
It's obvious that if there exists two real numbers <math>x</math> and <math>y</math>, which satisfies <cmath>f(x)=\frac{a^x-1}{a^x+1}</cmath> and <cmath>f(y)=\frac{a^y-1}{a^y+1}</cmath> | It's obvious that if there exists two real numbers <math>x</math> and <math>y</math>, which satisfies <cmath>f(x)=\frac{a^x-1}{a^x+1}</cmath> and <cmath>f(y)=\frac{a^y-1}{a^y+1}</cmath> | ||
| − | Then, for <math>f(x+y)</math>, <cmath>f(x+y)=\frac{f(x)+f(y)}{1+f(x)*f(y)}</cmath>, <cmath>f(x+y)=\frac{2*a^x+y-2}{2*a^x+y+2}</cmath> | + | Then, for <math>f(x+y)</math>, <cmath>f(x+y)=\frac{f(x)+f(y)}{1+f(x)*f(y)}</cmath>, <cmath>f(x+y)=\frac{2*a^x+y -2}{2*a^x+y +2}</cmath> |
| − | Then, <cmath>f(x+y)=\frac{a^x+y-1}{a^x+y+1}</cmath> | + | Then, <cmath>f(x+y)=\frac{a^x+y -1}{a^x+y +1}</cmath> |
The fraction is also satisfies for <math>f(x+y)</math> | The fraction is also satisfies for <math>f(x+y)</math> | ||
Revision as of 09:27, 3 October 2023
Problem 1
Let 
(meaning 
takes positive real numbers to positive real numbers) be a nonconstant function such that for any positive real numbers 
and 
, ![\[f(x)f(y)f(x+y)=f(x)+f(y)-f(x+y).\]](http://latex.artofproblemsolving.com/f/b/f/fbfc1781f9e0e0468cfe295fe4242ac8843e1cd2.png)
Prove that there is a constant 
such that ![\[f(x)=\frac{a^x-1}{a^x+1}\]](http://latex.artofproblemsolving.com/f/5/3/f53c551cb8a20af7ac1680b0189a9195cae3b7fe.png)
for all positive real numbers .
Solution
This problem needs a solution. If you have a solution for it, please help us out by adding it.
Because 
, we can find that ![\[f(x+y)=\frac{f(x)+f(y)}{1+f(x)*f(y)}\]](http://latex.artofproblemsolving.com/c/c/1/cc11c69980791dce2103382a323199dff5ecde09.png)
It's obvious that if there exists two real numbers and , which satisfies and ![\[f(y)=\frac{a^y-1}{a^y+1}\]](http://latex.artofproblemsolving.com/f/d/3/fd3de3e55e294e80aa44dca9f7ab616ac963ec0f.png)
Then, for 
, , ![\[f(x+y)=\frac{2*a^x+y -2}{2*a^x+y +2}\]](http://latex.artofproblemsolving.com/a/1/4/a1477b20749b560ac6433946c5e5eef0f6127ecc.png)
Then, ![\[f(x+y)=\frac{a^x+y -1}{a^x+y +1}\]](http://latex.artofproblemsolving.com/6/6/9/669848f7152c76ee2a523ed3cd2257952d8826c1.png)
The fraction is also satisfies for
Then, we can solve this problem using mathematical induction
~~Andy666
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 | ||
The problems on this page are copyrighted by the MAC's Christmas Mathematics Competitions. 
