Difference between revisions of "2016 APMO Problems/Problem 5"
(Created page with "==Problem== Find all functions <math>f: \mathbb{R}^+ \to \mathbb{R}^+</math> such that <cmath>(z + 1)f(x + y) = f(xf(z) + y) + f(yf(z) + x),</cmath>for all positive real numb...") |
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| + | We claim that <math>f(x)=x</math> is the only solution. It is easy to check that it works. Now, we will break things down in several claims. | ||
| + | |||
| + | [b]Claim 1:[/b] <math>f</math> is injective. | ||
Revision as of 23:31, 12 July 2021
Problem
Find all functions 
such that
![\[(z + 1)f(x + y) = f(xf(z) + y) + f(yf(z) + x),\]](http://latex.artofproblemsolving.com/7/9/2/79270c212bc2861a5e55e9316294cbd32ef66eaf.png)
for all positive real numbers 
.
Solution
We claim that 
is the only solution. It is easy to check that it works. Now, we will break things down in several claims.
[b]Claim 1:[/b] 
is injective.
