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Difference between revisions of "2016 APMO Problems/Problem 5"

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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.
 
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.
+
<b>Claim 1:</b> <math>f</math> is injective.
 +
 
 +
<i>Proof:</i>

Revision as of 23:34, 12 July 2021

Problem

Find all functions $f: \mathbb{R}^+ \to \mathbb{R}^+$ such that \[(z + 1)f(x + y) = f(xf(z) + y) + f(yf(z) + x),\]for all positive real numbers $x, y, z$.

Solution

We claim that $f(x)=x$ is the only solution. It is easy to check that it works. Now, we will break things down in several claims.

Claim 1: $f$ is injective.

Proof:

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