Find all functions  $f: \mathbb{Z} \rightarrow \mathbb{Z}$ such that  $f(2n)+2f(2m)=f(f(n+m))$,   $\forall m,n\in \mathbb{Z}$

Find all functions $f: \mathbb{Z} \rightarrow \mathbb{Z}$ such that $f(2n)+2f(2m)=f(f(n+m))$, $\forall m,n\in \mathbb{Z}$.

Math Competitions Putnam Competition Calculus
Paul F Paul F
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  • Paul F Paul F

    There is a typo: f(2n)+2f(m)=f(f(n+m))

  • Kr1Staps Kr1Staps

    just for clarification you're saying that the typo, that is the *incorrect* relation is: f(2n)+2f(2m)=f(f(n+m)), and that the CORRECT relation is: f(2n)+2f(m)=f(f(n+m))?

  • Paul F Paul F

    Yes

Answer

PDF attached

Kr1Staps Kr1Staps
Matchmaticians Find all functions  $f: \mathbb{Z} \rightarrow \mathbb{Z}$ such that  $f(2n)+2f(2m)=f(f(n+m))$,   $\forall m,n\in \mathbb{Z}$ File #1 File #1 (pdf)
  • Paul F Paul F

    Thank you so much. This is amazing.

The answer is accepted.