Question

22 Saint-Petersburg - Russia, 18th-28th September 2020 A6. Determine all functions \( f: \mathbb{Z} \rightarrow \mathbb{Z} \) such that \[ f^{a^{2}+b^{2}}(a+b)=a f(a)+b f(b) \quad \text { for every } a, b \in \mathbb{Z} \] Here, \( f^{n} \) denotes the \( n^{\text {th }} \) iteration of \( f \), i.e., \( f^{0}(x)=x \) and \( f^{n+1}(x)=f\left(f^{n}(x)\right) \) for all \( n \geqslant 0 \). (Slovakia)

          22
Saint-Petersburg - Russia, 18th-28th September 2020

A6. Determine all functions \( f: \mathbb{Z} \rightarrow \mathbb{Z} \) such that
\[
f^{a^{2}+b^{2}}(a+b)=a f(a)+b f(b) \quad \text { for every } a, b \in \mathbb{Z}
\]

Here, \( f^{n} \) denotes the \( n^{\text {th }} \) iteration of \( f \), i.e., \( f^{0}(x)=x \) and \( f^{n+1}(x)=f\left(f^{n}(x)\right) \) for all \( n \geqslant 0 \).
(Slovakia)

22
Saint-Petersburg - Russia, 18th-28th September 2020

A6. Determine all functions f: ℤ→ℤ such that

f^a^2+b^2(a+b)=a f(a)+b f(b) for every a, b ∈ℤ

Here, f^n denotes the n^th iteration of f, i.e., f^0(x)=x and f^n+1(x)=f(f^n(x)) for all n ⩾ 0.
(Slovakia)

Added by Darnell C.

Elementary and Intermediate Algebra

Alan S. Tussy, R. David Gustafson 5th Edition

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Solved by Expert Gaurav Gupta

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Step 1: Understand the problem We need to find all functions \( f: \mathbb{Z} \rightarrow \mathbb{Z} \) such that for every \( a, b \in \mathbb{Z} \), the equation \[ f^{a^{2}+b^{2}}(a+b)=a f(a)+b f(b) \] holds, where \( f^{n} \) denotes the \( n \)-th iteration of Show more…

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22 Saint-Petersburg - Russia, 18th-28th September 2020 A6. Determine all functions \( f: \mathbb{Z} \rightarrow \mathbb{Z} \) such that \[ f^{a^{2}+b^{2}}(a+b)=a f(a)+b f(b) \quad \text { for every } a, b \in \mathbb{Z} \] Here, \( f^{n} \) denotes the \( n^{\text {th }} \) iteration of \( f \), i.e., \( f^{0}(x)=x \) and \( f^{n+1}(x)=f\left(f^{n}(x)\right) \) for all \( n \geqslant 0 \). (Slovakia)