3 variable FE with divisibility condition

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3 variable FE with divisibility condition

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Source: Revenge JOM 2025 Problem 1, Revenge JOMSL 2025 N2, Own

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pithon_with_an_i

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pithon_with_an_i

#1 h May 13, 2025, 4:11 PM

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Find all functions $f:\mathbb{N} \rightarrow \mathbb{N}$ such that $f(a)+f(b)+f(c) \mid a^2 + af(b) + cf(a)$ for all $a,b,c \in \mathbb{N}$ .

This post has been edited 1 time. Last edited by pithon_with_an_i, May 13, 2025, 4:14 PM
Reason: Typo

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Primeniyazidayi

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Primeniyazidayi

#2 h May 13, 2025, 4:47 PM • 1 Y

Y by pithon_with_an_i

Idea is actually pretty simple.
(1,1,1) gives $3f(1) \mid 2f(1)+1$ and $f(1)=1$
(1,1,a) gives also $f(a)+2 \mid a+2$ .Putting $a=p-2$ where $p$ is prime gives that $f$ is above unbounded.
(a,b,1) gives $f(a)+f(b)+1 \mid a^2+af(b)+f(a)$ and by subtracting $a(f(a)+f(b)+1)$ $f(a)+f(b)+1 \mid a^2-af(a)+f(a)-a=(a-1)(a-f(a))$ .Note that $f(b)$ converges to $\infty$ for well-chosen $b$ so RHS must be zero which gives that $f(a)=a$ is the only solution.

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ATM_

#3 h May 13, 2025, 5:53 PM • 1 Y

Y by Rayanelba

If 0 is not included then proceed as above
If it is you can easily find that $f(0)=0$ ; then take $P(p;0;0)$ and do some case work to get that $f(p)=p$ for all primes
Now $P(a;p;0):f(a)+p \mid a^2+ap \implies f(a)+p \mid a+p \implies f(a)+p \mid a-f(a)$
Set p to be large enough and it follows that: $f(a)=a$

This post has been edited 1 time. Last edited by ATM_, May 13, 2025, 5:54 PM
Reason: english mistakes

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