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Omni-MATH / Determine all functions f defined on the set of all positive integers and taking non-negative integer values, satisfying the three conditions: [list] [*] (i) f(n)≠0 for at least on…

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problem

Determine all functions ff defined on the set of all positive integers and taking non-negative integer values, satisfying the three conditions: [list] [*] (i)(i) f(n)≠0f(n) \neq 0 for at least one nn; [*] (ii)(ii) f(xy)=f(x)+f(y)f(x y)=f(x)+f(y) for every positive integers xx and yy; [*] (iii)(iii) there are infinitely many positive integers nn such that f(k)=f(n−k)f(k)=f(n-k) for all k<nk<n. [/list]
Plain-text mathematical notation (without MathML)
Determine all functions f defined on the set of all positive integers and taking non-negative integer values, satisfying the three conditions:
[list]
[*] (i) f(n)≠0 for at least one n;
[*] (ii) f(xy)=f(x)+f(y) for every positive integers x and y;
[*] (iii) there are infinitely many positive integers n such that f(k)=f(n−k) for all k<n.
[/list]
Original LaTeX notation
Determine all functions $f$ defined on the set of all positive integers and taking non-negative integer values, satisfying the three conditions:
[list]
[*] $(i)$ $f(n) \neq 0$ for at least one $n$;
[*] $(ii)$ $f(x y)=f(x)+f(y)$ for every positive integers $x$ and $y$;
[*] $(iii)$ there are infinitely many positive integers $n$ such that $f(k)=f(n-k)$ for all $k<n$.
[/list]

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Source and history

task_id
2cb0a2fd-5bcd-5d16-a809-0803f9245c32
task_key
test--2cb0a2fd-5bcd-5d16-a809-0803f9245c32
task_revision_id
1
config / split
/ test

Official source

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