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OlympiadBench / 2177 / Find the smallest positive integer k for which there exist a colouring of the…

Problem

Answer published by the source. Consult the official source to check your work against its answer.

question

Find the smallest positive integer kk for which there exist a colouring of the positive integers Z>0\mathbb{Z}_{>0} with kk colours and a function f:Z>0→Z>0f: \mathbb{Z}_{>0} \rightarrow \mathbb{Z}_{>0} with the following two properties: (i) For all positive integers m,nm, n of the same colour, f(m+n)=f(m)+f(n)f(m+n)=f(m)+f(n). (ii) There are positive integers m,nm, n such that f(m+n)≠f(m)+f(n)f(m+n) \neq f(m)+f(n). In a colouring of Z>0\mathbb{Z}_{>0} with kk colours, every integer is coloured in exactly one of the kk colours. In both (i) and (ii) the positive integers m,nm, n are not necessarily different.
Plain-text mathematical notation (without MathML)
Find the smallest positive integer k for which there exist a colouring of the positive integers Z_(>0) with k colours and a function f:Z_(>0)→Z_(>0) with the following two properties:

(i) For all positive integers m,n of the same colour, f(m+n)=f(m)+f(n).

(ii) There are positive integers m,n such that f(m+n)≠f(m)+f(n).

In a colouring of Z_(>0) with k colours, every integer is coloured in exactly one of the k colours. In both (i) and (ii) the positive integers m,n are not necessarily different.
Original LaTeX notation
Find the smallest positive integer $k$ for which there exist a colouring of the positive integers $\mathbb{Z}_{>0}$ with $k$ colours and a function $f: \mathbb{Z}_{>0} \rightarrow \mathbb{Z}_{>0}$ with the following two properties:

(i) For all positive integers $m, n$ of the same colour, $f(m+n)=f(m)+f(n)$.

(ii) There are positive integers $m, n$ such that $f(m+n) \neq f(m)+f(n)$.

In a colouring of $\mathbb{Z}_{>0}$ with $k$ colours, every integer is coloured in exactly one of the $k$ colours. In both (i) and (ii) the positive integers $m, n$ are not necessarily different.

answer type

Numerical

is multiple answer

false

language

English

question type

Open-ended

subject

Math

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Source and history

upstream_id / Original problem ID
2177
task_id
63c97aa8-eef1-509a-89fd-6cee3c85adac
task_key
OE~5fTO~5fmaths~5fen~5fCOMP--train--2177
task_revision_id
4
config / split
OE_TO_maths_en_COMP / train

Official source

initial import