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OlympiadBench / 1664 / Find all positive integers n, for which the numbers in the set …

Problem

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

answer type

Numerical

is multiple answer

true

language

English

question

Find all positive integers

n

, for which the numbers in the set

S=\{1,2, \ldots, n\}

can be colored red and blue, with the following condition being satisfied: the set

S \times S \times S

contains exactly 2007 ordered triples

(x, y, z)

such that (i)

x, y, z

are of the same color and (ii)

x+y+z

is divisible by

n

Plain-text mathematical notation (without MathML)

Find all positive integers n, for which the numbers in the set S={1,2,…,n} can be colored red and blue, with the following condition being satisfied: the set S×S×S contains exactly 2007 ordered triples (x,y,z) such that (i) x,y,z are of the same color and (ii) x+y+z is divisible by n.

Original LaTeX notation

Find all positive integers $n$, for which the numbers in the set $S=\{1,2, \ldots, n\}$ can be colored red and blue, with the following condition being satisfied: the set $S \times S \times S$ contains exactly 2007 ordered triples $(x, y, z)$ such that (i) $x, y, z$ are of the same color and (ii) $x+y+z$ is divisible by $n$.

question type

Open-ended

subject

Math

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Source and history

upstream_id / Original problem ID: 1664
task_id: 41362e25-d82d-5fc9-8a4f-71b02ee05e95
task_key: OE~5fTO~5fmaths~5fen~5fCOMP--train--1664
task_revision_id: 1
config / split: OE_TO_maths_en_COMP / train

Official source

initial import