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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.

question

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

answer type

Numerical

is multiple answer

true

language

English

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
4
config / split
OE_TO_maths_en_COMP / train

Official source

initial import