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Omni-MATH / Find all positive integers n for which the numbers in the set S equal{} {1,2, ldots,n } can be colored red and blue, with the following condition being satisfied: The set S×S×S co…

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problem

Find all positive integers n n for which the numbers in the set $ S \equal{} \{1,2, \ldots,n \}$ can be colored red and blue, with the following condition being satisfied: The set S×S×S S \times S \times S contains exactly 2007 2007 ordered triples (x,y,z) \left(x, y, z\right) such that: [b](i)[/b] the numbers x x, y y, z z are of the same color, and [b](ii)[/b] the number $ x \plus{} y \plus{} z$ is divisible by n n. [i]Author: Gerhard W?ginger, Netherlands[/i]
Plain-text mathematical notation (without MathML)
Find all positive integers n for which the numbers in the set $ S \equal{} \{1,2, \ldots,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:

[b](i)[/b] the numbers x, y, z are of the same color,
and
[b](ii)[/b] the number $ x \plus{} y \plus{} z$ is divisible by n.

[i]Author: Gerhard W?ginger, Netherlands[/i]
Original LaTeX notation
Find all positive integers $ n$ for which the numbers in the set $ S \equal{} \{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 $ \left(x, y, z\right)$ such that:

[b](i)[/b] the numbers $ x$, $ y$, $ z$ are of the same color,
and
[b](ii)[/b] the number $ x \plus{} y \plus{} z$ is divisible by $ n$.

[i]Author: Gerhard W?ginger, Netherlands[/i]

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Source and history

task_id
c0609057-2018-54af-8227-c8929cf8353c
task_key
test--c0609057-2018-54af-8227-c8929cf8353c
task_revision_id
1
config / split
/ test

Official source

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