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OlympiadBench / 2000 / Find all positive integers n≥2 for which there exist n real…

Problem

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

question

Find all positive integers

n \geqslant 2

for which there exist

n

real numbers

a_{1}<\cdots<a_{n}

and a real number

r>0

such that the

\frac{1}{2} n(n-1)

differences

a_{j}-a_{i}

for

1 \leqslant i<j \leqslant n

are equal, in some order, to the numbers

r^{1}, r^{2}, \ldots, r^{\frac{1}{2} n(n-1)}

Plain-text mathematical notation (without MathML)

Find all positive integers n≥2 for which there exist n real numbers a₁<⋯<a_(n) and a real number r>0 such that the (1)/(2)n(n−1) differences a_(j)−a_(i) for 1≤i<j≤n are equal, in some order, to the numbers r¹,r²,…,r^((1)/(2)n(n−1)).

Original LaTeX notation

Find all positive integers $n \geqslant 2$ for which there exist $n$ real numbers $a_{1}<\cdots<a_{n}$ and a real number $r>0$ such that the $\frac{1}{2} n(n-1)$ differences $a_{j}-a_{i}$ for $1 \leqslant i<j \leqslant n$ are equal, in some order, to the numbers $r^{1}, r^{2}, \ldots, r^{\frac{1}{2} n(n-1)}$.

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: 2000
task_id: d52cdf98-ea74-5f12-90ed-cd154f2119c6
task_key: OE~5fTO~5fmaths~5fen~5fCOMP--train--2000
task_revision_id: 3
config / split: OE_TO_maths_en_COMP / train

Official source

initial import