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OlympiadBench / 2212 / Given a positive integer n≥2, determine the largest positive integer N…

Problem

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

question

Given a positive integer

n \geq 2

, determine the largest positive integer

N

for which there exist

N+1

real numbers

a_{0}, a_{1}, \ldots, a_{N}

such that (1)

a_{0}+a_{1}=-\frac{1}{n}

, and (2)

\left(a_{k}+a_{k-1}\right)\left(a_{k}+a_{k+1}\right)=a_{k-1}-a_{k+1}

for

1 \leq k \leq N-1

Plain-text mathematical notation (without MathML)

Given a positive integer n≥2, determine the largest positive integer N for which there exist N+1 real numbers a₀,a₁,…,a_(N) such that

(1) a₀+a₁=−(1)/(n), and

(2) (a_(k)+a_(k−1))(a_(k)+a_(k+1))=a_(k−1)−a_(k+1) for 1≤k≤N−1.

Original LaTeX notation

Given a positive integer $n \geq 2$, determine the largest positive integer $N$ for which there exist $N+1$ real numbers $a_{0}, a_{1}, \ldots, a_{N}$ such that

(1) $a_{0}+a_{1}=-\frac{1}{n}$, and

(2) $\left(a_{k}+a_{k-1}\right)\left(a_{k}+a_{k+1}\right)=a_{k-1}-a_{k+1}$ for $1 \leq k \leq N-1$.

answer type

Expression

is multiple answer

false

language

English

question type

Open-ended

subject

Math

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Source and history

upstream_id / Original problem ID: 2212
task_id: dddb58e1-65ed-5f91-b478-1c51ce281343
task_key: OE~5fTO~5fmaths~5fen~5fCOMP--train--2212
task_revision_id: 3
config / split: OE_TO_maths_en_COMP / train

Official source

initial import