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OlympiadBench / 1834 / Find the smallest positive integer n, or show that no such n exists, with…

Problem

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

question

Find the smallest positive integer

n

, or show that no such

n

exists, with the following property: there are infinitely many distinct

n

-tuples of positive rational numbers

\left(a_{1}, a_{2}, \ldots, a_{n}\right)

such that both

a_{1}+a_{2}+\cdots+a_{n} \quad \text { and } \quad \frac{1}{a_{1}}+\frac{1}{a_{2}}+\cdots+\frac{1}{a_{n}}

are integers.

Plain-text mathematical notation (without MathML)

Find the smallest positive integer n, or show that no such n exists, with the following property: there are infinitely many distinct n-tuples of positive rational numbers (a₁,a₂,…,a_(n)) such that both

a₁+a₂+⋯+a_(n)  and  (1)/(a₁)+(1)/(a₂)+⋯+(1)/(a_(n))

are integers.

Original LaTeX notation

Find the smallest positive integer $n$, or show that no such $n$ exists, with the following property: there are infinitely many distinct $n$-tuples of positive rational numbers $\left(a_{1}, a_{2}, \ldots, a_{n}\right)$ such that both

$$
a_{1}+a_{2}+\cdots+a_{n} \quad \text { and } \quad \frac{1}{a_{1}}+\frac{1}{a_{2}}+\cdots+\frac{1}{a_{n}}
$$

are integers.

answer type

Numerical

is multiple answer

false

language

English

question type

Open-ended

subject

Math

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Source and history

upstream_id / Original problem ID: 1834
task_id: a9d6eaa6-4377-5d6a-a07b-0d4b2170912d
task_key: OE~5fTO~5fmaths~5fen~5fCOMP--train--1834
task_revision_id: 3
config / split: OE_TO_maths_en_COMP / train

Official source

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