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

Problem

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problem

Find the smallest positive integer nn or show no such nn exists, with the following property: there are infinitely many distinct nn-tuples of positive rational numbers (a1,a2,…,an)(a_1, a_2, \ldots, a_n) such that both a1+a2+…+anand1a1+1a2+…+1ana_1+a_2+\dots +a_n \quad \text{and} \quad \frac{1}{a_1} + \frac{1}{a_2} + \dots + \frac{1}{a_n} are integers.
Plain-text mathematical notation (without MathML)
Find the smallest positive integer n or show 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 no such $n$ exists, with the following property: there are infinitely many distinct $n$-tuples of positive rational numbers $(a_1, a_2, \ldots, a_n)$ such that both
$$a_1+a_2+\dots +a_n \quad \text{and} \quad \frac{1}{a_1} + \frac{1}{a_2} + \dots + \frac{1}{a_n}$$
are integers.

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Source and history

task_id
d543eacf-a4dc-5b99-ade3-f19d3cf3e6e0
task_key
test--d543eacf-a4dc-5b99-ade3-f19d3cf3e6e0
task_revision_id
1
config / split
/ test

Official source

initial import