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Omni-MATH / Determine all positive integers n, n≥2, such that the following statement is true: If (a₁,a₂,...,a_(n)) is a sequence of positive integers with a₁+a₂+⋯+a_(n)=2n−1, then there is bl…

Problem

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problem

Determine all positive integers nn, n2n\ge2, such that the following statement is true: If (a1,a2,...,an)(a_1,a_2,...,a_n) is a sequence of positive integers with a1+a2++an=2n1a_1+a_2+\cdots+a_n=2n-1, then there is block of (at least two) consecutive terms in the sequence with their (arithmetic) mean being an integer.
Plain-text mathematical notation (without MathML)
Determine all positive integers n, n≥2, such that the following statement is true:
If (a₁,a₂,...,a_(n)) is a sequence of positive integers with a₁+a₂+⋯+a_(n)=2n−1, then there is block of (at least two) consecutive terms in the sequence with their (arithmetic) mean being an integer.
Original LaTeX notation
Determine all positive integers $n$, $n\ge2$, such that the following statement is true:
If $(a_1,a_2,...,a_n)$ is a sequence of positive integers with $a_1+a_2+\cdots+a_n=2n-1$, then there is block of (at least two) consecutive terms in the sequence with their (arithmetic) mean being an integer.

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