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Omni-MATH / Determine all integers n≥2 having the following property: for any integers a₁,a₂,…,a_(n) whose sum is not divisible by n, there exists an index 1≤i≤n such that none of the numbers …

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problem

Determine all integers n≥2 n\geq 2 having the following property: for any integers a1,a2,…,ana_1,a_2,\ldots, a_n whose sum is not divisible by nn, there exists an index 1≤i≤n1 \leq i \leq n such that none of the numbers ai,ai+ai+1,…,ai+ai+1+…+ai+n−1a_i,a_i+a_{i+1},\ldots,a_i+a_{i+1}+\ldots+a_{i+n-1} is divisible by nn. Here, we let ai=ai−na_i=a_{i-n} when i>ni >n. [i]
Plain-text mathematical notation (without MathML)
Determine all integers n≥2 having the following property: for any integers a₁,a₂,…,a_(n) whose sum is not divisible by n, there exists an index 1≤i≤n such that none of the numbers a_(i),a_(i)+a_(i+1),…,a_(i)+a_(i+1)+…+a_(i+n−1) is divisible by n. Here, we let a_(i)=a_(i−n) when i>n.

[i]
Original LaTeX notation
Determine all integers $ n\geq 2$ having the following property: for any integers $a_1,a_2,\ldots, a_n$ whose sum is not divisible by $n$, there exists an index $1 \leq i \leq n$ such that none of the numbers $$a_i,a_i+a_{i+1},\ldots,a_i+a_{i+1}+\ldots+a_{i+n-1}$$ is divisible by $n$. Here, we let $a_i=a_{i-n}$ when $i >n$.

[i]

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Source and history

task_id
3bd30c80-bf8e-564e-9adf-b456954c98dd
task_key
test--3bd30c80-bf8e-564e-9adf-b456954c98dd
task_revision_id
1
config / split
/ test

Official source

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