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Omni-MATH / In a town every two residents who are not friends have a friend in common, and…

Problem

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problem

In a town every two residents who are not friends have a friend in common, and no one is a friend of everyone else. Let us number the residents from 1 to nn and let aia_{i} be the number of friends of the ii-th resident. Suppose that ∑i=1nai2=n2−n\sum_{i=1}^{n} a_{i}^{2}=n^{2}-n. Let kk be the smallest number of residents (at least three) who can be seated at a round table in such a way that any two neighbors are friends. Determine all possible values of kk.
Plain-text mathematical notation (without MathML)
In a town every two residents who are not friends have a friend in common, and no one is a friend of everyone else. Let us number the residents from 1 to n and let a_(i) be the number of friends of the i-th resident. Suppose that ∑_(i=1)^(n)a_(i)²=n²−n. Let k be the smallest number of residents (at least three) who can be seated at a round table in such a way that any two neighbors are friends. Determine all possible values of k.
Original LaTeX notation
In a town every two residents who are not friends have a friend in common, and no one is a friend of everyone else. Let us number the residents from 1 to $n$ and let $a_{i}$ be the number of friends of the $i$-th resident. Suppose that $\sum_{i=1}^{n} a_{i}^{2}=n^{2}-n$. Let $k$ be the smallest number of residents (at least three) who can be seated at a round table in such a way that any two neighbors are friends. Determine all possible values of $k$.

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Source and history

task_id
07912873-495b-5c4e-81e1-8a22bf92f78c
task_key
test--07912873-495b-5c4e-81e1-8a22bf92f78c
task_revision_id
1
config / split
/ test

Official source

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