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Omni-MATH / Let n>1 be an integer. Find, with proof, all sequences x₁,x₂,…,x_(n−1) of positive integers with the following three properties: (a). x₁<x₂<⋯<x_(n−1) ; (b). x_(i)+x_(n−i)=2n for al…

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problem

Let n>1n > 1 be an integer. Find, with proof, all sequences x1,x2,…,xn−1x_1 , x_2 , \ldots , x_{n-1} of positive integers with the following three properties: (a). x1<x2<⋯<xn−1x_1 < x_2 < \cdots < x_{n-1} ; (b). xi+xn−i=2nx_i + x_{n-i} = 2n for all i=1,2,…,n−1i = 1, 2, \ldots , n - 1; (c). given any two indices ii and jj (not necessarily distinct) for which xi+xj<2nx_i + x_j < 2n, there is an index kk such that xi+xj=xkx_i + x_j = x_k.
Plain-text mathematical notation (without MathML)
Let n>1 be an integer. Find, with proof, all sequences x₁,x₂,…,x_(n−1) of positive integers with the following three properties: 
(a). x₁<x₂<⋯<x_(n−1) ; 
(b). x_(i)+x_(n−i)=2n for all i=1,2,…,n−1; 
(c). given any two indices i and j (not necessarily distinct) for which x_(i)+x_(j)<2n, there is an index k such that x_(i)+x_(j)=x_(k).
Original LaTeX notation
Let $n > 1$ be an integer. Find, with proof, all sequences $x_1 , x_2 , \ldots , x_{n-1}$ of positive integers with the following three properties: 
(a). $x_1 < x_2 < \cdots < x_{n-1}$ ; 
(b). $x_i + x_{n-i} = 2n$ for all $i = 1, 2, \ldots , n - 1$; 
(c). given any two indices $i$ and $j$ (not necessarily distinct) for which $x_i + x_j < 2n$, there is an index $k$ such that $x_i + x_j = x_k$.

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Source and history

task_id
a787e77c-41a7-5c97-93af-8f76e3d8260e
task_key
test--a787e77c-41a7-5c97-93af-8f76e3d8260e
task_revision_id
1
config / split
/ test

Official source

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