MathDB

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

Problem Statement