USAJMO 2010 Question 2

Let $n\gt 1$ be an integer. Find all positive integer sequences $x_1,\ldots,x_{n-1}$ such that (a) $x_1\lt \cdots\lt x_{n-1}$; (b) $x_i+x_{n-i}=2n$ for $1\le i\le n-1$; (c) whenever $x_i+x_j\lt 2n$ (indices may coincide), the sum is one of the terms of the sequence.

Proposed by Răzvan Gelca (credit in the supplied notes).