Problem

Source: Mexico National Olympiad Mock Exam 2018 Problem 6

Tags: combinatorics, elements of set



Let $A$ be a finite set of positive integers, and for each positive integer $n$ we define \[S_n = \{x_1 + x_2 + \cdots + x_n \;\vert\; x_i \in A \text{ for } i = 1, 2, \dots, n\}\] That is, $S_n$ is the set of all positive integers which can be expressed as sum of exactly $n$ elements of $A$, not necessarily different. Prove that there exist positive integers $N$ and $k$ such that $$\left\vert S_{n + 1} \right\vert = \left\vert S_n \right\vert + k \text{ for all } n\geq N.$$ Proposed by Ariel GarcĂ­a