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Expressing integers as difference of two elements

Source: China second round 2018 (A) Q3

May 5, 2019

combinatorics

Problem Statement

Let

n,k,m

be positive integers, where

k\ge 2

and

n\le m < \frac{2k-1}{k}n

. Let

A

be a subset of

\{1,2,\ldots ,m\}

with

n

elements. Prove that every integer in the range

\left(0,\frac{n}{k-1}\right)

can be expressed as

a-b

, where

a,b\in A

.

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