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r_k is largest quantity of different elements of A so that they differ not k

Source: Cono Sur 2002 P5

August 1, 2018

combinatoricsnumber theorymaximumDifferencecono sur

Problem Statement

Consider the set

A = \{1, 2, ..., n\}

. For each integer

k

, let

r_k

be the largest quantity of different elements of

A

that we can choose so that the difference between two numbers chosen is always different from

k

. Determine the highest value possible of

r_k

, where

1 \le k \le \frac{n}{2}