MathDB
17 numbers

Source: Canada 1999

March 4, 2006
combinatorics

Problem Statement

Suppose a1,a2,⋯ ,a8a_1,a_2,\cdots,a_8 are eight distinct integers from {1,2,⋯ ,16,17}\{1,2,\cdots,16,17\}. Show that there is an integer k>0k > 0 such that the equation ai−aj=ka_i - a_j = k has at least three different solutions. Also, find a specific set of 7 distinct integers from {1,2,…,16,17}\{1,2,\ldots,16,17\} such that the equation ai−aj=ka_i - a_j = k does not have three distinct solutions for any k>0k > 0.
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