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3 of 8 integers from {1, 2, . . . , 16, 17} such that a_i - a_j = k

Source: New Zealand NZMOC Camp Selection Problems 2010 Seniors 6

September 18, 2021
number theory

Problem Statement

Suppose a1,a2,...,a8a_1, a_2, . . . , a_8 are eight distinct integers from {1,2,...,16,17}\{1, 2, . . . , 16, 17\}. Show that there is an integer k>0k > 0 such that there are at least three different (not necessarily disjoint) pairs (i,j)(i, j) such that ai−aj=ka_i - a_j = k. Also find a set of seven distinct integers from {1,2,...,16,17}\{1, 2, . . . , 16, 17\} such that there is no integer k>0k > 0 with that property.
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