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arithmetic progression of a, a + d, a + 2d,.. with 3,4,7 of given numbers

Source: Singapore Senior Math Olympiad 2017 2nd Round p5 SMO

March 26, 2020
Arithmetic ProgressionalgebracombinatoricsIntegersSequencenumber theory

Problem Statement

Given 77 distinct positive integers, prove that there is an infinite arithmetic progression of positive integers a,a+d,a+2d,..a, a + d, a + 2d,.. with a<da < d, that contains exactly 33 or 44 of the 77 given integers.
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