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Yummy segments

Source: St Petersburg Olympiad 2010, Grade 11, P7

September 14, 2017
combinatoricsnumber theory

Problem Statement

600600 integer numbers from [1,1000][1,1000] colored in red. Natural segment [n,k][n,k] is called yummy if for every natural tt from [1,k−n][1,k-n] there are two red numbers a,ba,b from [n,k][n,k] and b−a=tb-a=t . Prove that there is yummy segment with [a,b][a,b] with b−a≥199b-a \geq 199
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