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IMO Shortlist 2013, Algebra #2

Source: IMO Shortlist 2013, Algebra #2

July 9, 2014
algebrabinomial theorempigeonhole principleIMO Shortlist

Problem Statement

Prove that in any set of 20002000 distinct real numbers there exist two pairs a>ba>b and c>dc>d with a≠ca \neq c or b≠db \neq d , such that ∣a−bc−d−1∣<1100000. \left| \frac{a-b}{c-d} - 1 \right|< \frac{1}{100000}.
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