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Triples (a,b,c) belong to s

Source: APMO 1989

March 10, 2006
floor functioninequalitiesgraph theorycombinatorics

Problem Statement

Let SS be a set consisting of mm pairs (a,b)(a,b) of positive integers with the property that 1≤a<b≤n1 \leq a < b \leq n. Show that there are at least 4m⋅(m−n24)3n 4m \cdot \dfrac{(m - \dfrac{n^2}{4})}{3n} triples (a,b,c)(a,b,c) such that (a,b)(a,b), (a,c)(a,c), and (b,c)(b,c) belong to SS.
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