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1989 APMO Problems/Problem 4

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Problem

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

Solution

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