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Min sum of products of number of points

Source: Japan TST 2018 P4

January 25, 2021
combinatoricsgeometryalgebra

Problem Statement

Let n≥3n\ge 3 be a positive integer. SS is a set of nn points on the plane, with no three collinear. LL is the set of all lines passing through any two points in SS. For any ℓ∈L\ell \in L, the separating number of ℓ\ell is the product of the number of points in SS in the two sides of ℓ\ell, excluding the two points on ℓ\ell itself. Determine the minimum possible total sum of all separating numbers.
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