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Isosceles triangles among a group of points

Source: Iran 3rd round 2012-Special Lesson exam-Part1-P2

July 27, 2012
combinatorics proposedcombinatorics

Problem Statement

Consider a set of nn points in plane. Prove that the number of isosceles triangles having their vertices among these nn points is O(n73)\mathcal O (n^{\frac{7}{3}}). Find a configuration of nn points in plane such that the number of equilateral triangles with vertices among these nn points is Ω(n2)\Omega (n^2).
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