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Amer average number barycentric coordinates base bichromatic graph blue edges blue triangle blue-empty graph cake candidate circle color columns cone conic Consider contain convergent series denote density diagonal Dirichlet principles divergent series ellipses inscribed equal equation example Exercise extremal graph fact Figure formula Fubini principle G.H. Hardy geometric given harmonic series Hence infinitely inscribed in ABC isoperimetric theorem latin square least logarithmic density logn Math mathematical matrix midpoint ellipse midpoint triangle n-by-n square n-digit integers number of occupied number theory occupied seats orthogonal projection partial sum participants pentagon piece plane positive integers prime numbers probability problem proof Prove quadrilateral Ramsey's Theorem random reciprocals red triangles regions result Riemann hypothesis ROSS HONSBERGER sequence set of integers singular pair solutions submatrix submatrix of O's subsets Suppose tangent Theorem total number transversal triangle ABC vertex zeros