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Page 69
... regions bounded by curves . A small portion of the map may look like Figure 15 : B A FIG . 15 Assume that at most three regions meet at a point . Call a point where three regions do meet a vertex . Call a common border of two regions ...
... regions bounded by curves . A small portion of the map may look like Figure 15 : B A FIG . 15 Assume that at most three regions meet at a point . Call a point where three regions do meet a vertex . Call a common border of two regions ...
Page 87
... regions meet . Prove that if one assumes , instead , that at each vertex at least three regions meet , there still must exist a triangle , quadrilateral , or pentagon . 5. Assume that the surface of a ball is cut up into 3 - sided regions ...
... regions meet . Prove that if one assumes , instead , that at each vertex at least three regions meet , there still must exist a triangle , quadrilateral , or pentagon . 5. Assume that the surface of a ball is cut up into 3 - sided regions ...
Page 90
... regions , where each region is bounded by a curve or polygon . Assume that any finite number of these regions can be colored with at most 5 colors in such a way that countries that share an edge have different colors . Prove that all of ...
... regions , where each region is bounded by a curve or polygon . Assume that any finite number of these regions can be colored with at most 5 colors in such a way that countries that share an edge have different colors . Prove that all of ...
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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