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AABC ABCD altitude angles circumcircle coefficients consecutive Consider construct contestants convex hull convex quadrilateral cos2 denote diameter digits distinct divisible edges equality holds equilateral equivalent factor Figure follows function geometric given equation Greitzer H. S. M. Coxeter Hence induction inequality inscribed integer International Mathematical Olympiads International Olympiad intersection isosceles l(mod law of cosines least lemma length locus Mathematical matrix midpoint multiple natural numbers NML vol non-negative obtain parallel perpendicular plane points polynomial positive integers positive numbers prismatoid problem proof Prove quadratic equation quadratic form radius real numbers relatively prime right triangle roots S. L. Greitzer satisfy Second solution segment BC sides solution due solve sphere square subsets Suppose tangent tetrahedron theorem triangle ABC unit circle unit vectors values vertex vertices volume yields zero