MathDB

1

Part of 1995 Vietnam National Olympiad

Problems(2)

Equation

Source: Vietnam MO 1995,problem 1

11/11/2007
Find all real solutions to x^3 \minus{} 3x^2 \minus{} 8x \plus{} 40 \minus{} 8\sqrt[4]{4x \plus{} 4} \equal{} 0
algebra proposedalgebra
concurrence of four planes

Source: Vietnam NMO 1995, problem 4

9/7/2008
Let a tetrahedron ABCD ABCD and A′,B′,C′,D′ A',B',C',D' be the circumcenters of triangles BCD,CDA,DAB,ABC BCD,CDA,DAB,ABC respectively. Denote planes (PA),(PB),(PC),(PD) (P_A),(P_B),(P_C),(P_D) be the planes which pass through A,B,C,D A,B,C,D and perpendicular to C′D′,D′A′,A′B′,B′C′ C'D',D'A',A'B',B'C' respectively. Prove that these planes have a common point called I. I. If P P is the center of the circumsphere of the tetrahedron, must this tetrahedron be regular?
geometry3D geometrytetrahedroncircumcirclegeometry proposed