MathDB

1

Part of 1993 Vietnam National Olympiad

Problems(2)

min and max

Source: 31-th Vietnamese Mathematical Olympiad 1993

2/17/2007
f:[−1995,1995]→Rf : [-\sqrt{1995},\sqrt{1995}] \to\mathbb{R} is defined by f(x)=x(1993+1995−x2)f(x) = x(1993+\sqrt{1995-x^{2}}). Find its maximum and minimum values.
calculustrigonometryreal analysisalgebra proposedalgebra
right tetrahedron at A

Source: 31-th Vietnamese Mathematical Olympiad 1993

2/17/2007
The tetrahedron ABCDABCD has its vertices on the fixed sphere SS. Prove that AB2+AC2+AD2−BC2−BD2−CD2AB^{2}+AC^{2}+AD^{2}-BC^{2}-BD^{2}-CD^{2} is minimum iff AB⊥AC,AC⊥AD,AD⊥ABAB\perp AC,AC\perp AD,AD\perp AB.
geometry3D geometrytetrahedronspheregeometry proposed