2

Part of 1998 Vietnam National Olympiad

Problems(2)

tetrahedron and circumsphere

Source: 36-th Vietnamese Mathematical Olympiad 1998

Let be given a tetrahedron whose circumcenter is

O

. Draw diameters

AA_{1},BB_{1},CC_{1},DD_{1}

of the circumsphere of

ABCD

A_{0},B_{0},C_{0},D_{0}

be the centroids of triangle

BCD,CDA,DAB,ABC

A_{0}A_{1},B_{0}B_{1},C_{0}C_{1},D_{0}D_{1}

are concurrent at a point, say,

F

. Prove that the line through

F

and a midpoint of a side of

ABCD

is perpendicular to the opposite side.

geometry3D geometrytetrahedroncircumcirclegeometric transformationreflectionparallelogram

Source: 36-th Vietnamese Mathematical Olympiad 1998

Find minimum value of

F(x,y)=\sqrt{(x+1)^{2}+(y-1)^{2}}+\sqrt{(x-1)^{2}+(y+1)^{2}}+\sqrt{(x+2)^{2}+(y+2)^{2}}

x,y\in\mathbb{R}

analytic geometrytrigonometryinequalities proposedinequalities