MathDB

3

Part of 1992 Romania Team Selection Test

Problems(2)

Tetrahedron ABCD

Source: Romanian TST 1992 - Day 1 - Problem 3

4/9/2012
Let ABCDABCD be a tetrahedron; B′,C′,D′B', C', D' be the midpoints of the edges AB,AC,ADAB, AC, AD; GA,GB,GC,GDG_A, G_B, G_C, G_D be the barycentres of the triangles BCD,ACD,ABD,ABCBCD, ACD, ABD, ABC, and GG be the barycentre of the tetrahedron. Show that A,G,GB,GC,GDA, G, G_B, G_C, G_D are all on a sphere if and only if A,G,B′,C′,D′A, G, B', C', D' are also on a sphere.
Dan Brânzei
geometry3D geometrytetrahedronspheregeometric transformationhomothetyreflection
f(\pi) is a square, f : \pi \to \pi, image of any triangle is a square

Source: Romania IMO TST 1992 p3

2/19/2020
Let π\pi be the set of points in a plane and f:π→πf : \pi \to \pi be a mapping such that the image of any triangle (as its polygonal line) is a square. Show that f(π)f(\pi) is a square.
functionTrianglealgebra