MathDB

2

Part of 1995 Belarus Team Selection Test

Problems(2)

Geometry

Source: Belarus IMO TST 1995 Day 1 P 2

11/28/2015
Circles S,S1,S2S,S_1,S_2 are given in a plane. S1S_1 and S2S_2 touch each other externally, and both touch SS internally at A1A_1 and A2A_2 respectively. The common internal tangent to S1S_1 and S2S_2 meets SS at PP and Q.Q. Let B1B_1 and B2B_2 be the intersections of PA1PA_1 and PA2PA_2 with S1S_1 and S2S_2, respectively. Prove that B1B2B_1B_2 is a common tangent to S1,S2S_1,S_2
geometrycirclestangent circlescommon tangents
Combinatorics

Source: 1995 Belarus IMO TST Day 2 P 2

11/28/2015
There is a room having a form of right-angled parallelepiped. Four maps of the same scale are hung (generally, on different levels over the floor) on four walls of the room, so that sides of the maps are parallel to sides of the wall. It is known that the four points corresponding to each of Stockholm, Moscow, and Istanbul are coplanar. Prove that the four points coresponding to Hong Kong are coplanar as well.

geometry3D geometryparallelepiped