MathDB

8

Part of 2011 Sharygin Geometry Olympiad

Problems(4)

concurrent lines out of reflections related to touchpoints of incircles

Source: 2011 Sharygin Geometry Olympiad Correspondence Round P8

4/2/2019
The incircle of right-angled triangle ABCABC (∠B=90o\angle B = 90^o) touches AB,BC,CAAB,BC,CA at points C1,A1,B1C_1,A_1,B_1 respectively. Points A2,C2A_2, C_2 are the reflections of B1B_1 in lines BC,ABBC, AB respectively. Prove that lines A1A2A_1A_2 and C1C2C_1C_2 meet on the median of triangle ABCABC.
geometryincircleconcurrencyReflectionsreflection
using only ruler, divide the side of a square table into n equal parts

Source: Sharygin 2011 Final 8.8

12/15/2018
Using only the ruler, divide the side of a square table into nn equal parts. All lines drawn must lie on the surface of the table.
geometryconstructionDivision
circumscribed convex n-gon dissected into equal triangles by diagonals non-inter

Source: Sharygin 2011 Final 9.8

12/20/2018
A convex nn-gon PP, where n>3n > 3, is dissected into equal triangles by diagonals non-intersecting inside it. Which values of nn are possible, if PP is circumscribed?
geometrydiagonalsconvex polygoncircumcircle
given a sheet of tin 6x 6, make a cube with edge 2 divided into unit cubes

Source: Sharygin 2011 Final 10.8

3/31/2019
Given a sheet of tin 6×66\times 6. It is allowed to bend it and to cut it but in such a way that it doesn’t fall to pieces. How to make a cube with edge 22, divided by partitions into unit cubes?
combinatorial geometrycubegeometry