MathDB

5

Part of 2016 Sharygin Geometry Olympiad

Problems(3)

folding a transparent paper with 3 points to create an equilateral

Source: Sharygin Geometry Olympiad 2016 Final Round problem 5 grade 8

7/22/2018
Three points are marked on the transparent sheet of paper. Prove that the sheet can be folded along some line in such a way that these points form an equilateral triangle.
by A.Khachaturyan
geometrypaperEquilateral Triangle
Polar bisects segment

Source: Sharygin Geometry Olympiad, Final Round 2016, Problem 5 grade 9

8/4/2016
The center of a circle ω2\omega_2 lies on a circle ω1\omega_1. Tangents XPXP and XQXQ to ω2\omega_2 from an arbitrary point XX of ω1\omega_1 (PP and QQ are the touching points) meet ω1\omega_1 for the second time at points RR and SS. Prove that the line PQPQ bisects the segment RSRS.
geometryangles
Peculiar convex polyhedrons

Source: Sharygin geometry olympiad 2016, grade 10, Final Round, Problem 5.

8/5/2016
Does there exist a convex polyhedron having equal number of edges and diagonals? (A diagonal of a polyhedron is a segment through two vertices not lying on the same face)
combinatoricsgeometry3D geometry