MathDB

7

Part of 2010 Sharygin Geometry Olympiad

Problems(4)

Prove that O_1M=O_2M (7)

Source:

10/28/2010
The line passing through the vertex BB of a triangle ABCABC and perpendicular to its median BMBM intersects the altitudes dropped from AA and CC (or their extensions) in points KK and N.N. Points O1O_1 and O2O_2 are the circumcenters of the triangles ABKABK and CBNCBN respectively. Prove that O1M=O2M.O_1M=O_2M.
geometrycircumcirclegeometry proposed
cuting 2 regular polygons, folding one piece from each to create another

Source: Sharygin 2010 Final 8.7

10/2/2018
Each of two regular polygons PP and QQ was divided by a line into two parts. One part of PP was attached to one part of QQ along the dividing line so that the resulting polygon was regular and not congruent to PP or QQ. How many sides can it have?
geometryregular polygonpaperPolygons
cutting 2 regular polyhedra by a plane, and putting pieces together

Source: Sharygin 2010 Final 10.7

11/25/2018
Each of two regular polyhedrons PP and QQ was divided by the plane into two parts. One part of PP was attached to one part of QQ along the dividing plane and formed a regular polyhedron not equal to PP and QQ. How many faces can it have?
geometrypolyhedroncutPlane
2 symmetrical (to the circumcircle) circles are tangent iff a triangle is right

Source: Sharygin 2010 Final 9.7

10/6/2018
Given triangle ABCABC. Lines ALaAL_a and AMaAM_a are the internal and the external bisectrix of angle AA. Let ωa\omega_a be the reflection of the circumcircle of △ALaMa\triangle AL_aM_a in the midpoint of BCBC. Circle ωb\omega_b is defined similarly. Prove that ωa\omega_a and ωb\omega_b touch if and only if △ABC\triangle ABC is right-angled.
geometrytangent circlesright trianglesymmetry