MathDB

8

Part of 2009 Sharygin Geometry Olympiad

Problems(4)

divide a polygon into two equal parts

Source: I.F.Sharygin contest 2009 - Correspondence round - Problem 8

5/31/2009
Some polygon can be divided into two equal parts by three different ways. Is it certainly valid that this polygon has an axis or a center of symmetry?
symmetrygeometrygeometry proposed
distance between touchpoints of side with incircle and excircle

Source: 2009 Sharygin Geometry Olympiad Final Round problem 8 grade 8

7/26/2018
A triangle ABCABC is given, in which the segment BCBC touches the incircle and the corresponding excircle in points MM and NN. If ∠BAC=2∠MAN\angle BAC = 2 \angle MAN, show that BC=2MNBC = 2MN.
(N.Beluhov)
geometryincircleexcircle
4 equal circles touch diagonals and internally circumcircle of ABCD

Source: 2009 Sharygin Geometry Olympiad Final Round problem 8 grade 9

7/26/2018
Given cyclic quadrilateral ABCDABCD. Four circles each touching its diagonals and the circumcircle internally are equal. Is ABCDABCD a square?
(C.Pohoata, A.Zaslavsky)
geometrycircumcirclecyclic quadrilateralcircles
regular octahedron inscribed into regular dodecahedron

Source: 2009 Sharygin Geometry Olympiad Final Round problem 8 grade 10

7/26/2018
Can the regular octahedron be inscribed into regular dodecahedron in such way that all vertices of octahedron be the vertices of dodecahedron?
(B.Frenkin)
geometry3D geometrydodecahedronoctahedron