MathDB

6

Part of 2016 Saint Petersburg Mathematical Olympiad

Problems(2)

100-part broken lines, odd no of self-intersections points

Source: St. Petetsburg 2016 10.6

5/1/2019
The circle contains a closed 100100-part broken line, such that no three segments pass through one point. All its corners are obtuse, and their sum in degrees is divided by 720720. Prove that this broken line has an odd number of self-intersection points.
combinatoricscombinatorial geometryoddpolygonclo
Geometry with incircles II

Source: St Petersburg Olympiad 2016, Grade 9, P6

9/8/2018
Incircle of △ABC\triangle ABC touch ACAC at DD. BDBD intersect incircle at EE. Points F,GF,G on incircle are such points, that FE∥BC,GE∥ABFE \parallel BC,GE \parallel AB. I1,I2I_1,I_2 are incenters of DEF,DEGDEF,DEG. Prove that angle bisector of ∠GDF\angle GDF passes though the midpoint of I1I2I_1I_2 .
geometryincircleincenterPetersburgGrade 9