MathDB

3

Part of 2022 Iranian Geometry Olympiad

Problems(3)

Convex pentagon with 3 equal sides

Source: IGO 2022 Elementary P3

12/14/2022
Let ABCDEABCDE be a convex pentagon such that AB=BC=CDAB = BC = CD and ∠BDE=∠EAC=30∘\angle BDE = \angle EAC = 30 ^{\circ}. Find the possible values of ∠BEC\angle BEC.
Proposed by Josef Tkadlec (Czech Republic)
pentagoniranian geometry olympiadgeometry
Configuration with three similar triangles

Source: IGO 2022 Intermediate P3

12/13/2022
Let OO be the circumcenter of triangle ABCABC. Arbitrary points MM and NN lie on the sides ACAC and BCBC, respectively. Points PP and QQ lie in the same half-plane as point CC with respect to the line MNMN, and satisfy △CMN∼△PAN∼△QMB\triangle CMN \sim \triangle PAN \sim \triangle QMB (in this exact order). Prove that OP=OQOP=OQ.
Proposed by Medeubek Kungozhin, Kazakhstan
geometrysimilar triangles
IGO 2022 P3

Source: Iranian Geometry Olympiad 2022 P3 Advanced, Free

12/13/2022
In triangle ABCABC (∠A≠90∘)(\angle A\neq 90^\circ), let OO, HH be the circumcenter and the foot of the altitude from AA respectively. Suppose MM, NN are the midpoints of BCBC, AHAH respectively. Let DD be the intersection of AOAO and BCBC and let H′H' be the reflection of HH about MM. Suppose that the circumcircle of OH′DOH'D intersects the circumcircle of BOCBOC at EE. Prove that NONO and AEAE are concurrent on the circumcircle of BOCBOC.
Proposed by Mehran Talaei
geometry