MathDB

3

Part of 2018 Iranian Geometry Olympiad

Problems(3)

2018 IGO Elementary Level P3

Source:

9/20/2018
In the given figure, ABCDABCD is a parallelogram. We know that ∠D=60∘\angle D = 60^\circ, AD=2AD = 2 and AB=3+1AB = \sqrt3 + 1. Point MM is the midpoint of ADAD. Segment CKCK is the angle bisector of CC. Find the angle CKBCKB.
Proposed by Mahdi Etesamifard
IranIGO2018 igogeometry
equal segments starting with 2 intersecting circles (junior)

Source: Iranian Geometry Olympiad 2018 IGO Intermediate p3

9/19/2018
Let ω1,ω2\omega_1,\omega_2 be two circles with centers O1O_1 and O2O_2, respectively. These two circles intersect each other at points AA and BB. Line O1BO_1B intersects ω2\omega_2 for the second time at point CC, and line O2AO_2A intersects ω1\omega_1 for the second time at point DD . Let XX be the second intersection of ACAC and ω1\omega_1. Also YY is the second intersection point of BDBD and ω2\omega_2. Prove that CX=DYCX = DY .
Proposed by Alireza Dadgarnia
geometryequal segmentscircles
find so that exists n-gon if each diagonal is perp. bisector of another diagonal

Source: Iranian Geometry Olympiad 2018 IGO Advanced p3

9/19/2018
Find all possible values of integer n>3n > 3 such that there is a convex nn-gon in which, each diagonal is the perpendicular bisector of at least one other diagonal.
Proposed by Mahdi Etesamifard
geometrydiagonalperpendicularpolygon