MathDB

3

Part of 2017 Iranian Geometry Olympiad

Problems(3)

2017 IGO Elementary P3

Source: 4th Iranian Geometry Olympiad (Elementary) P3

9/15/2017
In the regular pentagon ABCDEABCDE, the perpendicular at CC to CDCD meets ABAB at FF. Prove that AE+AF=BEAE+AF=BE.
Proposed by Alireza Cheraghi
IGOIrangeometry
2017 IGO Intermediate P3

Source: 4th Iranian Geometry Olympiad (Intermediate) P3

9/15/2017
On the plane, nn points are given (n>2n>2). No three of them are collinear. Through each two of them the line is drawn, and among the other given points, the one nearest to this line is marked (in each case this point occurred to be unique). What is the maximal possible number of marked points for each given nn?
Proposed by Boris Frenkin (Russia)
IGOIrangeometry
2017 IGO Advanced P3

Source: 4th Iranian Geometry Olympiad (Advanced) P3

9/15/2017
Let OO be the circumcenter of triangle ABCABC. Line COCO intersects the altitude from AA at point KK. Let P,MP,M be the midpoints of AKAK, ACAC respectively. If POPO intersects BCBC at YY, and the circumcircle of triangle BCMBCM meets ABAB at XX, prove that BXOYBXOY is cyclic.
Proposed by Ali Daeinabi - Hamid Pardazi