MathDB

3

Part of 2020 Iranian Geometry Olympiad

Problems(3)

2020 IGO Intermediate P3

Source: 7th Iranian Geometry Olympiad (Intermediate) P3

11/4/2020
In acute-angled triangle ABCABC (AC>ABAC > AB), point HH is the orthocenter and point MM is the midpoint of the segment BCBC. The median AMAM intersects the circumcircle of triangle ABCABC at XX. The line CHCH intersects the perpendicular bisector of BCBC at EE and the circumcircle of the triangle ABCABC again at FF. Point JJ lies on circle ω\omega, passing through X,E,X, E, and FF, such that BCHJBCHJ is a trapezoid (CB∥HJCB \parallel HJ). Prove that JBJB and EMEM meet on ω\omega.
Proposed by Alireza Dadgarnia
geometrycircumcircleorthocenterIGO
2020 IGO Elementary P3

Source: 7th Iranian Geometry Olympiad (Elementary) P3

11/4/2020
According to the figure, three equilateral triangles with side lengths a,b,ca,b,c have one common vertex and do not have any other common point. The lengths x,yx, y, and zz are defined as in the figure. Prove that 3(x+y+z)>2(a+b+c)3(x+y+z)>2(a+b+c). Proposed by Mahdi Etesamifard
geometryIGO
2020 IGO Advanced P3

Source: 7th Iranian Geometry Olympiad (Advanced) P3

11/4/2020
Assume three circles mutually outside each other with the property that every line separating two of them have intersection with the interior of the third one. Prove that the sum of pairwise distances between their centers is at most 222\sqrt{2} times the sum of their radii. (A line separates two circles, whenever the circles do not have intersection with the line and are on different sides of it.) [color=#45818E]Note. Weaker results with 222\sqrt{2} replaced by some other cc may be awarded points depending on the value of c>22c>2\sqrt{2} Proposed by Morteza Saghafian
geometryIGOiranian geometry olympiad