MathDB

2

Part of 2020 Iranian Geometry Olympiad

Problems(3)

2020 IGO Elementary P2

Source: 7th Iranian Geometry Olympiad (Elementary) P2

11/4/2020
A parallelogram ABCDABCD is given (AB≠BCAB \neq BC). Points EE and GG are chosen on the line CD‾\overline{CD} such that AC‾\overline{AC} is the angle bisector of both angles ∠EAD\angle EAD and ∠BAG\angle BAG. The line BC‾\overline{BC} intersects AE‾\overline{AE} and AG‾\overline{AG} at FF and HH, respectively. Prove that the line FG‾\overline{FG} passes through the midpoint of HEHE. Proposed by Mahdi Etesamifard
geometryparallelogramangle bisector
2020 IGO Intermediate P2

Source: 7th Iranian Geometry Olympiad (Intermediate) P2

11/4/2020
Let ABCABC be an isosceles triangle (AB=ACAB = AC) with its circumcenter OO. Point NN is the midpoint of the segment BCBC and point MM is the reflection of the point NN with respect to the side ACAC. Suppose that TT is a point so that ANBTANBT is a rectangle. Prove that ∠OMT=12∠BAC\angle OMT = \frac{1}{2} \angle BAC.
Proposed by Ali Zamani
rectanglecircumcircleanglesTrianglegeometryIGO
2020 IGO Advanced P2

Source: 7th Iranian Geometry Olympiad (Advanced) P2

11/4/2020
Let △ABC\triangle ABC be an acute-angled triangle with its incenter II. Suppose that NN is the midpoint of the arc \overarc{BAC} of the circumcircle of triangle △ABC\triangle ABC, and PP is a point such that ABPCABPC is a parallelogram.Let QQ be the reflection of AA over NN and RR the projection of AA on QI‾\overline{QI}. Show that the line AI‾\overline{AI} is tangent to the circumcircle of triangle △PQR\triangle PQR Proposed by Patrik Bak - Slovakia
geometryIGOiranian geometry olympiadincentercircumcirclereflectionComputer problems