MathDB

5

Part of 2019 Iranian Geometry Olympiad

Problems(3)

2019 IGO Elementary P5

Source: 6th Iranian Geometry Olympiad (Elementary) P5

9/20/2019
For a convex polygon (i.e. all angles less than 180∘180^\circ) call a diagonal bisector if its bisects both area and perimeter of the polygon. What is the maximum number of bisector diagonals for a convex pentagon?
Proposed by Morteza Saghafian
IGOIrangeometrycombinatorics
2019 IGO Intermediate P5

Source: 6th Iranian Geometry Olympiad (Intermediate) P5

9/20/2019
Let ABCABC be a triangle with ∠A=60∘\angle A = 60^\circ. Points EE and FF are the foot of angle bisectors of vertices BB and CC respectively. Points PP and QQ are considered such that quadrilaterals BFPEBFPE and CEQFCEQF are parallelograms. Prove that ∠PAQ>150∘\angle PAQ > 150^\circ. (Consider the angle PAQPAQ that does not contain side ABAB of the triangle.)
Proposed by Alireza Dadgarnia
IGOIrangeometry
2019 IGO Advanced P5

Source: 6th Iranian Geometry Olympiad (Advanced) P5

9/20/2019
Let points A,BA, B and CC lie on the parabola Δ\Delta such that the point HH, orthocenter of triangle ABCABC, coincides with the focus of parabola Δ\Delta. Prove that by changing the position of points A,BA, B and CC on Δ\Delta so that the orthocenter remain at HH, inradius of triangle ABCABC remains unchanged.
Proposed by Mahdi Etesamifard
IGOIrangeometryparabola