MathDB

5

Part of 2023 Iranian Geometry Olympiad

Problems(3)

circumcircle of every triangle used in triangulation contains entire polygo

Source: 2023 IGO Elementary P5

1/18/2024
A polygon is decomposed into triangles by drawing some non-intersecting interior diagonals in such a way that for every pair of triangles of the triangulation sharing a common side, the sum of the angles opposite to this common side is greater than 180o180^o. a) Prove that this polygon is convex. b) Prove that the circumcircle of every triangle used in the decomposition contains the entire polygon.
Proposed by Morteza Saghafian - Iran
geometrycombinatoricscombinatorial geometrytriangulationconvex polygon
IGO 2023 Intermidiate P5

Source: IGO 2023 Intermidiate P5

1/18/2024
There are nn points in the plane such that at least 99%99\% of quadrilaterals with vertices from these points are convex. Can we find a convex polygon in the plane having at least 90%90\% of the points as vertices?
Proposed by Morteza Saghafian - Iran
geometry
tangent (ART) , (PEF) wanted

Source: 2023 IGO Adcanced P5

1/18/2024
In triangle ABCABC points MM and NN are the midpoints of sides ACAC and ABAB, respectively and DD is the projection of AA into BCBC. Point OO is the circumcenter of ABCABC and circumcircles of BOCBOC, DMNDMN intersect at points R,TR, T. Lines DTDT, DRDR intersect line MNMN at EE and FF, respectively. Lines CTCT, BRBR intersect at KK. A point PP lies on KDKD such that PKPK is the angle bisector of ∠BPC\angle BPC. Prove that the circumcircles of ARTART and PEFPEF are tangent.
Proposed by Mehran Talaei - Iran
geometrytangent circles