MathDB

5

Part of 2020 Iranian Geometry Olympiad

Problems(3)

2020 IGO Elementary P5

Source: 7th Iranian Geometry Olympiad (Elementary) P5

11/4/2020
We say two vertices of a simple polygon are visible from each other if either they are adjacent, or the segment joining them is completely inside the polygon (except two endpoints that lie on the boundary). Find all positive integers nn such that there exists a simple polygon with nn vertices in which every vertex is visible from exactly 44 other vertices. (A simple polygon is a polygon without hole that does not intersect itself.) Proposed by Morteza Saghafian
geometryIGO
2020 IGO Intermediate P5

Source: 7th Iranian Geometry Olympiad (Intermediate) P5

11/4/2020
Find all numbers n≥4n \geq 4 such that there exists a convex polyhedron with exactly nn faces, whose all faces are right-angled triangles. (Note that the angle between any pair of adjacent faces in a convex polyhedron is less than 180∘180^\circ.)
Proposed by Hesam Rajabzadeh
3D geometrygeometryIGO
2020 IGO Advanced P5

Source: 7th Iranian Geometry Olympiad (Advanced) P5

11/4/2020
Consider an acute-angled triangle △ABC\triangle ABC (AC>ABAC>AB) with its orthocenter HH and circumcircle Γ\Gamma.Points MM,PP are midpoints of BCBC and AHAH respectively.The line AM‾\overline{AM} meets Γ\Gamma again at XX and point NN lies on the line BC‾\overline{BC} so that NX‾\overline{NX} is tangent to Γ\Gamma. Points JJ and KK lie on the circle with diameter MPMP such that ∠AJP=∠HNM\angle AJP=\angle HNM (BB and JJ lie one the same side of AH‾\overline{AH}) and circle ω1\omega_1, passing through K,HK,H, and JJ, and circle ω2\omega_2 passing through K,MK,M, and NN, are externally tangent to each other. Prove that the common external tangents of ω1\omega_1 and ω2\omega_2 meet on the line NH‾\overline{NH}. Proposed by Alireza Dadgarnia
geometrycircumcircleIGO