MathDB

5

Part of 2017 Iranian Geometry Olympiad

Problems(2)

2017 IGO Intermediate P5

Source: 4th Iranian Geometry Olympiad (Intermediate) P5

9/15/2017
Let X,YX,Y be two points on the side BCBC of triangle ABCABC such that 2XY=BC2XY=BC (XX is between B,YB,Y). Let AA′AA' be the diameter of the circumcirle of triangle AXYAXY. Let PP be the point where AXAX meets the perpendicular from BB to BCBC, and QQ be the point where AYAY meets the perpendicular from CC to BCBC. Prove that the tangent line from A′A' to the circumcircle of AXYAXY passes through the circumcenter of triangle APQAPQ.
Proposed by Iman Maghsoudi
IGOIrangeometrycircumcircle
2017 IGO Advanced P5

Source: 4th Iranian Geometry Olympiad (Advanced) P5

9/15/2017
Sphere SS touches a plane. Let A,B,C,DA,B,C,D be four points on the plane such that no three of them are collinear. Consider the point A′A' such that SS in tangent to the faces of tetrahedron A′BCDA'BCD. Points B′,C′,D′B',C',D' are defined similarly. Prove that A′,B′,C′,D′A',B',C',D' are coplanar and the plane A′B′C′D′A'B'C'D' touches SS.
Proposed by Alexey Zaslavsky (Russia)
IGOIrangeometry3D geometryspheretetrahedron