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Part of 2012 Oral Moscow Geometry Olympiad

Problems(2)

OI \perp AC, where I incenter and O circumcenter of excenters I_A,I_C, and I

Source: 2012 Oral Moscow Geometry Olympiad grades 8-9 p4

9/8/2019
In triangle ABCABC, point II is the center of the inscribed circle points, points IAI_A and ICI_C are the centers of the excircles, tangent to sides BCBC and ABAB, respectively. Point OO is the center of the circumscribed circle of triangle IIAICII_AI_C. Prove that OI⊥ACOI \perp AC
geometryincenterexcentersexcenterperpendicularcircumcircle
lines forming largeast angles with polyhedron faces and concurrent lines

Source: 2012 Oral Moscow Geometry Olympiad grades 10-11 p4

9/25/2019
Inside the convex polyhedron, the point PP and several lines ℓ1,ℓ2,...,ℓn\ell_1,\ell_2, ..., \ell_n passing through PP and not lying in the same plane. To each face of the polyhedron we associate one of the lines l1,l2,...,lnl_1, l_2, ..., l_n that forms the largest angle with the plane of this face (if there are there are several direct ones, we will choose any of them). Prove that there is a face that intersects with its corresponding line.
geometry3D geometrypolyhedron