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Part of 2015 Sharygin Geometry Olympiad

Problems(3)

any convex quadrilateral can be divided into 5 polygons with symmetry axes

Source: Sharygin Geometry Olympiad 2015 Final 8.4

8/1/2018
Prove that an arbitrary convex quadrilateral can be divided into five polygons having symmetry axes.
(N. Belukhov)
geometryconvex quadrilateralpartitioncutting the paper
fixed point for Russian 9graders

Source: Sharygin Geometry Olympiad 2015 Final 9.4

8/1/2018
A fixed triangle ABCABC is given. Point PP moves on its circumcircle so that segments BCBC and APAP intersect. Line APAP divides triangle BPCBPC into two triangles with incenters I1I_1 and I2I_2. Line I1I2I_1I_2 meets BCBC at point ZZ. Prove that all lines ZPZP pass through a fixed point.
(R. Krutovsky, A. Yakubov)
geometryFixed point
Touching points geometry

Source: Sharygin geometry olympiad 2015, grade 10, Final Round, Problem 4

7/17/2018
Let AA1AA_1, BB1BB_1, CC1CC_1 be the altitudes of an acute-angled, nonisosceles triangle ABCABC, and A2A_2, B2B_2, C2C_2 be the touching points of sides BCBC, CACA, ABAB with the correspondent excircles. It is known that line B1C1B_1C_1 touches the incircle of ABCABC. Prove that A1A_1 lies on the circumcircle of A2B2C2A_2B_2C_2.
geometry