MathDB

8

Part of 2016 Sharygin Geometry Olympiad

Problems(3)

sould policemen know geometry? a criminal problem

Source: Sharygin Geometry Olympiad 2016 Final Round problem 8 grade 8

7/22/2018
A criminal is at point XX, and three policemen at points A,BA, B and CC block him up, i.e. the point XX lies inside the triangle ABCABC. Each evening one of the policemen is replaced in the following way: a new policeman takes the position equidistant from three former policemen, after this one of the former policemen goes away so that three remaining policemen block up the criminal too. May the policemen after some time occupy again the points A,BA, B and CC (it is known that at any moment XX does not lie on a side of the triangle)?
by V.Protasov
geometryproblem
Radical axis passes through curvilinear touch point

Source: Sharygin Geometry Olympiad, Final Round 2016, Problem 8 grade 9

8/4/2016
The diagonals of a cyclic quadrilateral meet at point MM. A circle ω\omega touches segments MAMA and MDMD at points P,QP,Q respectively and touches the circumcircle of ABCDABCD at point XX. Prove that XX lies on the radical axis of circles ACQACQ and BDPBDP.
(Proposed by Ivan Frolov)
geometrygeometry proposedHi
Radical Center lies on OI

Source: Sharygin geometry olympiad 2016, grade 10, Final Round, Problem 8.

8/5/2016
Let ABCABC be a non-isosceles triangle, let AA1AA_1 be its angle bisector and A2A_2 be the touching point of the incircle with side BCBC. The points B1,B2,C1,C2B_1,B_2,C_1,C_2 are defined similarly. Let OO and II be the circumcenter and the incenter of triangle ABCABC. Prove that the radical center of the circumcircle of the triangles AA1A2,BB1B2,CC1C2AA_1A_2, BB_1B_2, CC_1C_2 lies on the line OIOI.
geometrygeometry proposed