MathDB

2

Part of 2019 Sharygin Geometry Olympiad

Problems(4)

Sharygin CR 2019 P2

Source:

3/6/2019
The circle ω1\omega_1 passes through the center OO of the circle ω2\omega_2 and meets it at points AA and BB. The circle ω3\omega_3 centered at AA with radius ABAB meets ω1\omega_1 and ω2\omega_2 at points CC and DD (distinct from BB). Prove that C,O,DC, O, D are collinear.
geometry
Weirdly placed

Source: Sharygin 2019 Finals Day 1 Grade 8 P2

7/30/2019
A point MM inside triangle ABCABC is such that AM=AB/2AM=AB/2 and CM=BC/2CM=BC/2. Points C0C_0 and A0A_0 lying on ABAB and CBCB respectively are such that BC0:AC0=BA0:CA0=3BC_0:AC_0 = BA_0:CA_0 = 3. Prove that the distances from MM to C0C_0 and A0A_0 are equal.
geometrySharygin Geometry Olympiad
Reflection about.... perpendicular bisector?

Source: Sharygin 2019 Finals Day 1 Grade 9 P2

7/30/2019
Let PP be a point on the circumcircle of triangle ABCABC. Let A1A_1 be the reflection of the orthocenter of triangle PBCPBC about the reflection of the perpendicular bisector of BCBC. Points B1B_1 and C1C_1 are defined similarly. Prove that A1,B1,C1A_1,B_1,C_1 are collinear.
geometrySharygin Geometry Olympiadperpendicular bisector
Circumcircles intersect B_1C_1 at isotomic points

Source: Sharygin 2019 Finals Day 1 Grade 10 P2

7/30/2019
Let A1A_1, B1B_1, C1C_1 be the midpoints of sides BCBC, ACAC and ABAB of triangle ABCABC, AKAK be the altitude from AA, and LL be the tangency point of the incircle γ\gamma with BCBC. Let the circumcircles of triangles LKB1LKB_1 and A1LC1A_1LC_1 meet B1C1B_1C_1 for the second time at points XX and YY respectively, and γ\gamma meet this line at points ZZ and TT. Prove that XZ=YTXZ = YT.
geometrycircumcircle