MathDB

3

Part of 2019 Sharygin Geometry Olympiad

Problems(4)

Sharygin CR 2019 P3

Source:

3/6/2019
The rectangle ABCDABCD lies inside a circle. The rays BABA and DADA meet this circle at points A1A_1 and A2A_2. Let A0A_0 be the midpoint of A1A2A_1A_2. Points B0B_0, C0,D0C_0, D_0 are defined similarly. Prove that A0C0=B0D0A_0C_0 = B_0D_0.
geometry
Construction, as usual.

Source: Sharygin 2019 Finals Day 1 Grade 8 P3

7/30/2019
Construct a regular triangle using a plywood square. (You can draw a line through pairs of points lying on the distance less than the side of the square, construct a perpendicular from a point to the line the distance between them does not exceed the side of the square, and measure segments on the constructed lines equal to the side or to the diagonal of the square)
geometrySharygin Geometry Olympiad
Finding Locus

Source: Sharygin 2019 Finals Day 1 Grade 9 P3

7/30/2019
Let ABCDABCD be a cyclic quadrilateral such that AD=BD=ACAD=BD=AC. A point PP moves along the circumcircle ω\omega of triangle ABCDABCD. The lined APAP and DPDP meet the lines CDCD and ABAB at points EE and FF respectively. The lines BEBE and CFCF meet point QQ. Find the locus of QQ.
geometrySharygin Geometry Olympiadmoving points
AA_1, BB_1, CC_1 concurrent in isogonal conjugate diagram + angle condition

Source: Sharygin 2019 Finals Day 1 Grade 10 P3

7/30/2019
Let PP and QQ be isogonal conjugates inside triangle ABCABC. Let ω\omega be the circumcircle of ABCABC. Let A1A_1 be a point on arc BCBC of ω\omega satisfying ∠BA1P=∠CA1Q\angle BA_1P = \angle CA_1Q. Points B1B_1 and C1C_1 are defined similarly. Prove that AA1AA_1, BB1BB_1, CC1CC_1 are concurrent.
geometry