MathDB

3

Part of 2018 Saint Petersburg Mathematical Olympiad

Problems(3)

Tangent circles

Source: St Petersburg Olympiad 2018, Grade 11, P3

6/21/2018
Point TT lies on the bisector of ∠B\angle B of acuteangled △ABC\triangle ABC. Circle SS with diameter BTBT intersects ABAB and BCBC at points PP and QQ. Circle, that goes through point AA and tangent to SS at PP intersects line ACAC at XX. Circle, that goes through point CC and tangent to SS at QQ intersects line ACAC at YY. Prove, that TX=TYTX=TY
geometry
Flipping coins

Source: St Petersburg Olympiad 2018, Grade 10, P3

6/22/2018
nn coins lies in the circle. If two neighbour coins lies both head up or both tail up, then we can flip both. How many variants of coins are available that can not be obtained from each other by applying such operations?
combinatorics
Variable points for triangle

Source: St Petersburg Olympiad 2018, Grade 9, P3

6/22/2018
ABCABC is acuteangled triangle. Variable point XX lies on segment ACAC, and variable point YY lies on the ray BCBC but not segment BCBC, such that ∠ABX+∠CXY=90\angle ABX+\angle CXY =90. TT is projection of BB on the XYXY. Prove that all points TT lies on the line.
geometry