MathDB

8

Part of 2019 Tuymaada Olympiad

Problems(2)

game with 2x1, 1x 2, 1x3, 3x1 rectangles in a 1000x1000 board

Source: Tuymaada Olympiad 2019 juniors p8

7/22/2019
Andy, Bess, Charley and Dick play on a 1000×10001000 \times 1000 board. They make moves in turn: Andy first, then Bess, then Charley and finally Dick, after that Andy moves again and so on. At each move a player must paint several unpainted squares forming 2×1,1×2,1×32 \times 1, 1 \times 2, 1 \times 3, or 3×13 \times 1 rectangle. The player that cannot move loses. Prove that some three players can cooperate to make the fourth player lose.
geometryrectanglegamegame strategycombinatoricscombinatorial geometry
Reflection of arc midpoint w.r.t side of triangle lies on some line

Source: Tuymaada 2019 P8

7/15/2019
In △ABC\triangle ABC ∠B\angle B is obtuse and AB≠BCAB \ne BC. Let OO is the circumcenter and ω\omega is the circumcircle of this triangle. NN is the midpoint of arc ABCABC. The circumcircle of △BON\triangle BON intersects ACAC on points XX and YY. Let BX∩ω=P≠BBX \cap \omega = P \ne B and BY∩ω=Q≠BBY \cap \omega = Q \ne B. Prove that P,QP, Q and reflection of NN with respect to line ACAC are collinear.
geometrygeometric transformationreflectionarc midpointcircumcirclemoving pointsInversion