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4

Part of 2023 Iran Team Selection Test

Problems(2)

Tiling game on regular hexonal grid

Source: Iran TST 2023 P4

9/16/2023
The game of Hive is played on a regular hexagonal grid (as shown in the figure) by 3 players. The grid consists of kk layers (where kk is a natural number) surrounding a regular hexagon, with each layer constructed around the previous layer. The figure below shows a grid with 2 layers.
The players, Ali, Shayan, and Sajad, take turns playing the game. In each turn, a player places a tile, similar to the one shown in the figure, on the empty cells of the grid (rotation of the tile is also allowed). The first player who is unable to place a tile on the grid loses the game. Prove that two players can collaborate in such a way that the third player always loses.
Proposed by Pouria Mahmoudkhan Shirazi.
combinatorics
Circle passing two fix point

Source: Iran TST 2023 ; Exam 2 Problem 4

3/16/2023
line ll through the point AA from triangle ABCABC . Point XX is on line ll.ωb{\omega}_b and ωc{\omega}_c are circles that through points X,AX,A and respectively tanget to ABAB adn ACAC. tangets from B,CB,C respectively to ωb{\omega}_b and ωc{\omega}_c meet them in Y,ZY,Z. Prove that by changing XX, the circumcircle of the circle XYZXYZ passes through two fixed points.
Proposed by Ali Zamani
geometryFixed point