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4

Part of 2023 Assara - South Russian Girl's MO

Problems(2)

100 colors for 50x50 table, 1x2 tile related condition

Source: 2023 South Russian Girls MO - Assara Juniors p4 / M2774 in no. 11-12 of Kvant, 2023

2/28/2024
In a 50×5050 \times 50 checkered square, each cell is painted in one of 100100 given colors so that all colors are present and it is impossible to cut a single-color domino from the square (i.e. a 1×21 \times 2 rectangle). Galiia wants to recolor all the cells of one of the colors into another color (out of the given 100100 colors) so that this condition is preserved (i.e., it is still impossible to cut out a domino of the same color). Is it true that Galiia will definitely be able to do this?
combinatoricsColoring
CD // AB, starting with equal intersecting circles

Source: 2023 South Russian Girls MO - Assara Seniors p4

3/1/2024
Two equal circles Ω1\Omega_1 and Ω2\Omega_2 intersect at points AA and BB, and MM is the midpoint of ABAB. Two rays were drawn from MM, lying in the same half-plane wrt ABAB (see figure). The first ray intersects the circles Ω1\Omega_1 and Ω2\Omega_2 at points X1X_1 and X2X_2, and the second ray intersects them at points Y1Y_1 and Y2Y_2, respectively. Let CC be the intersection point of straight lines AX1AX_1 and BY2BY_2, and let DD be the intersection point of straight lines AX2AX_2 and BY1BY_1. Prove that CD∥ABCD \parallel AB. https://cdn.artofproblemsolving.com/attachments/4/a/fae047c3956d8b30f15a9d88e8d12e5f4d48ec.png
geometryparallelequal circles