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P1

Part of 2022 3rd Memorial "Aleksandar Blazhevski-Cane"

Problems(2)

Chesslike squares with a Connect Four condition

Source: 3rd Memorial Mathematical Competition "Aleksandar Blazhevski - Cane"- Junior D1 P2/ Senior D1 P1

1/17/2022
A 6×66 \times 6 board is given such that each unit square is either red or green. It is known that there are no 44 adjacent unit squares of the same color in a horizontal, vertical, or diagonal line. A 2×22 \times 2 subsquare of the board is chesslike if it has one red and one green diagonal. Find the maximal possible number of chesslike squares on the board.
Proposed by Nikola Velov
combinatoricsboardColoringchesslikemaximum
Perpendicular bisector of an altitude in an isosceles triangle

Source: 3rd Memorial Mathematical Competition "Aleksandar Blazhevski - Cane"- Junior D1 P1

1/17/2022
Let ABCABC be an acute triangle with altitude ADAD (D∈BCD \in BC). The line through CC parallel to ABAB meets the perpendicular bisector of ADAD at GG. Show that AC=BCAC = BC if and only if ∠AGC=90∘\angle AGC = 90^{\circ}.
geometrycaneiffright angle