MathDB

3

Part of 2023 Middle European Mathematical Olympiad

Problems(2)

Easy incircle geo

Source: MEMO 2023 I3

8/24/2023
Let ABCABC be a triangle with incenter II, and the incircle touches BCBC at DD. The points E,FE, F are such that BE∥AI∥CFBE \parallel AI \parallel CF and ∠BEI=∠CFI=90∘\angle BEI=\angle CFI=90^{\circ}. If DE,DFDE, DF meet the incircle at E′,F′E', F', show that E′F′⊥AIE'F' \perp AI.
geometryMEMO2023
Bishops on a chessboard

Source: MEMO 2023 T3

8/25/2023
Find the smallest integer bb with the following property: For each way of colouring exactly bb squares of an 8×88 \times 8 chessboard green, one can place 77 bishops on 77 green squares so that no two bishops attack each other.
combinatorics