MathDB

4

Part of 2021 Cyprus JBMO TST

Problems(2)

Colouring of 4 \times 19 chessboard with three colours

Source: Cyprus 2021Junior TST-2 Problem 4

5/26/2021
We colour every square of a 4×194\times 19 chess board with one of the colours red, green and blue. Prove that however this colouring is done, we can always find two horizontal rows and two vertical columns such that the 44 squares on the intersections of these lines all have the same colour.
combinatoricspigeonhole principleRamsey Theory
Midpoints and equal angles

Source: Cyprus 2021Junior TST-3 Problem 4

5/19/2021
Let △ABΓ\triangle AB\varGamma be an acute-angled triangle with AB<AΓAB < A\varGamma, and let OO be the center of the circumcircle of the triangle. On the sides ABAB and AΓA \varGamma we select points TT and PP respectively such that OT=OPOT=OP. Let M,KM,K and Λ\varLambda be the midpoints of PT,PBPT,PB and ΓT\varGamma T respectively. Prove that ∠TMK=∠MΛK\angle TMK = \angle M\varLambda K.
geometry