MathDB

2021 Cyprus JBMO TST

Part of Cyprus JBMO TST

Subcontests

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Colouring of 4 \times 19 chessboard with three colours

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.

Midpoints and equal angles

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.
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Operations on triples of integers on blackboard

George plays the following game: At every step he can replace a triple of integers (x,y,z)(x,y,z) which is written on the blackboard, with any of the following triples:
(i) (x,z,y)(x,z,y) (ii) (−x,y,z)(-x,y,z) (iii) (x+y,y,2x+y+z)(x+y,y,2x+y+z) (iv) (x−y,y,y+z−2x)(x-y,y,y+z-2x)
Initially, the triple (1,1,1)(1,1,1) is written on the blackboard. Determine whether George can, with a sequence of allowed steps, end up at the triple (2021,2019,2023)(2021,2019,2023), fully justifying your answer.

Trapezium and circle inscribed in rhombus

Let ABΓΔAB\varGamma\varDelta be a rhombus.
(a) Prove that you can construct a circle (c)(c) which is inscribed in the rhombus and is tangent to its sides. (b) The points Θ,H,K,I\varTheta,H,K,I are on the sides ΔΓ,BΓ,AB,AΔ\varDelta\varGamma,B\varGamma,AB,A\varDelta of the rhombus respectively, such that the line segments KHKH and IΘI\varTheta are tangent on the circle (c)(c). Prove that the quadrilateral defined from the points Θ,H,K,I\varTheta,H,K,I is a trapezium.
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