MathDB

6

Part of 2022 239 Open Mathematical Olympiad

Problems(2)

Problem 6

Source: 239-School Open Olympiad (Senior Level)

4/25/2022
239239 wise men stand in a circle near an opaque baobab. The king put on the head of each of these wise men a hat og one of 1616 colors. Each wise men does nor know the color of his hat and can only see the two nearest wise men on each side around the circle. Without communicating, these wise men must at the same time make a guess about the color of their hat ((i.e, tell one color)). These wise men were allowed to consult in advance, while they are afraid of being too lucky. What is the maximum kk for which, in any arrangement of hats, they can certainly ensure that no more than kk wise men guess the color of their hats??
combinatoricshatColors239
Geometry with rectangle

Source: 239-School Open Olympiad 2022, Junior League P6

4/19/2023
On the side BCBC of the rectangle ABCDABCD, a point PP{} is marked so that ∠APD=90∘\angle APD = 90^\circ. On the straight line ADAD, points QQ{} and RR{} are selected outside the segment ADAD such that AQ=BPAQ = BP and CP=DRCP = DR. The circle ω\omega passes through the points Q,DQ, D and the circumcenter of the triangle PDQPDQ. The circle γ\gamma passes through the points A,RA, R and the circumcenter of the triangle APRAPR. Prove that the radius of one of the circles touching the line ADAD and the circles ω\omega and γ\gamma is 2AB2AB.
geometrycircles