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3

Part of 2019 Saint Petersburg Mathematical Olympiad

Problems(3)

2019 Saint Petersburg Grade 11 P3

Source:

4/14/2019
Kid and Karlsson play a game. Initially they have a square piece of chocolate 2019×20192019\times 2019 grid with 1×11\times 1 cells . On every turn Kid divides an arbitrary piece of chololate into three rectanglular pieces by cells, and then Karlsson chooses one of them and eats it. The game finishes when it's impossible to make a legal move. Kid wins if there was made an even number of moves, Karlsson wins if there was made an odd number of moves. Who has the winning strategy?
(Д. Ширяев)
Thanks to the user Vlados021 for translating the problem.
combinatorics
Simple inequality

Source: St. Petersburg Mathematical Olympiad 2019

4/13/2019
Let a,ba, b and cc be non-zero natural numbers such that c≥bc \geq b . Show that ab(a+b)c>cbac.a^b\left(a+b\right)^c>c^b a^c.
inequalities
inequality with 2 midpoints, a chord one and the corresponding arc

Source: St. Petersburg 2019 9.3

5/1/2019
Prove that the distance between the midpoint of side BCBC of triangle ABCABC and the midpoint of the arc ABCABC of its circumscribed circle is not less than AB/2AB / 2
inequalitiesgeometric inequalitymidpointarc midpoint