MathDB

Problem 2

Part of 2022 Kyiv City MO Round 2

Problems(4)

Hedgehog recoloring points on circle

Source: Kyiv City MO 2022 Round 2, Problem 9.2, 10.2

1/30/2022
20222022 points are arranged in a circle, one of which is colored in black, and others in white. In one operation, The Hedgehog can do one of the following actions:
1) Choose two adjacent points of the same color and flip the color of both of them (white becomes black, black becomes white)
2) Choose two points of opposite colors with exactly one point in between them, and flip the color of both of them
Is it possible to achieve a configuration where each point has a color opposite to its initial color with these operations?
(Proposed by Oleksii Masalitin)
combinatoricsColoring
Trains go wild

Source: Kyiv City MO 2022 Round 2, Problem 7.2

1/30/2022
There is a central train station in point OO, which is connected to other train stations A1,A2,…,A8A_1, A_2, \ldots, A_8 with tracks. There is also a track between stations AiA_i and Ai+1A_{i+1} for each ii from 11 to 88 (here A9=A1A_9 = A_1). The length of each track AiAi+1A_iA_{i+1} is equal to 11, and the length of each track OAiOA_i is equal to 22, for each ii from 11 to 88.
There are also 88 trains B1,B2,…,B8B_1, B_2, \ldots, B_8, with speeds 1,2,…,81, 2, \ldots, 8 correspondently. Trains can move only by the tracks above, in both directions. No time is wasted on changing directions. If two or more trains meet at some point, they will move together from now on, with the speed equal to that of the fastest of them.
Is it possible to arrange trains into stations A1,A2,…,A8A_1, A_2, \ldots, A_8 (each station has to contain one train initially), and to organize their movement in such a way, that all trains arrive at OO in time t<12t < \frac{1}{2}?
(Proposed by Bogdan Rublov)
combinatorics
Bogdan and Monica playing weird games...

Source: Kyiv City MO 2022 Round 2, Problem 8.2, 11.1

1/30/2022
Monica and Bogdan are playing a game, depending on given integers n,kn, k. First, Monica writes some kk positive numbers. Bogdan wins, if he is able to find nn points on the plane with the following property: for any number mm written by Monica, there are some two points chosen by Bogdan with distance exactly mm between them. Otherwise, Monica wins.
Determine who has a winning strategy depending on n,kn, k.
(Proposed by Fedir Yudin)
combinatoricsgamegeometry
Computer making crazy operations with polynomials

Source: Kyiv City MO 2022 Round 2, Problem 11.2

1/30/2022
Initially memory of computer contained a single polynomial x2−1x^2-1. Every minute computer chooses any polynomial f(x)f(x) from its memory and writes f(x2−1)f(x^2-1) and f(x)2−1f(x)^2-1 to it, or chooses any two distinct polynomials g(x),h(x)g(x), h(x) from its memory and writes polynomial g(x)+h(x)2\frac{g(x) + h(x)}{2} to it (no polynomial is ever erased from its memory). Can it happen that after some time, memory of computer contains P(x)=11024(x2−1)2048−1P(x) = \frac{1}{1024}(x^2-1)^{2048} - 1?
(Proposed by Arsenii Nikolaiev)
algebrapolynomial