Problems(3)
Toggling most of rows and columns to get chessboard coloring
Source: St Petersburg 2021 11.2
12/23/2021
The cells of a table are colored white. In one move, it is allowed to select some cells from the same row or column and recolor each of them with the opposite color. What is the smallest number of moves needed to get a table with a chessboard coloring?S. Berlov
combinatoricsgrid
Primes number in a row with differences 6 or 12
Source: St Petersburg 2021 10.2
12/24/2021
Given are prime numbers written in a row. Each number, except for those in the two ends, differs from its two adjacent numbers with and . Prove that there are at least two equal numbers.
number theory
Rooks on a grid
Source: St. Petersburg 2021 9.2
12/23/2021
Misha has a x chessboard and a bag with rooks. In one move he can either put one rook from the bag on the lower left cell of the grid, or remove two rooks which are on the same cell, put one of them on the adjacent square which is above it or right to it, and put the other in the bag. Misha wants to place exactly rooks on the board, which don't beat each other. Will he be able to achieve such arrangement?
combinatorics