MathDB

4

Part of 2023 Saint Petersburg Mathematical Olympiad

Problems(2)

Grid operations

Source: St. Petersburg 2023 11.4

8/12/2023
What is the minimal number of operations needed to repaint a entirely white grid 100×100100 \times 100 to be entirely black, if on one move we can choose 9999 cells from any row or column and change their color?
combinatorics
Two-side grid with numbers

Source: St. Petersburg 2023 9.4=10.4

8/12/2023
One side of a square sheet of paper is colored red, the other - in blue. On both sides, the sheet is divided into n2n^2 identical square cells. In each of these 2n22n^2 cells is written a number from 11 to kk. Find the smallest kk,for which the following properties hold simultaneously: (i) on the red side, any two numbers in different rows are distinct; (ii) on the blue side, any two numbers in different columns are different; (iii) for each of the n2n^2 squares of the partition, the number on the blue side is not equal to the number on the red side.
combinatorics