Problems(3)
Rearrange numbers such a sum in all 1 \times 3 rectangles doesn't change
Source: Saint Petersburg olympiad 2024, 11.1
9/22/2024
The table is filled with numbers from to as shown in the figure. Is it possible to rearrange some numbers so that there is still one number in each cell, and so that the sum of the numbers does not change in all rectangles of three cells?
combinatoricsalgebra
Sum of all numbers is divisible by 13
Source: Saint Petersburg olympiad 2024, 10.1
9/22/2024
In the cells of the board, integers are arranged so that in any rectangle (vertical or horizontal) with one cut corner cell that does not go beyond the board, the sum of the numbers is divided by . Prove that the sum of all the numbers on the board is divisible by .
combinatoricsnumber theory
(Ir)rational points on line are colored
Source: Saint Petersburg olympiad 2024, 9.1
9/21/2024
Dima has red and blue felt—tip pens, with one of them he paints rational points on the numerical axis, and with the other - irrational ones. Dima colored rational and irrational points, after which he erased the signatures that allowed to find out where the origin was and what the scale was. Sergey has a compass with which he can measure the distance between any two colored points and , and then mark on the axis a point located at a measured distance from any colored point (left or right); at the same time, Dima immediately paints it with the appropriate felt-tip pen. How Sergei can find out what color Dima paints rational points and what color he paints irrational ones?
number theory