MathDB

4

Part of 2021 OMpD

Problems(2)

Numbers in black and white squares are in a specific ratio

Source: 2021 2nd OMpD L3 P4 - Brazil - Olimphída Matemáticos por Diversão

7/8/2023
Determine the smallest positive integer nn with the following property: on a board n×nn \times n, whose squares are painted in checkerboard pattern (that is, for any two squares with a common edge, one of them is black and the other is white), it is possible to place the numbers 1,2,3,...,n21,2,3 , ... , n^2, a number in each square, so if BB is the sum of the numbers written in the white squares and PP is the sum of the numbers written in the black squares, then BP=20214321\frac {B}{P} = \frac{2021}{4321}.
checkerboardcombinatoricsratio
Davi Lopes vs Lavi Dopes, a legendary battle

Source: 2021 2nd OMpD L2 P4 - Brazil - Olimpíada Matemáticos por Diversão

7/8/2023
Let nn be a positive integer. Lavi Dopes has two boards n×nn \times n. On the first board, he writes an integer in each of his n2n^2 squares (the written numbers are not necessarily distinct). On the second board, he writes, on each square, the sum of the numbers corresponding, on the first board, to that square and to all its adjacent squares (that is, those that share a common vertex). For example, if n=3n = 3 and if Lavi Dopes writes the numbers on the first board, as shown below, the second board will look like this.
Next, Davi Lopes receives only the second board, and from it, he tries to discover the numbers written by Lavi Dopes on the first board.
(a) If n=4n = 4, is it possible that Davi Lopes always manages to find the numbers written by Lavi Dopes on the first board?
(b) If n=5n = 5, is it possible that Davi Lopes always manages to find the numbers written by Lavi Dopes on the first board?
combinatoricsgamegame strategy