Olympiad problems

2021 USEMO, 1

Let $n$ be a fixed positive integer and consider an $n\times n$ grid of real numbers. Determine the greatest possible number of cells $c$ in the grid such that the entry in $c$ is both strictly greater than the average of $c$'s column and strictly less than the average of $c$'s row. [i]Proposed by Holden Mui[/i]

combinatorics grid problem square grid grid algebra double counting mathematical induction number theory

2019 Moroccan TST, 5

Let $n$ be a nonzero even integer. We fill up all the cells of an $n\times n$ grid with $+$ and $-$ signs ensuring that the number of $+$ signs equals the number of $-$ signs. Show that there exists two rows with the same number of $+$ signs or two collumns with the same number of $+$ signs.

combinatorics grid problem

KoMaL A Problems 2023/2024, A. 865

A crossword is a grid of black and white cells such that every white cell belongs to some $2\times 2$ square of white cells. A word in the crossword is a contiguous sequence of two or more white cells in the same row or column, delimited on each side by either a black cell or the boundary of the grid. Show that the total number of words in an $n\times n$ crossword cannot exceed $(n+1)^2/2$. [i]Proposed by Nikolai Beluhov, Bulgaria[/i]

combinatorics grid problem casework sequences