Olympiad problems

2023 Harvard-MIT Mathematics Tournament, 6

Tags: grid , hmmt

Each cell of a $3 $ × $3$ grid is labeled with a digit in the set {$1, 2, 3, 4, 5$} Then, the maximum entry in each row and each column is recorded. Compute the number of labelings for which every digit from $1$ to $5$ is recorded at least once.

1998 Austrian-Polish Competition, 8

Tags: polygon , combinatorial geometry , combinatorics , grid , area

In each unit square of an infinite square grid a natural number is written. The polygons of area $n$ with sides going along the gridlines are called [i]admissible[/i], where $n > 2$ is a given natural number. The [i]value [/i] of an admissible polygon is defined as the sum of the numbers inside it. Prove that if the values of any two congruent admissible polygons are equal, then all the numbers written in the unit squares of the grid are equal. (We recall that a symmetric image of polygon $P$ is congruent to $P$.)

2018 Estonia Team Selection Test, 2

Tags: combinatorics , combinatorial geometry , square table , grid

Find the greatest number of depicted pieces composed of $4$ unit squares that can be placed without overlapping on an $n \times n$ grid (where n is a positive integer) in such a way that it is possible to move from some corner to the opposite corner via uncovered squares (moving between squares requires a common edge). The shapes can be rotated and reflected. [img]https://cdn.artofproblemsolving.com/attachments/b/d/f2978a24fdd737edfafa5927a8d2129eb586ee.png[/img]

2024 Turkey EGMO TST, 5

Tags: grid , prime number , combinatorics

Let $p$ be a given prime number. For positive integers $n,k\geq2$ let $S_1, S_2,\dots, S_n$ be unit square sets constructed by choosing exactly one unit square from each of the columns from $p\times k$ chess board. If $|S_i \cap S_j|=1$ for all $1\leq i < j \leq n$ and for any duo of unit squares which are located at different columns there exists $S_i$ such that both of these unit squares are in $S_i$ find all duos of $(n,k)$ in terms of $p$. Note: Here we denote the number of rows by $p$ and the number of columns by $k$.

2024 China Western Mathematical Olympiad, 4

Tags: grid , combinatorics

Given positive integer $n \geq 2$. Now Cindy fills each cell of the $n*n$ grid with a positive integer not greater than $n$ such that the numbers in each row are in a non-decreasing order (from left to right) and numbers in each column is also in a non-decreasing order (from top to bottom). We call two adjacant cells form a $domino$ , if they are filled with the same number. Now Cindy wants the number of $domino$s as small as possible. Find the smallest number of $dominos$ Cindy can reach. (Here, two cells are called adjacant if they share one common side)

Kvant 2023, M2766

Tags: combinatorics , grid

Let $n{}$ be a natural number. The playing field for a "Master Sudoku" is composed of the $n(n+1)/2$ cells located on or below the main diagonal of an $n\times n$ square. A teacher secretly selects $n{}$ cells of the playing field and tells his student [list] [*]the number of selected cells on each row, and [*]that there is one selected cell on each column. [/list]The teacher's selected cells form a Master Sudoku if his student can determine them with the given information. How many Master Sudokus are there? [i]Proposed by T. Amdeberkhan, M. Ruby and F. Petrov[/i]

Durer Math Competition CD Finals - geometry, 2008.D1

Tags: geometry , lattice points , distance , grid , number theory

Given a square grid where the distance between two adjacent grid points is $1$. Can the distance between two grid points be $\sqrt5, \sqrt6, \sqrt7$ or $\sqrt{2007}$ ?

2024 Singapore MO Open, Q5

Tags: combinatorics , grid , number theory , prime number , prime

Let $p$ be a prime number. Determine the largest possible $n$ such that the following holds: it is possible to fill an $n\times n$ table with integers $a_{ik}$ in the $i$th row and $k$th column, for $1\le i,k\le n$, such that for any quadruple $i,j,k,l$ with $1\le i<j\le n$ and $1\le k<l\le n$, the number $a_{ik}a_{jl}-a_{il}a_{jk}$ is not divisible by $p$. [i]Proposed by oneplusone[/i]

2025 Bangladesh Mathematical Olympiad, P5

Tags: combinatorics , coloring , pigeonhole principle , grid

In an $N \times N$ table consisting of small unit squares, some squares are coloured black and the other squares are coloured white. For each pair of columns and each pair of rows, the four squares on the intersections of these rows and columns must not all be of the same colour. What is the largest possible value of $N$?

1976 Swedish Mathematical Competition, 4

Tags: combinatorics , grid , square table

A number is placed in each cell of an $n \times n$ board so that the following holds: (A) the cells on the boundary all contain 0; (B) other cells on the main diagonal are each1 greater than the mean of the numbers to the left and right; (C) other cells are the mean of the numbers to the left and right. Show that (B) and (C) remain true if ''left and right'' is replaced by ''above and below''.

Russian TST 2018, P1

Tags: combinatorics , grid

Let $n$ be an odd positive integer, and consider an infinite square grid. Prove that it is impossible to fill in one of $1,2$ or $3$ in every cell, which simultaneously satisfies the following conditions: (1) Any two cells which share a common side does not have the same number filled in them. (2) For any $1\times 3$ or $3\times 1$ subgrid, the numbers filled does not contain $1,2,3$ in that order be it reading from top to bottom, bottom to top, or left to right, or right to left. (3) The sum of numbers of any $n\times n$ subgrid is the same.