Olympiad problems

2024 Junior Balkan Team Selection Tests - Romania, P5

Tags: combinatorics , board

An [i]$n$-type triangle[/i] where $n\geqslant 2$ is formed by the cells of a $(2n+1)\times(2n+1)$ board, situated under both main diagonals. For instance, a $3$-type triangle looks like this:[img]https://i.ibb.co/k4fmwWY/Screenshot-2024-07-31-153932.png[/img]Determine the maximal length of a sequence with pairwise distinct cells in an $n$-type triangle, such that, beggining with the second one, any cell of the sequence has a common side with the previous one. [i]Cristi Săvescu[/i]

2022/2023 Tournament of Towns, P2

Tags: combinatorics , board

There is a bacterium in one of the cells of a $10 \times 10{}$ checkered board. At the first move, the bacterium shifts to a cell adjacent by side to the original one, and divides into two bacteria (both stay in the same cell). Then again, one of the bacteria on the board shifts to a cell adjacent by side and divides into two bacteria, and so on. Is it possible that after some number of such moves the number of bacteria in each cell of the board is the same? [i]Alexandr Gribalko[/i]

2015 Rioplatense Mathematical Olympiad, Level 3, 4

Tags: combinatorics , board , coloring

You have a $9 \times 9$ board with white squares. A tile can be moved from one square to another neighbor (tiles that share one side). If we paint some squares of black, we say that such coloration is [i]good [/i] if there is a white square where we can place a chip that moving through white squares can return to the initial square having passed through at least $3$ boxes, without passing the same square twice. Find the highest possible value of $k$ such that any form of painting $k$ squares of black are a [i]good [/i] coloring.

2021 New Zealand MO, 8

Tags: combinatorics , board

Two cells in a $20 \times 20$ board are adjacent if they have a common edge (a cell is not considered adjacent to itself). What is the maximum number of cells that can be marked in a $20 \times 20$ board such that every cell is adjacent to at most one marked cell?

2024 Rioplatense Mathematical Olympiad, 1

Tags: grid , board , rioplatense , combinatorics

Ana draws a checkered board that has at least 20 rows and at least 24 columns. Then, Beto must completely cover that board, without holes or overlaps, using only pieces of the following two types: Each piece must cover exactly 4 or 3 squares of the board, as shown in the figure, without leaving the board. It is permitted to rotate the pieces and it is not necessary to use all types of pieces. Explain why, regardless of how many rows and how many columns Ana's board has, Beto can always complete his task.

2020 Lusophon Mathematical Olympiad, 5

Tags: board , combinatorics

In how many ways can we fill the cells of a $4\times4$ grid such that each cell contains exactly one positive integer and the product of the numbers in each row and each column is $2020$?

2021 Regional Olympiad of Mexico Center Zone, 2

Tags: combinatorics , board

The Mictlán is an $n\times n$ board and each border of each $1\times 1$ cell is painted either purple or orange. Initially, a catrina lies inside a $1\times 1$ cell and may move in four directions (up, down, left, right) into another cell with the condition that she may move from one cell to another only if the border that joins them is painted orange. We know that no matter which cell the catrina chooses as a starting point, she may reach all other cells through a sequence of valid movements and she may not abandon the Mictlán (she may not leave the board). What is the maximum number of borders that could have been colored purple? [i]Proposed by CDMX[/i]

2020 Bosnia and Herzegovina Junior BMO TST, 2

Tags: combinatorics , number , board

A board $n \times n$ is divided into $n^2$ unit squares and a number is written in each unit square. Such a board is called [i] interesting[/i] if the following conditions hold: $\circ$ In all unit squares below the main diagonal, the number $0$ is written; $\circ$ Positive integers are written in all other unit squares. $\circ$ When we look at the sums in all $n$ rows, and the sums in all $n$ columns, those $2n$ numbers are actually the numbers $1,2,...,2n$ (not necessarily in that order). $a)$ Determine the largest number that can appear in a $6 \times 6$ [i]interesting[/i] board. $b)$ Prove that there is no [i]interesting[/i] board of dimensions $7\times 7$.

2021 Regional Olympiad of Mexico Southeast, 4

Tags: combinatorics , board

Hernan wants to paint a $8\times 8$ board such that every square is painted with blue or red. Also wants to every $3\times 3$ subsquare have exactly $a$ blue squares and every $2\times 4$ or $4\times 2$ rectangle have exactly $b$ blue squares. Find all couples $(a,b)$ such that Hernan can do the required.