Olympiad problems

2007 Estonia National Olympiad, 5

In a grid of dimensions $n \times n$, a part of the squares is marked with crosses such that in each at least half of the $4 \times 4$ squares are marked. Find the least possible the total number of marked squares in the grid.

1985 Tournament Of Towns, (089) 5

Tags: combinatorics , digit , square table

The digits $0, 1 , 2, ..., 9$ are written in a $10 x 10$ table , each number appearing $10$ times . (a) Is it possible to write them in such a way that in any row or column there would be not more than $4$ different digits? (b) Prove that there must be a row or column containing more than $3$ different digits . { L . D . Kurlyandchik , Leningrad)

2013 Tournament of Towns, 4

Tags: chessboard , combinatorics , square table , coloring

There is a $8\times 8$ table, drawn in a plane and painted in a chess board fashion. Peter mentally chooses a square and an interior point in it. Basil can draws any polygon (without self-intersections) in the plane and ask Peter whether the chosen point is inside or outside this polygon. What is the minimal number of questions suffcient to determine whether the chosen point is black or white?

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]

1991 All Soviet Union Mathematical Olympiad, 549

Tags: combinatorics , square table , table

An $h \times k$ minor of an $n \times n$ table is the $hk$ cells which lie in $h$ rows and $k$ columns. The semiperimeter of the minor is $h + k$. A number of minors each with semiperimeter at least $n$ together include all the cells on the main diagonal. Show that they include at least half the cells in the table.

2019 Saudi Arabia Pre-TST + Training Tests, 2.2

Tags: combinatorics , square table

A sequence $(a_1, a_2,...,a_k)$ consisting of pairwise different cells of an $n\times n$ board is called a cycle if $k \ge 4$ and cell ai shares a side with cell $a_{i+1}$ for every $i = 1,2,..., k$, where $a_{k+1} = a_1$. We will say that a subset $X$ of the set of cells of a board is [i]malicious [/i] if every cycle on the board contains at least one cell belonging to $X$. Determine all real numbers $C$ with the following property: for every integer $n \ge 2$ on an $n\times n$ board there exists a malicious set containing at most $Cn^2$ cells.

2010 Estonia Team Selection Test, 6

Tags: combinatorics , coloring , square table

Every unit square of a $n \times n$ board is colored either red or blue so that among all 2 $\times 2$ squares on this board all possible colorings of $2 \times 2$ squares with these two colors are represented (colorings obtained from each other by rotation and reflection are considered different). a) Find the least possible value of $n$. b) For the least possible value of $n$ find the least possible number of red unit squares

2019 Saudi Arabia Pre-TST + Training Tests, 4.1

Tags: combinatorics , coloring , square table

Find the smallest positive integer $n$ with the following property: After painting black exactly $n$ cells of a $7\times 7$ board there always exists a $2\times 2$ square with at least three black cells.

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''.