This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

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Found problems: 169

2009 Peru MO (ONEM), 4
Tags: combinatorics , tiles , dominoes , Tiling
Let $ n$ be a positive integer. A $4\times n$ rectangular grid is divided in$ 2\times 1$ or $1\times 2$ rectangles (as if it were completely covered with tiles of domino, no overlaps or gaps). Then all the grid points which are vertices of one of the $2\times 1$ or $1\times 2$ rectangles, are painted red. What is the least amount of red points you can get?

KoMaL A Problems 2017/2018, A. 715
Tags: Tiling , combinatorics , counting
Let $a$ and $b$ be positive integers. We tile a rectangle with dimensions $a$ and $b$ using squares whose side-length is a power of $2$, i.e. the tiling may include squares of dimensions $1\times 1, 2\times 2, 4\times 4$ etc. Denote by $M$ the minimal number of squares in such a tiling. Numbers $a$ and $b$ can be uniquely represented as the sum of distinct powers of $2$: $a=2^{a_1}+\cdots+2^{a_k},\; b=2^{b_1}+\cdots +2^{b_\ell}.$ Show that $$M=\sum_{i=1}^k \;\sum_{j=1}^{\ell} 2^{|a_i-b_j|}.$$

2020 Dutch IMO TST, 4
Tags: combinatorics , tiles , Tiling
Given are two positive integers $k$ and $n$ with $k \le n \le 2k - 1$. Julian has a large stack of rectangular $k \times 1$ tiles. Merlin calls a positive integer $m$ and receives $m$ tiles from Julian to place on an $n \times n$ board. Julian first writes on every tile whether it should be a horizontal or a vertical tile. Tiles may be used the board should not overlap or protrude. What is the largest number $m$ that Merlin can call if he wants to make sure that he has all tiles according to the rule of Julian can put on the plate?

2018 Germany Team Selection Test, 1
Tags: rectangle , IMO Shortlist , combinatorics , Tiling
A rectangle $\mathcal{R}$ with odd integer side lengths is divided into small rectangles with integer side lengths. Prove that there is at least one among the small rectangles whose distances from the four sides of $\mathcal{R}$ are either all odd or all even. [i]Proposed by Jeck Lim, Singapore[/i]

1995 Poland - Second Round, 6
Tags: Tiling , combinatorics , combinatorial geometry
Determine all positive integers $n$ for which the square $n \times n$ can be cut into squares $2\times 2$ and $3\times3$ (with the sides parallel to the sides of the big square).

2020 Thailand TSTST, 6
Tags: combinatorics , table , Tiling
Prove that the unit square can be tiled with rectangles (not necessarily of the same size) similar to a rectangle of size $1\times(3+\sqrt[3]{3})$.

1998 Finnish National High School Mathematics Competition, 5
Tags: combinatorics unsolved , combinatorics , Tiling
$15\times 36$-checkerboard is covered with square tiles. There are two kinds of tiles, with side $7$ or $5.$ Tiles are supposed to cover whole squares of the board and be non-overlapping. What is the maximum number of squares to be covered?

1999 Tournament Of Towns, 5
Tags: combinatorial geometry , combinatorics , rectangle , Tiling
Is it possible to divide a $8 \times 8$ chessboard into $32$ rectangles, each either $1 \times 2$ or $2 \times 1$, and to draw exactly one diagonal on each rectangle such that no two of these diagonals have a common endpoint? (A Shapovalov)

1999 Tournament Of Towns, 4
Tags: combinatorial geometry , combinatorics , tlies , Tiling , Equilateral , Equilateral Triangle
A black unit equilateral triangle is drawn on the plane. How can we place nine tiles, each a unit equilateral triangle, on the plane so that they do not overlap, and each tile covers at least one interior point of the black triangle? (Folklore)

2002 IMO Shortlist, 2
Tags: combinatorics , Tiling , dissection , IMO Shortlist
For $n$ an odd positive integer, the unit squares of an $n\times n$ chessboard are coloured alternately black and white, with the four corners coloured black. A it tromino is an $L$-shape formed by three connected unit squares. For which values of $n$ is it possible to cover all the black squares with non-overlapping trominos? When it is possible, what is the minimum number of trominos needed?

1990 Austrian-Polish Competition, 8
Tags: rectangle , tiles , Tiling , combinatorics , combinatorial geometry
We are given a supply of $a \times b$ tiles with $a$ and $b$ distinct positive integers. The tiles are to be used to tile a $28 \times 48$ rectangle. Find $a, b$ such that the tile has the smallest possible area and there is only one possible tiling. (If there are two distinct tilings, one of which is a reflection of the other, then we treat that as more than one possible tiling. Similarly for other symmetries.) Find $a, b$ such that the tile has the largest possible area and there is more than one possible tiling.

2012 Czech-Polish-Slovak Junior Match, 6
Tags: combinatorics , Tiling
The $8 \times 8$ board is covered with the same shape as in the picture to the right (each of the shapes can be rotated $90^o$) so that any two do not overlap or extend beyond the edge of the chessboard. Determine the largest possible number of fields of this chessboard can be covered as described above. [img]https://cdn.artofproblemsolving.com/attachments/e/5/d7f44f37857eb115edad5ea26400cdca04e107.png[/img]

1998 Tournament Of Towns, 2
Tags: combinatorics , combinatorial geometry , Tiling , Rectangles , geometry , rectangle
A square of side $1$ is divided into rectangles . We choose one of the two smaller sides of each rectangle (if the rectangle is a square, then we choose any of the four sides) . Prove that the sum of the lengths of all the chosen sides is at least $1$ . (Folklore)

2015 Caucasus Mathematical Olympiad, 3
Tags: Tiling , tiles , combinatorics , combinatorial geometry
The workers laid a floor of size $n \times n$ with tiles of two types: $2 \times 2$ and $3 \times 1$. It turned out that they were able to completely lay the floor in such a way that the same number of tiles of each type was used. Under what conditions could this happen? (You can’t cut tiles and also put them on top of each other.)

1941 Moscow Mathematical Olympiad, 082
Tags: combinatorial geometry , minimum , geometry , Triangle , Tiling
* Given $\vartriangle ABC$, divide it into the minimal number of parts so that after being flipped over these parts can constitute the same $\vartriangle ABC$.

2003 Kazakhstan National Olympiad, 7
Tags: combinatorics , Tiling , dissection , IMO Shortlist
For $n$ an odd positive integer, the unit squares of an $n\times n$ chessboard are coloured alternately black and white, with the four corners coloured black. A it tromino is an $L$-shape formed by three connected unit squares. For which values of $n$ is it possible to cover all the black squares with non-overlapping trominos? When it is possible, what is the minimum number of trominos needed?

1994 North Macedonia National Olympiad, 5
Tags: Tiling , combinatorics , combinatorial geometry
A square with the dimension $ 1 \times1 $ has been removed from a square board $ 3 ^n \times 3 ^n $ ($ n \in \mathbb {N}, $ $ n> 1 $). a) Prove that any defective board with the dimension $ 3 ^ n \times3 ^ n $ can be covered with shaped figures of shape 1 (the 3 squares' one) and of shape 2 (the 5 squares' one). Figures covering the board must not overlap each other and must not cross the edge of the board. Also the squares removed from the board must not be covered. (b) How many small figures in shape 2 must be used to cover the board? [img]https://cdn.artofproblemsolving.com/attachments/4/7/e970fadd7acc7fd6f5897f1766a84787f37acc.png[/img]

2004 Switzerland - Final Round, 10
Tags: combinatorics , tiles , Tiling
Let $n > 1$ be an odd natural number. The squares of an $n \times n$ chessboard are alternately colored white and black so that the four corner squares are black. An $L$-triomino is an $L$-shaped piece that covers exactly three squares of the board. For which values ​​of $n$ is it possible to cover all black squares with $L$-triominoes, so that no two $L$-triominos overlap? For these values ​​of $n$ determine the smallest possible number of $L$-triominoes that are necessary for this.

2010 Belarus Team Selection Test, 6.3
Tags: Tiling , combinatorial geometry , combinatorics , tiles
A $50 \times 50$ square board is tiled by the tetrominoes of the following three types: [img]https://cdn.artofproblemsolving.com/attachments/2/9/62c0bce6356ea3edd8a2ebfe0269559b7527f1.png[/img] Find the greatest and the smallest possible number of $L$ -shaped tetrominoes In the tiling. (Folklore)

2004 BAMO, 1
Tags: Tiling , combinatorics
A tiling of the plane with polygons consists of placing the polygons in the plane so that interiors of polygons do not overlap, each vertex of one polygon coincides with a vertex of another polygon, and no point of the plane is left uncovered. A unit polygon is a polygon with all sides of length one. It is quite easy to tile the plane with infinitely many unit squares. Likewise, it is easy to tile the plane with infinitely many unit equilateral triangles. (a) Prove that there is a tiling of the plane with infinitely many unit squares and infinitely many unit equilateral triangles in the same tiling. (b) Prove that it is impossible to find a tiling of the plane with infinitely many unit squares and finitely many (and at least one) unit equilateral triangles in the same tiling.

2021 Iranian Combinatorics Olympiad, P2
Tags: combinatorics , Tiling
We assume a truck as a $1 \times (k + 1)$ tile. Our parking is a $(2k + 1) \times (2k + 1)$ table and there are $t$ trucks parked in it. Some trucks are parked horizontally and some trucks are parked vertically in the parking. The vertical trucks can only move vertically (in their column) and the horizontal trucks can only move horizontally (in their row). Another truck is willing to enter the parking lot (it can only enter from somewhere on the boundary). For $3k + 1 < t < 4k$, prove that we can move other trucks forward or backward in such a way that the new truck would be able to enter the lot. Prove that the statement is not necessarily true for $t = 3k + 1$.

1990 Tournament Of Towns, (266) 4
Tags: combinatorics , combinatorial geometry , Tiling
A square board with dimensions $100 \times 100$ is divided into $10 000 $unit squares. One of the squares is cut out. Is it possible to cover the rest of the board by isosceles right angled triangles which have hypotenuses of length $2$, and in such a way that their hypotenuses lie on sides of the squares and their other two sides lie on diagonals? The triangles must not overlap each other or extend beyond the edges of the board. (S Fomin, Leningrad)

1985 Polish MO Finals, 2
Tags: combinatorics , combinatorial geometry , Tiling
Given a square side $1$ and $2n$ positive reals $a_1, b_1, ... , a_n, b_n$ each $\le 1$ and satisfying $\sum a_ib_i \ge 100$. Show that the square can be covered with rectangles $R_i$ with sides length $(a_i, b_i)$ parallel to the square sides.

2004 Denmark MO - Mohr Contest, 5
Tags: combinatorics , combinatorial geometry , tiles , Tiling
Determine for which natural numbers $n$ you can cover a $2n \times 2n$ chessboard with non-overlapping $L$ pieces. An $L$ piece covers four spaces and has appearance like the letter $L$. The piece may be rotated and mirrored at will.

2011 Bundeswettbewerb Mathematik, 1
Tags: square , hexagon , Tiling , combinatorial geometry , tiles , angles , geometry
Prove that you can't split a square into finitely many hexagons, whose inner angles are all less than $180^o$.