Olympiad problems

2017 IMO Shortlist, C1

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]

2016 Czech-Polish-Slovak Junior Match, 4

Tags: combinatorial geometry , tiling , combinatorics

Several tiles congruent to the one shown in the picture below are to be fit inside a $11 \times 11$ square table, with each tile covering $6$ whole unit squares, no sticking out the square and no overlapping. (a) Determine the greatest number of tiles which can be placed this way. (b) Find, with a proof, all unit squares which have to be covered in any tiling with the maximal number of tiles. [img]https://cdn.artofproblemsolving.com/attachments/c/d/23d93e9d05eab94925fc54006fe05123f0dba9.png[/img] Poland

2004 Denmark MO - Mohr Contest, 5

Tags: combinatorics , combinatorial geometry , tiling , tiles

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.

2016 Baltic Way, 12

Tags: tiling , combinatorics

Does there exist a hexagon (not necessarily convex) with side lengths $1, 2, 3, 4, 5, 6$ (not necessarily in this order) that can be tiled with a) $31$ b) $32$ equilateral triangles with side length $1?$

2024 Czech and Slovak Olympiad III A, 3

Tags: tiling , tiles , combinatorics

Find the largest natural number $n$ such that any set of $n$ tetraminoes, each of which is one of the four shapes in the picture, can be placed without overlapping in a $20 \times 20$ table (no tetramino extends beyond the borders of the table), such that each tetramino covers exactly 4 cells of the 20x20 table. An individual tetramino is allowed to turn and flip at will. [img]https://cdn.artofproblemsolving.com/attachments/b/9/0dddb25c2aa07536b711ded8363679e47972d6.png[/img]

1987 Spain Mathematical Olympiad, 3

Tags: combinatorial geometry , triangle , tiling , combinatorics

A given triangle is divided into $n$ triangles in such a way that any line segment which is a side of a tiling triangle is either a side of another tiling triangle or a side of the given triangle. Let $s$ be the total number of sides and $v$ be the total number of vertices of the tiling triangles (counted without multiplicity). (a) Show that if $n$ is odd then such divisions are possible, but each of them has the same number $v$ of vertices and the same number $s$ of sides. Express $v$ and $s$ as functions of $n$. (b) Show that, for $n$ even, no such tiling is possible

2004 Switzerland - Final Round, 10

Tags: tiling , tiles , combinatorics

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.

1989 All Soviet Union Mathematical Olympiad, 502

Tags: rectangle , lattice points , tiling , polygon

Show that for each integer $n > 0$, there is a polygon with vertices at lattice points and all sides parallel to the axes, which can be dissected into $1 \times 2$ (and / or $2 \times 1$) rectangles in exactly $n$ ways.

2017 Junior Balkan Team Selection Tests - Romania, 4

Tags: combinatorial geometry , tiling , tiles , combinatorics

Two right isosceles triangles of legs equal to $1$ are glued together to form either an isosceles triangle - called [i]t-shape[/i] - of leg $\sqrt2$, or a parallelogram - called [i]p-shape[/i] - of sides $1$ and $\sqrt2$. Find all integers $m$ and $n, m, n \ge 2$, such that a rectangle $m \times n$ can be tilled with t-shapes and p-shapes.

2022 Saudi Arabia BMO + EGMO TST, 1.4

Tags: combinatorial geometry , tiling , tiles , combinatorics

The sword is a figure consisting of $6$ unit squares presented in the picture below (and any other figure obtained from it by rotation). [img]https://cdn.artofproblemsolving.com/attachments/4/3/08494627d043ea575703564e9e6b5ba63dc2ef.png[/img] Determine the largest number of swords that can be cut from a $6\times 11$ piece of paper divided into unit squares (each sword should consist of six such squares).

2018 Peru IMO TST, 1

Tags: rectangle , 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]

2011 Brazil Team Selection Test, 1

Tags: tiling , tiles , combinatorics

Find the smallest positive integer $n$ such that it is possible to paint each of the $64$ squares of an $8 \times 8$ board of one of $n$ colors so that any four squares that form an $L$ as in the following figure (or congruent figures obtained through rotations and/or reflections) have different colors. [img]https://cdn.artofproblemsolving.com/attachments/a/2/c8049b1be8f37657c058949e11faf041856da4.png[/img]

2000 Denmark MO - Mohr Contest, 4

Tags: geometry , tiling , rectangle

A rectangular floor is covered by a certain number of equally large quadratic tiles. The tiles along the edge are red, and the rest are white. There are equally many red and white tiles. How many tiles can there be?

1998 Tournament Of Towns, 2

Tags: coloring , tiling , combinatorics , combinatorial geometry

John and Mary each have a white $8 \times 8$ square divided into $1 \times 1$ cells. They have painted an equal number of cells on their respective squares in blue. Prove that one can cut up each of the two squares into $2 \times 1 $ dominoes so that it is possible to reassemble John's dominoes into a new square and Mary's dominoes into another square with the same pattern of blue cells. (A Shapovalov)

2016 Miklós Schweitzer, 8

Tags: geometry , tiling , combinatorics , rectangle

For which integers $n>1$ does there exist a rectangle that can be subdivided into $n$ pairwise noncongruent rectangles similar to the original rectangle?

2013 Greece Team Selection Test, 4

Tags: tiling , combinatorics , trapezoid

Let $n$ be a positive integer. An equilateral triangle with side $n$ will be denoted by $T_n$ and is divided in $n^2$ unit equilateral triangles with sides parallel to the initial, forming a grid. We will call "trapezoid" the trapezoid which is formed by three equilateral triangles (one base is equal to one and the other is equal to two). Let also $m$ be a positive integer with $m<n$ and suppose that $T_n$ and $T_m$ can be tiled with "trapezoids". Prove that, if from $T_n$ we remove a $T_m$ with the same orientation, then the rest can be tiled with "trapezoids".

2009 Peru MO (ONEM), 4

Tags: domino , tiling , tiles , combinatorics

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?

1999 Tournament Of Towns, 5

Tags: combinatorial geometry , tiling , rectangle , combinatorics

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)

2021 Iranian Combinatorics Olympiad, P2

Tags: tiling , combinatorics

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 $t = 3k + 1$.

1994 Austrian-Polish Competition, 3

Tags: rectangle , tiling , square , combinatorics , combinatorial geometry

A rectangular building consists of $30$ square rooms situated like the cells of a $2 \times 15$ board. In each room there are three doors, each of which leads to another room (not necessarily different). How many ways are there to distribute the doors between the rooms so that it is possible to get from any room to any other one without leaving the building?

2012 Peru MO (ONEM), 3

Tags: combinatorics , tiles , tiling , domino

A domino is a $1\times2$ or $2\times 1$ rectangle. Diego wants to completely cover a $6\times 6$ board using $18$ dominoes. Determine the smallest positive integer $k$ for which Diego can place $k$ dominoes on the board (without overlapping) such that what remains of the board can be covered uniquely using the remaining dominoes.

1940 Moscow Mathematical Olympiad, 064

Tags: combinatorial geometry , combinatorics , tiling

How does one tile a plane, without gaps or overlappings, with the tiles equal to a given irregular quadrilateral?

2019 Iran RMM TST, 3

Tags: tiling , combinatorics

An infinite network is partitioned with dominos. Prove there exist three other tilings with dominos, have neither common domino with the existing tiling nor with each other. Clarifications for network: It means an infinite board consisting of square cells.

2019 Canada National Olympiad, 3

Tags: checkerboard , combinatorics , grid , tiling , bijection , induction

You have a $2m$ by $2n$ grid of squares coloured in the same way as a standard checkerboard. Find the total number of ways to place $mn$ counters on white squares so that each square contains at most one counter and no two counters are in diagonally adjacent white squares.

2001 Slovenia National Olympiad, Problem 4

Tags: tiling , combinatorics

Find the smallest number of squares on an $8\times8$ board that should be colored so that every $L$-tromino on the board contains at least one colored square.