Olympiad problems

2025 Kyiv City MO Round 2, Problem 4

Tags: combinatorics , Tiling

A square $ K = 2025 \times 2025 $ is given. We define a [i]stick[/i] as a rectangle where one of its sides is $ 1 $, and the other side is a positive integer from $ 1 $ to $ 2025 $. Find the largest positive integer $ C $ such that the following condition holds: [list] [*] If several sticks with a total area not exceeding $ C $ are taken, it is always possible to place them inside the square $ K $ so that each stick fully completely covers an integer number of $ 1 \times 1 $ squares, and no $ 1 \times 1 $ square is covered by more than one stick. [/list] [i](Basically, you can rotate sticks, but they have to be aligned by lines of the grid)[/i] [i]Proposed by Anton Trygub[/i]

2018 Morocco TST., 2

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]

2022 Swedish Mathematical Competition, 1

Tags: combinatorics , Tiling , tiles , combinatorial geometry

What sizes of squares with integer sides can be completely covered without overlap by identical tiles consisting of three squares with side $1$ joined together in one $L$ shape? [center][img]https://cdn.artofproblemsolving.com/attachments/3/f/9fe95b05527857f7e44dfd033e6fb01e5d25a2.png[/img][/center]

2021-IMOC, C7

Tags: combinatorics , Tiling

Given a positive integer $n$, an $n$-gun is a $2n$-mino that is formed by putting a $1 \times n$ grid and an $n \times 1$ grid side by side so that one of the corner unit squares of the first grid is next to one of the corner unit squares of the second grid. Find the minimum possible $k$ such that it is possible to color the infinite planar grid with $k$ colors such that any $n$-gun cannot cover two different squares with the same color. [i]Itf0501[/i]

2020 Tournament Of Towns, 5

Tags: geometry , 3D geometry , sphere , combinatorial geometry , Tiling , covering , combinatorics

A triangle is given on a sphere of radius $1$, the sides of which are arcs of three different circles of radius $1$ centered in the center of a sphere having less than $\pi$ in length and an area equal to a quarter of the area of the sphere. Prove that four copies of such a triangle can cover the entire sphere. A. Zaslavsky

2022 Austrian Junior Regional Competition, 2

Tags: combinatorics , tiles , Tiling

You are given a rectangular playing field of size $13 \times 2$ and any number of dominoes of sizes $2\times 1$ and $3\times 1$. The playing field should be seamless with such dominoes and without overlapping, with no domino protruding beyond the playing field may. Furthermore, all dominoes must be aligned in the same way, i. e. their long sides must be parallel to each other. How many such coverings are possible? (Walther Janous)

1940 Moscow Mathematical Olympiad, 064

Tags: combinatorics , combinatorial geometry , Tiling

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

2017 Romanian Masters In Mathematics, 5

Tags: combinatorics , RMM , RMM 2017 , Tiling

Fix an integer $n \geq 2$. An $n\times n$ sieve is an $n\times n$ array with $n$ cells removed so that exactly one cell is removed from every row and every column. A stick is a $1\times k$ or $k\times 1$ array for any positive integer $k$. For any sieve $A$, let $m(A)$ be the minimal number of sticks required to partition $A$. Find all possible values of $m(A)$, as $A$ varies over all possible $n\times n$ sieves. [i]Palmer Mebane[/i]

2012 Peru MO (ONEM), 3

Tags: combinatorics , dominoes , tiles , Tiling

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.

1992 All Soviet Union Mathematical Olympiad, 578

Tags: combinatorics , equilaterals , Tiling , Equilateral , combinatorial geometry

An equilateral triangle side $10$ is divided into $100$ equilateral triangles of side $1$ by lines parallel to its sides. There are m equilateral tiles of $4$ unit triangles and $25 - m$ straight tiles of $4$ unit triangles (as shown below). For which values of $m$ can they be used to tile the original triangle. [The straight tiles may be turned over.]

2022 Austrian MO National Competition, 6

Tags: combinatorics , tiles , Tiling

(a) Prove that a square with sides $1000$ divided into $31$ squares tiles, at least one of which has a side length less than $1$. (b) Show that a corresponding decomposition into $30$ squares is also possible. [i](Walther Janous)[/i]

1997 Denmark MO - Mohr Contest, 5

Tags: combinatorics , combinatorial geometry , tiles , Tiling

A $7\times 7$ square is cut into pieces following types: [img]https://cdn.artofproblemsolving.com/attachments/e/d/458b252c719946062b655340cbe8415d1bdaf9.png[/img] Show that exactly one of the pieces is of type (b). [img]https://cdn.artofproblemsolving.com/attachments/4/9/f3dd0e13fed9838969335c82f5fe866edc83e8.png[/img]

1978 Dutch Mathematical Olympiad, 2

Tags: combinatorics , combinatorial geometry , tiles , Tiling

One tiles a floor of $a \times b$ dm$^2$ with square tiles, $a,b \in N$. Tiles do not overlap, and sides of floor and tiles are parallel. Using tiles of $2\times 2$ dm$^2$ leaves the same amount of floor uncovered as using tiles of $4\times 4$ dm$^2$. Using $3\times 3$ dm$^2$ tiles leaves $29$ dm$^2$ floor uncovered. Determine $a$ and $b$.

2019 India IMO Training Camp, P2

Tags: combinatorics , Tiling

Let $n$ be a natural number. A tiling of a $2n \times 2n$ board is a placing of $2n^2$ dominos (of size $2 \times 1$ or $1 \times 2$) such that each of them covers exactly two squares of the board and they cover all the board.Consider now two [i]sepearate tilings[/i] of a $2n \times 2n$ board: one with red dominos and the other with blue dominos. We say two squares are red neighbours if they are covered by the same red domino in the red tiling; similarly define blue neighbours. Suppose we can assign a non-zero integer to each of the squares such that the number on any square equals the difference between the numbers on it's red and blue neighbours i.e the number on it's red neigbhbour minus the number on its blue neighbour. Show that $n$ is divisible by $3$ [i] Proposed by Tejaswi Navilarekallu [/i]

2022 Federal Competition For Advanced Students, P2, 6

Tags: combinatorics , tiles , Tiling

(a) Prove that a square with sides $1000$ divided into $31$ squares tiles, at least one of which has a side length less than $1$. (b) Show that a corresponding decomposition into $30$ squares is also possible. [i](Walther Janous)[/i]

2014 May Olympiad, 5

Tags: Tiling , tiles , combinatorial geometry , combinatorics , Coloring

Each square on a $ n \times n$ board, with $n \ge 3$, is colored with one of $ 8$ colors. For what values of $n$ it can be said that some of these figures included in the board, does it contain two squares of the same color. [img]https://cdn.artofproblemsolving.com/attachments/3/9/6af58460585772f39dd9e8ef1a2d9f37521317.png[/img]

1989 All Soviet Union Mathematical Olympiad, 502

Tags: lattice points , polygon , rectangle , Tiling

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.

1994 Tournament Of Towns, (409) 7

Tags: combinatorics , combinatorial geometry , Tiling

In a $10$ by $10$ square grid (which we call “the bay”) you are requested to place ten “ships”: one $1$ by $4$ ship, two $1$ by $3$ ships, three $1$ by $2$ ships and four $1$ by $1$ ships. The ships may not have common points (even corners) but may touch the “shore” of the bay. Prove that (a) by placing the ships one after the other arbitrarily but in the order indicated above, it is always possible to complete the process; (b) by placing the ships in reverse order (beginning with the smaller ones), it is possible to reach a situation where the next ship cannot be placed (give an example). (KN Ignatjev)

2011 Dutch IMO TST, 2

Tags: combinatorics , Tiling , tiles , tilings

We consider tilings of a rectangular $m \times n$-board with $1\times2$-tiles. The tiles can be placed either horizontally, or vertically, but they aren't allowed to overlap and to be placed partially outside of the board. All squares on theboard must be covered by a tile. (a) Prove that for every tiling of a $4 \times 2010$-board with $1\times2$-tiles there is a straight line cutting the board into two pieces such that every tile completely lies within one of the pieces. (b) Prove that there exists a tiling of a $5 \times 2010$-board with $1\times 2$-tiles such that there is no straight line cutting the board into two pieces such that every tile completely lies within one of the pieces.

2003 Germany Team Selection Test, 3

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?

2020 India National Olympiad, 6

Tags: Tiling , combinatorics , Coloring

A stromino is a $3 \times 1$ rectangle. Show that a $5 \times 5$ board divided into twenty-five $1 \times 1$ squares cannot be covered by $16$ strominos such that each stromino covers exactly three squares of the board, and every square is covered by one or two strominos. (A stromino can be placed either horizontally or vertically on the board.) [i]Proposed by Navilarekallu Tejaswi[/i]

1996 Tournament Of Towns, (511) 4

Tags: combinatorics , tiles , Tiling , combinatorial geometry

(a) A square is cut into right triangles with legs of lengths $3$ and $4$. Prove that the total number of the triangles is even. (b) A rectangle is cut into right triangles with legs of lengths $1$ and $2$. Prove that the total number of the triangles is even. (A Shapovalov)

1996 Austrian-Polish Competition, 9

Tags: combinatorics , Tiling , covering

For any triple $(a, b, c)$ of positive integers, not all equal, We are given sufficiently many rectangular blocks of size $a \times b \times c$. We use these blocks to fill up a cubic box of edge $10$. (a) Assume we have used at least $100$ blocks. Show that there are two blocks, one of which is a translate of the other. (b) Find a number smaller than $100$ (the smaller, the better) for which the above statement still holds.

2001 Saint Petersburg Mathematical Olympiad, 10.4

Tags: combinatorics , Tiling , board , table , Coloring , rectangle , domino

Rectangles $1\times20$, $1\times 19$, ..., $1\times 1$ were cut out of $20\times20$ table. Prove that from the remaining part of the table $36$ $1\times2$ dominos can be cut [I]Proposed by S. Berlov[/i]

2018 Peru IMO TST, 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]