Found problems: 169
2015 Caucasus Mathematical Olympiad, 3
What is the smallest number of $3$-cell corners that you need to paint in a $5 \times5$ square so that you cannot paint more than one corner of one it? (Shaded corners should not overlap.)
2024 Sharygin Geometry Olympiad, 13
Can an arbitrary polygon be cut into isosceles trapezoids?
2003 Estonia National Olympiad, 1
Jiiri and Mari both wish to tile an $n \times n$ chessboard with cards shown in the picture (each card covers exactly one square). Jiiri wants that for each two cards that have a common edge, the neighbouring parts are of different color, and Mari wants that the neighbouring parts are always of the same color. How many possibilities does Jiiri have to tile the chessboard and how many possibilities does Mari have?
[img]https://cdn.artofproblemsolving.com/attachments/7/3/9c076eb17ba7ae7c000a2893c83288a94df384.png[/img]
2017 Romanian Master of Mathematics, 5
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]
1991 Chile National Olympiad, 3
A board of $6\times 6$ is totally covered by $18$ dominoes (of $2\times 1$), that is, there are no overlaps, gaps, and the tiles do not come off the board. Prove that, regardless of the arrangement of the tiles, there is always a line that divides the board into two non-empty parts, and without cutting tiles.
1999 Tournament Of Towns, 5
Is it possible to divide a $6 \times 6$ chessboard into $18$ 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)
2019 Tournament Of Towns, 3
Prove that any triangle can be cut into $2019$ quadrilaterals such that each quadrilateral is both inscribed and circumscribed.
(Nairi Sedrakyan)
2011 IFYM, Sozopol, 5
Let $n\geq 2$ be a natural number. A unit square is removed from a square $n$ x $n$ and the remaining figure is cut into squares 2 x 2 and 3 x 3. Determine all possible values of $n$.
2020 Durer Math Competition Finals, 1
How many ways are there to tile a $3 \times 3$ square with $4$ dominoes of size $1 \times 2$ and $1$ domino of size $1 \times 1$?
Tilings that can be obtained from each other by rotating the square are considered different. Dominoes of the same size are completely identical
2020 March Advanced Contest, 3
A [i]simple polygon[/i] is a polygon whose perimeter does not self-intersect. Suppose a simple polygon $\mathcal P$ can be tiled with a finite number of parallelograms. Prove that regardless of the tiling, the sum of the areas of all rectangles in the tiling is fixed.\\
[i]Note:[/i] Points will be awarded depending on the generality of the polygons for which the result is proven.
1993 Italy TST, 4
An $m \times n$ chessboard with $m,n \ge 2$ is given.
Some dominoes are placed on the chessboard so that the following conditions are satisfied:
(i) Each domino occupies two adjacent squares of the chessboard,
(ii) It is not possible to put another domino onto the chessboard without overlapping,
(iii) It is not possible to slide a domino horizontally or vertically without overlapping.
Prove that the number of squares that are not covered by a domino is less than $\frac15 mn$.
1998 Tournament Of Towns, 1
Pinocchio claims that he can take some non-right-angled triangles , all of which are similar to one another and some of which may be congruent to one another, and put them together to form a rectangle. Is Pinocchio lying?
(A Fedotov)
2013 Greece Team Selection Test, 4
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".
2022 Nigerian MO round 3, Problem 3
A unit square is removed from the corner of an $n \times n$ grid, where $n \geq 2$. Prove that the remainder can be covered by copies of the figures of $3$ or $5$ unit squares depicted in the drawing below.
[asy]
import geometry;
draw((-1.5,0)--(-3.5,0)--(-3.5,2)--(-2.5,2)--(-2.5,1)--(-1.5,1)--cycle);
draw((-3.5,1)--(-2.5,1)--(-2.5,0));
draw((0.5,0)--(0.5,3)--(1.5,3)--(1.5,1)--(3.5,1)--(3.5,0)--cycle);
draw((1.5,0)--(1.5,1));
draw((2.5,0)--(2.5,1));
draw((0.5,1)--(1.5,1));
draw((0.5,2)--(1.5,2));
[/asy]
[b]Note:[/b] Every square must be covered once and figures must not go over the bounds of the grid.
1984 Tournament Of Towns, (079) 5
A $7 \times 7$ square is made up of $16$ $1 \times 3$ tiles and $1$ $1 \times 1$ tile. Prove that the $1 \times 1$ tile lies either at the centre of the square or adjoins one of its boundaries .
2017 Junior Balkan Team Selection Tests - Romania, 4
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.
2009 Switzerland - Final Round, 8
Given is a floor plan composed of $n$ unit squares. Albert and Berta want to cover this floor with tiles, with all tiles having the shape of a $1\times 2$ domino or a $T$-tetromino. Albert only has tiles from one color, while Berta has two-color dominoes and tetrominoes available in four colors. Albert can use this floor plan in $a$ ways to cover tiles, Berta in $ b$ ways. Assuming that $a \ne 0$, determine the ratio $b/a$.
1998 Tournament Of Towns, 5
A square is divided into $25$ small squares. We draw diagonals of some of the small squares so that no two diagonals share a common point (not even a common endpoint). What is the largest possible number of diagonals that we can draw?
(I Rubanov)
2022 Austrian MO Beginners' Competition, 2
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)
2014 Switzerland - Final Round, 4
The checkered plane (infinitely large house paper) is given. For which pairs (a,, b) one can color each of the squares with one of $a \cdot b$ colors, so that each rectangle of size $ a \times b$ or $b \times a$, placed appropriately in the checkered plane, always contains a unit square with each color ?
2020 Ukraine Team Selection Test, 1
Square $600\times 600$ is divided into figures of four types, shown in figure. In the figures of the two types, shown on the left, in painted black, the cells recorded number $2^k$, where $k$ is the number of the column, where is this cell (columns numbered from left to right by numbers from $1$ to $600$). Prove that the sum of all recorded numbers are divisible by $9$.
[asy]
// Set up the drawing area
size(10cm,0);
defaultpen(fontsize(10pt));
unitsize(0.8cm);
// A helper function to draw a single unit square
// c = coordinates of the lower-left corner
// p = fill color (default is white)
void drawsq(pair c, pen p=white) {
fill(shift(c)*unitsquare, p);
draw(shift(c)*unitsquare);
}
// --- Shape 1 (left) ---
// 2 columns, 3 rows, black square in the middle-left
drawsq((1,1), black); // middle-left black
drawsq((2,0)); // bottom-right
drawsq((2,1)); // middle-right
drawsq((2,2)); // top-right
// --- Shape 2 (next to the first) ---
// 2 columns, 3 rows, black square in the middle-right
drawsq((4,0));
drawsq((4,1));
drawsq((4,2));
drawsq((5,1), black); // middle-right black
// --- Shape 3 (the "T" shape, 3 across the bottom + 1 in the middle top) ---
drawsq((7,0));
drawsq((8,0));
drawsq((9,0));
drawsq((8,1));
// --- Shape 4 (the "T" shape, 3 across the top + 1 in the middle bottom) ---
drawsq((11,1));
drawsq((12,1));
drawsq((13,1));
drawsq((12,0));
[/asy]
2016 Peru MO (ONEM), 2
How many dominoes can be placed on a at least $3 \times 12$ board, such so that it is impossible to place a $1\times 3$, $3 \times 1$, or $ 2 \times 2$ tile on what remains of the board?
Clarification: Each domino covers exactly two squares on the board. The chips cannot overlap.
2023 Israel TST, P1
Toph wants to tile a rectangular $m\times n$ square grid with the $6$ types of tiles in the picture (moving the tiles is allowed, but rotating and reflecting is not). For which pairs $(m,n)$ is this possible?
2017 SG Originals, 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]
1995 ITAMO, 1
Determine for which values of $n$ it is possible to tile a square of side $n$ with figures of the type shown in the picture
[asy]
unitsize(0.4 cm);
draw((0,0)--(5,0));
draw((0,1)--(5,1));
draw((1,2)--(4,2));
draw((2,3)--(3,3));
draw((0,0)--(0,1));
draw((1,0)--(1,2));
draw((2,0)--(2,3));
draw((3,0)--(3,3));
draw((4,0)--(4,2));
draw((5,0)--(5,1));
[/asy]