Found problems: 178
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)
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
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]
(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]
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]
Determine for which natural numbers $n$ it is possible to completely cover a board of $ n \times n$, divided into $1 \times 1$ squares, with pieces like the one in the figure, without gaps or overlays and without leaving the board. Each of the pieces covers exactly six boxes.
Note: Parts can be rotated.
[img]https://cdn.artofproblemsolving.com/attachments/c/2/d87d234b7f9799da873bebec845c721e4567f9.png[/img]
The $U$-tile is made up of $1 \times 1$ squares and has the following shape:
[img]https://cdn.artofproblemsolving.com/attachments/8/7/5795ee33444055794119a99e675ef977add483.png[/img]
where there are two vertical rows of a squares, one horizontal row of $b$ squares, and also $a \ge 2$ and $b \ge 3$.
Notice that there are different types of tile $U$ .
For example, some types of $U$ tiles are as follows:
[img]https://cdn.artofproblemsolving.com/attachments/0/3/ca340686403739ffbbbb578d73af76e81a630e.png[/img]
Prove that for each integer $n \ge 6$, the board of $n\times n$ can be completely covered with $U$-tiles ,
with no gaps and no overlapping clicks.
Clarifications: The $U$-tiles can be rotated. Any amount can be used in the covering of tiles of each type.
Does there exist a rectangle which can be dissected into $15$ congruent polygons which are not rectangles?
Can a square be dissected into $15$ congruent polygons which are not rectangles?
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 ?
Rectangle on a checked paper with length of a unit square side being $1$ Is divided into domino figures( two unit square sharing a common edge). Prove that you colour all corners of squares on the edge of rectangle and inside rectangle with $3$ colours such that for any two corners with distance $1$ the following conditions hold: they are coloured in different colour if the line connecting the two corners is on the border of two domino figures and coloured in same colour if the line connecting the two corners is inside a domino figure.
Show that there exists an integer $N$ such that for all $n \ge N$ a square can be partitioned into $n$ smaller squares.
Let $k$ and $n$ be integers such that $k\ge 2$ and $k \le n \le 2k-1$. Place rectangular tiles, each of size $1 \times k$, or $k \times 1$ on a $n \times n$ chessboard so that each tile covers exactly $k$ cells and no two tiles overlap. Do this until no further tile can be placed in this way. For each such $k$ and $n$, determine the minimum number of tiles that such an arrangement may contain.
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
Can an arbitrary polygon be cut into isosceles trapezoids?
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.
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 $t = 3k + 1$.
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.
We call a tiling of an $m\times$ n rectangle with arabos (see figure below) [i]regular[/i] if there is no sub-rectangle which is tiled with arabos. Prove that if for some $m$ and $n$ there exists a [i]regular[/i] tiling of the $m\times n$ rectangle then there exists a [i]regular[/i] tiling also for the $2m \times 2n$ rectangle.
[img]https://cdn.artofproblemsolving.com/attachments/1/1/2ab41cc5107a21760392253ed52d9e4ecb22d1.png[/img]
By a $\emph{tile}$ we mean a polyomino (i.e. a finite edge-connected set of cells in the infinite grid). There are many ways to place a tile in the infinite table (rotation is allowed but we cannot flip the tile). We call a tile $\textbf{T}$ special if we can place a permutation of the positive integers on all cells of the infinite table in such a way that each number would be maximum between all the numbers that tile covers in at most one placement of the tile.
1. Prove that each square is a special tile.
2. Prove that each non-square rectangle is not a special tile.
3. Prove that tile $\textbf{T}$ is special if and only if it looks the same after $90^\circ$ rotation.
The plane is divided into unit squares. Each box should be be colored in one of $n$ colors , so that if four squares can be covered with an $L$-tetromino, then these squares have four different colors (the $L$-Tetromino may be rotated and be mirrored). Find the smallest value of $n$ for which this is possible.
You have a large number of congruent equilateral triangular tiles on a table and you want to fit $n$ of them together to make a convex equiangular hexagon (i.e. one whose interior angles are $120^o$) . Obviously, $n$ cannot be any positive integer. The first three feasible $n$ are $6, 10$ and $13$. Show that $12$ is not feasible but $14$ is.
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]
A set of identical square tiles with side length $1$ is placed on a (very large) floor. Every tile after the first shares an entire edge with at least one tile that has already been placed.
- What is the largest possible perimeter for a figure made of $10$ tiles?
- What is the smallest possible perimeter for a figure made of $10$ tiles?
- What is the largest possible perimeter for a figure made of $2011$ tiles?
- What is the smallest possible perimeter for a figure made of $2011$ tiles?
Prove that your answers are correct.
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]
Infinitely many equidistant parallel lines are drawn in the plane. A positive integer $n \geqslant 3$ is called frameable if it is possible to draw a regular polygon with $n$ sides all whose vertices lie on these lines, and no line contains more than one vertex of the polygon.
(a) Show that $3, 4, 6$ are frameable.
(b) Show that any integer $n \geqslant 7$ is not frameable.
(c) Determine whether $5$ is frameable.
[i]Proposed by Muralidharan[/i]