Found problems: 178
There is a hole in the roof with dimensions $23 \times 19$ cm. Can August fill the the roof with tiles of dimensions $5 \times 24 \times 30$ cm?
Let $n \ge 3$ be a positive integer. We call a $3 \times 3$ grid [i]beautiful[/i] if the cell located at the center is colored white and all other cells are colored black, or if it is colored black and all other cells are colored white. Determine the minimum value of $a+b$ such that there exist positive integers $a$, $b$ and a coloring of an $a \times b$ grid with black and white, so that it contains $n^2 - n$ [i]beautiful[/i] subgrids.
It is known that a cells square can be cut into $n$ equal figures of $k$ cells.
Prove that it is possible to cut it into $k$ equal figures of $n$ cells.
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 $\frac15 mn$.
A grid measuring $10 \times 11$ is given. How many "crosses" covering five unit squares can be placed on the grid?
(pictured right) so that no two of them cover the same square?
[img]https://cdn.artofproblemsolving.com/attachments/a/7/8a5944233785d960f6670e34ca7c90080f0bd6.png[/img]
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.
Can $77$ blocks each $3 \times 3 \times1$ be assembled to form a $7 \times 9 \times 11$ block?
A holey triangle is an upward equilateral triangle of side length $n$ with $n$ upward unit triangular holes cut out. A diamond is a $60^\circ-120^\circ$ unit rhombus.
Prove that a holey triangle $T$ can be tiled with diamonds if and only if the following condition holds: Every upward equilateral triangle of side length $k$ in $T$ contains at most $k$ holes, for $1\leq k\leq n$.
[i]Proposed by Federico Ardila, Colombia [/i]
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 $$M=\sum_{i=1}^k \;\sum_{j=1}^{\ell} 2^{|a_i-b_j|}.$$
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?
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 $1?$
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)
For which integers $n>1$ does there exist a rectangle that can be subdivided into $n$ pairwise noncongruent rectangles similar to the original rectangle?
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?
Rectangles $1\times20$, $1\times 19$, ..., $1\times 1$ were cut out of $20\times20$ table. Prove that at least 85 dominoes(1×2 rectangle) can be removed from the remainder.
Proposed by S. Berlov
Basil has an unrestricted supply of straight bricks $1 \times 1 \times 3$ and Γ-shape bricks made of three cubes $1\times 1\times 1$. Basil filled a whole box $m \times n \times k$ with these bricks, where $m, n$ and $k$ are integers greater than $1$. Prove that it was sufficient to use only Γ-shape bricks.
(Mikhail Evdokimov)
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.
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.
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)
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?
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)
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 "L-shapes" consisting of $3$ or $5$ unit square, as depicted in the figure below. Every square must be covered once and the L-shapes must not go over the bounds of the grid.
[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][i]Estonian Olympiad, 2009[/i]
We say that a polygon is rectangular when all of its angles are $90^\circ$ or $270^\circ$. Is it true that each rectangular polygon, which sides are with length equal to odd numbers only, [u]can't[/u] be covered with 2x1 domino tiles?
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 $13$. Determine if $19$ and $20$ are feasible .
Show that we cannot tile a $10 x 10$ board with $25$ pieces of type $A$, or with $25$ pieces of type $B$, or with $25$ pieces of type $C$.