Found problems: 1342
A rectangle is divided into $9$ small rectangles if by parallel lines to its sides, as shown in the figure.
[img]https://cdn.artofproblemsolving.com/attachments/e/d/1fd545862a3c7950249ec54a631c74e59fb9ed.png[/img]
The four numbers written indicate the areas of the four corresponding rectangles.
Prove that the total area of the rectangle is greater than or equal to $90$.
Tanya and Serezha take turns putting chips in empty squares of a chessboard. Tanya starts with a chip in an arbitrary square. At every next move, Serezha must put a chip in the column where Tanya put her last chip, while Tanya must put a chip in the row where Serezha put his last chip. The player who cannot make a move loses. Which of the players has a winning strategy?
[i]Proposed by A. Golovanov[/i]
A rectangle $3 \times 5$ is divided into $15$ $1 \times 1$ cells. The middle $3$ cells that have no common points with the border of the rectangle are deleted. Is it possible to put in the remaining $12$ cells numbers $1, 2, \ldots, 12$ in some order, so that the sums of the numbers in the cells along each of the four sides of the rectangle are equal?
[i]Proposed by Mariia Rozhkova[/i]
Inside the $7\times8$ rectangle below, one point is chosen a distance $\sqrt2$ from the left side and a distance $\sqrt7$ from the bottom side. The line segments from that point to the four vertices of the rectangle are drawn. Find the area of the shaded region.
[asy]
import graph;
size(4cm);
pair A = (0,0);
pair B = (9,0);
pair C = (9,7);
pair D = (0,7);
pair P = (1.5,3);
draw(A--B--C--D--cycle,linewidth(1.5));
filldraw(A--B--P--cycle,rgb(.76,.76,.76),linewidth(1.5));
filldraw(C--D--P--cycle,rgb(.76,.76,.76),linewidth(1.5));
[/asy]
Let $n$ be a positive integer. A 4-by-$n$ rectangle is divided into $4n$ unit squares in the usual way. Each unit square is colored black or white. Suppose that every white unit square shares an edge with at least one black unit square. Prove that there are at least $n$ black unit squares.
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 unit square is divided into two rectangles in such a way that the smaller rectangle can be put on the greater rectangle with every vertex of the smaller on exactly one of the edges of the greater. Calculate the dimensions of the smaller rectangle.
On each side of triangle $ABC$, two distinct points are marked. It is known that these points are the feet of the altitudes and of the bisectors.
a) Using only a ruler determine which points are the feet of the altitudes and which points are the feet of the bisectors.
b) Solve p.a) drawing only three lines.
The perimeter of a rectangle is $100$ and its diagonal has length $x$. What is the area of this rectangle?
$\textbf{(A) }625-x^2\qquad\textbf{(B) }625-\dfrac{x^2}2\qquad\textbf{(C) }1250-x^2\qquad\textbf{(D) }1250-\dfrac{x^2}2\qquad\textbf{(E) }2500-\dfrac{x^2}2$
A rectangular region is bounded by the graphs of the equations $y=a, y=-b, x=-c,$ and $x=d$, where $a,b,c,$ and $d$ are all positive numbers. Which of the following represents the area of this region?
$ \textbf{(A)}\ ac+ad+bc+bd\qquad\textbf{(B)}\ ac-ad+bc-bd\qquad\textbf{(C)}\ ac+ad-bc-bd \quad\quad\qquad\textbf{(D)}\ -ac-ad+bc+bd\qquad\textbf{(E)}\ ac-ad-bc+bd $
Three points inside a rectangle determine a triangle. A fourth point is taken inside the triangle.
i) Prove at least one of the three concave quadrilaterals formed by these four points has perimeter lesser than that of the rectangle.
ii) Assuming the three points inside the rectangle are three corners of it, prove at least two of the three concave quadrilaterals formed by these four points have perimeters lesser than that of the rectangle.
[i](Dan Schwarz)[/i]
How many ways are there to remove an $11\times11$ square from a $2011\times2011$ square so that the remaining part can be tiled with dominoes ($1\times 2$ rectangles)?
Given a rectangle, circumscribe a rectangle of maximum area.
We call a coloring $f$ of the elements in the set $M = \{(x, y) | x = 0, 1, \dots , kn - 1; y = 0, 1, \dots , ln - 1\}$ with $n$ colors allowable if every color appears exactly $k$ and $ l$ times in each row and column and there are no rectangles with sides parallel to the coordinate axes such that all the vertices in $M$ have the same color. Prove that every allowable coloring $f$ satisfies $kl \leq n(n + 1).$
$1390$ ants are placed near a line, such that the distance between their heads and the line is less than $1\text{cm}$ and the distance between the heads of two ants is always larger than $2\text{cm}$. Show that there is at least one pair of ants such that the distance between their heads is at least $10$ meters (consider the head of an ant as point).
Two congruent rectangles are located as shown in the figure. Find the area of the shaded part.
[img]https://cdn.artofproblemsolving.com/attachments/2/e/10b164535ab5b3a3b98ce1a0b84892cd11d76f.png[/img]
A $2003\times 2004$ rectangle consists of unit squares. We consider rhombi formed by four diagonals of unit squares.
What maximum number of such rhombi can be arranged in this rectangle so that no two of them have any common points except vertices?
[i]Proposed by A. Golovanov[/i]
Peter has many squares of equal side. Some of the squares are black, some are white. Peter wants to assemble
a big square, with side equal to $n$ sides of the small squares, so that the big square has no rectangle formed by the small squares such that all the squares in the vertices of the rectangle are of equal colour. How big a square is Peter able to assemble?
Let $ m,n\in \mathbb{N}^*$. Find the least $ n$ for which exists $ m$, such that rectangle $ (3m \plus{} 2)\times(4m \plus{} 3)$ can be covered with $ \dfrac{n(n \plus{} 1)}{2}$ squares, among which exist $ n$ squares of length $ 1$, $ n \minus{} 1$ of length $ 2$, $ ...$, $ 1$ square of length $ n$. For the found value of $ n$ give the example of covering.
A triangle has side lengths 10, 10, and 12. A rectangle has width 4 and area equal to the area of the triangle. What is the perimeter of this rectangle?
$ \textbf{(A)}\ 16\qquad\textbf{(B)}\ 24\qquad\textbf{(C)}\ 28\qquad\textbf{(D)}\ 32\qquad\textbf{(E)}\ 36$
Let $n$ be a positive integer. Nancy is given a rectangular table in which each entry is a positive integer. She is permitted to make either of the following two moves:
(1) select a row and multiply each entry in this row by $n$;
(2) select a column and subtract $n$ from each entry in this column.
Find all possible values of $n$ for which the following statement is true:
Given any rectangular table, it is possible for Nancy to perform a finite sequence of moves to create a table in which each entry is $0$.
Let $ ABCD$ be a convex quadrilateral with $ AB\equal{}AD$ and $ BC\equal{}CD$. On the sides $ AB,BC,CD,DA$ we consider points $ K,L,L_1,K_1$ such that quadrilateral $ KLL_1K_1$ is rectangle. Then consider rectangles $ MNPQ$ inscribed in the triangle $ BLK$, where $ M\in KB,N\in BL,P,Q\in LK$ and $ M_1N_1P_1Q_1$ inscribed in triangle $ DK_1L_1$ where $ P_1$ and $ Q_1$ are situated on the $ L_1K_1$, $ M$ on the $ DK_1$ and $ N_1$ on the $ DL_1$. Let $ S,S_1,S_2,S_3$ be the areas of the $ ABCD,KLL_1K_1,MNPQ,M_1N_1P_1Q_1$ respectively. Find the maximum possible value of the expression:
$ \frac{S_1\plus{}S_2\plus{}S_3}{S}$
What the smallest number of circles of radius $\sqrt{2}$ that are needed to cover a rectangle
$(a)$ of size $6\times 3$?
$(b)$ of size $5\times 3$?
A square is cut into $100$ rectangles by $9$ straight lines parallel to one of the sides and $9$ lines parallel to another. If exactly $9$ of the rectangles are actually squares, prove that at least two of these $9$ squares are of the same size .
(V Proizvolov)
The lateral sides of a box with base $a\times b$ and height $c$ (where $a$; $b$;$ c$ are natural numbers) are completely covered without overlap by rectangles whose edges are parallel to the edges of the box, each containing an even number of unit squares. (Rectangles may cross the lateral edges of the box.) Prove that if $c$ is odd, then
the number of possible coverings is even.
[i]D. Karpov, C. Gukshin, D. Fon-der-Flaas[/i]