Found problems: 1342
Let $ABCD$ be a rectangle. Two perpendicular lines pass through point $B$. One of them meets segment $AD$ at point $K$, and the second one meets the extension of side $CD$ at point $L$. Let $F$ be the common point of $KL$ and $AC$. Prove that $BF\perp KL$.
A point $ (x,y)$ is randomly picked from inside the rectangle with vertices $ (0,0)$, $ (4,0)$, $ (4,1)$, and $ (0,1)$. What is the probability that $ x<y$?
$ \textbf{(A)}\ \frac{1}{8} \qquad
\textbf{(B)}\ \frac{1}{4} \qquad
\textbf{(C)}\ \frac{3}{8} \qquad
\textbf{(D)}\ \frac{1}{2} \qquad
\textbf{(E)}\ \frac{3}{4}$
The rectangle is cut into $10$ squares as shown in the figure on the right. Find its sides if the side of the smallest square is $3$.[img]https://cdn.artofproblemsolving.com/attachments/e/5/1fe3a0e41b2d3182338a557d3d44ff5ef9385d.png[/img]
There is a $11\times 11$ square grid. We divided this in $5$ rectangles along unit squares. How many ways that one of the rectangles doesn’t have a edge on basic circumference.
Note that we count as different ways that one way coincides with another way by rotating or reversing.
A rectangle has area $1100$. If the length is increased by ten percent and the width is
decreased by ten percent, what is the area of the new rectangle?
Let $P$ be a variable point on arc $BC$ of the circumcircle of triangle $ABC$ not containing $A$. Let $I_1$ and $I_2$ be the incenters of the triangles $PAB$ and $PAC$, respectively. Prove that:
[b](a)[/b] The circumcircle of $?PI_1I_2$ passes through a fixed point.
[b](b)[/b] The circle with diameter $I_1I_2$ passes through a fixed point.
[b](c)[/b] The midpoint of $I_1I_2$ lies on a fixed circle.
A table $2 \times 2010$ is divided to unit cells. Ivan and Peter are playing the following game. Ivan starts, and puts horizontal $2 \times 1$ domino that covers exactly two unit table cells. Then Peter puts vertical $1 \times 2$ domino that covers exactly two unit table cells. Then Ivan puts horizontal domino. Then Peter puts vertical domino, etc. The person who cannot put his domino will lose the game. Find who have winning strategy.
Cut the $10$ cm $\times 20$ cm rectangle into two pieces with one straight cut so that they can fit inside the $19.5$ cm diameter circle without intersecting.
Ana and Beatriz take turns in a game that starts with a square of side $1$ drawn on an infinite grid. Each turn consists of drawing a square that does not overlap with the rectangle already drawn, in such a way that one of its sides is a (complete) side of the figure already drawn. A player wins if she completes a rectangle whose area is a multiple of $5$. If Ana goes first, does either player have a winning strategy?
$(BUL 3)$ One hundred convex polygons are placed on a square with edge of length $38 cm.$ The area of each of the polygons is smaller than $\pi cm^2,$ and the perimeter of each of the polygons is smaller than $2\pi cm.$ Prove that there exists a disk with radius $1$ in the square that does not intersect any of the polygons.
A rectangular floor measures $ a$ by $ b$ feet, where $ a$ and $ b$ are positive integers with $ b > a$. An artist paints a rectangle on the floor with the sides of the rectangle parallel to the sides of the floor. The unpainted part of the floor forms a border of width $ 1$ foot around the painted rectangle and occupies half of the area of the entire floor. How many possibilities are there for the ordered pair $ (a,b)$?
$ \textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 3\qquad\textbf{(D)}\ 4\qquad\textbf{(E)}\ 5$
Let $ABCD$ be a trapezoid such that side $[AB]$ and side $[CD]$ are perpendicular to side $[BC]$. Let $E$ be a point on side $[BC]$ such that $\triangle AED$ is equilateral. If $|AB|=7$ and $|CD|=5$, what is the area of trapezoid $ABCD$?
$
\textbf{(A)}\ 27\sqrt{3}
\qquad\textbf{(B)}\ 42
\qquad\textbf{(C)}\ 24\sqrt{3}
\qquad\textbf{(D)}\ 40
\qquad\textbf{(E)}\ 36
$
Given a matrix $\{a_{ij}\}_{i,j=0}^{9}$, $a_{ij}=10i+j+1$. Andrei is going to cover its entries by $50$ rectangles $1\times 2$ (each such rectangle contains two adjacent entries) so that the sum of $50$ products in these rectangles is minimal possible. Help him.
[i]A. Badzyan[/i]
Let $ABCD$ be a rectangle and let $E$ and $F$ be points on $CD$ and $BC$ respectively such that area $(ADE) = 16$, area $(CEF) = 9$ and area $(ABF) = 25$. What is the area of triangle $AEF$ ?
Given a $8$x$8$ square board
a) Prove that: for any ways to color the board, we are always be able to find a rectangle consists of $8$ squares such that these squares are not colored.
b) Prove that: we can color $7$ squares on the board such that for any rectangles formed by $\geq 9$ squares, there are at least $1$ colored square.
Suppose from $(m+2)\times(n+2)$ rectangle we cut $4$, $1\times1$ corners. Now on first and last row first and last columns we write $2(m+n)$ real numbers. Prove we can fill the interior $m\times n$ rectangle with real numbers that every number is average of it's $4$ neighbors.
Let the points $O$ and $H$ be the circumcenter and orthocenter of an acute angled triangle $ABC.$ Let $D$ be the midpoint of $BC.$ Let $E$ be the point on the angle bisector of $\angle BAC$ such that $AE\perp HE.$ Let $F$ be the point such that $AEHF$ is a rectangle. Prove that $D,E,F$ are collinear.
Find all values of $n$ such that there exists a rectangle with integer side lengths, perimeter $n$, and area $2n$.
Two opposite sides of a rectangle are each divided into $n$ congruent segments, and the endpoints of one segment are joined to the center to form triangle $A$. The other sides are each divided into $m$ congruent segments, and the endpoints of one of these segments are joined to the center to form triangle $B$. [See figure for $n = 5, m = 7$.] What is the ratio of the area of triangle $A$ to the area of triangle $B$?
[asy]
int i;
for(i=0; i<8; i=i+1) {
dot((i,0)^^(i,5));
}
for(i=1; i<5; i=i+1) {
dot((0,i)^^(7,i));
}
draw(origin--(7,0)--(7,5)--(0,5)--cycle, linewidth(0.8));
pair P=(3.5, 2.5);
draw((0,4)--P--(0,3)^^(2,0)--P--(3,0));
label("$B$", (2.3,0), NE);
label("$A$", (0,3.7), SE);[/asy]
$\text{(A)} \ 1 \qquad \text{(B)} \ m/n \qquad \text{(C)} \ n/m \qquad \text{(D)} \ 2m/n \qquad \text{(E)} \ 2n/m$
For an integer $m\geq 1$, we consider partitions of a $2^m\times 2^m$ chessboard into rectangles consisting of cells of chessboard, in which each of the $2^m$ cells along one diagonal forms a separate rectangle of side length $1$. Determine the smallest possible sum of rectangle perimeters in such a partition.
[i]Proposed by Gerhard Woeginger, Netherlands[/i]
When the unit squares at the four corners are removed from a three by three squares, the resulting shape is called a cross. What is the maximum number of non-overlapping crosses placed within the boundary of a $ 10\times 11$ chessboard? (Each cross covers exactly five unit squares on the board.)
Two rectangles of unit area overlap to form a convex octagon. Show that the area of the octagon is at least $\dfrac {1} {2}$.
[i]Kvant Magazine [/i]
There are many $a\times b$ rectangular cardboard pieces ($a,b\in\mathbb{N}$ such that $a<b$). It is given that by putting such pieces together without overlapping one can make $49\times 51$ rectangle, and $99\times 101$ rectangle. Can one uniquely determine $a,b$ from this?
$12.$ In a rectangle $ABCD$ $AB=8$ and $BC=20.$ Let $P$ be a point on $AD$ such that $\angle{BPC}=90^o.$ If $r_1,r_2,r_3.$ are the radii of the incircles of triangles $APB,$ $BPC,$ and $CPD.$ what is the value of $r_1+r_2+r_3 ?$
Several pupils wrote a solution of a math problem on the blackboard on the break. When the teacher came in, a pupil was just clearing the blackboard, so the teacher could only observe that there was a rectangle with the sides of integer lenghts and a diagonal of lenght $ 2002$. Then the teacher pointed out that there was a computation error in pupils' solution. Why did he conclude that?