Found problems: 1342
In rectangle $ABCD$, $AB=6$ and $BC=3$. Point $E$ between $B$ and $C$, and point $F$ between $E$ and $C$ are such that $BE=EF=FC$. Segments $\overline{AE}$ and $\overline{AF}$ intersect $\overline{BD}$ at $P$ and $Q$, respectively. The ratio $BP:PQ:QD$ can be written as $r:s:t$, where the greatest common factor of $r,s$ and $t$ is $1$. What is $r+s+t$?
$\textbf{(A) } 7
\qquad \textbf{(B) } 9
\qquad \textbf{(C) } 12
\qquad \textbf{(D) } 15
\qquad \textbf{(E) } 20$
Let $ ABCD$ be a trapezoid with $ AB\parallel{}CD$, $ AB\equal{}11$, $ BC\equal{}5$, $ CD\equal{}19$, and $ DA\equal{}7$. Bisectors of $ \angle A$ and $ \angle D$ meet at $ P$, and bisectors of $ \angle B$ and $ \angle C$ meet at $ Q$. What is the area of hexagon $ ABQCDP$?
$ \textbf{(A)}\ 28\sqrt{3}\qquad
\textbf{(B)}\ 30\sqrt{3}\qquad
\textbf{(C)}\ 32\sqrt{3}\qquad
\textbf{(D)}\ 35\sqrt{3}\qquad
\textbf{(E)}\ 36\sqrt{3}$
The square was cut into $n^2$ rectangles with sides $a_i \times b_j, i , j= 1,..., n$.
For what is the smallest $n$ in the set $\{a_1, b_1, ..., a_n, b_n\}$ all the numbers can be different?
Rectangle $ ABCD$, pictured below, shares $50\%$ of its area with square $ EFGH$. Square $ EFGH$ shares $20\%$ of its area with rectangle $ ABCD$. What is $ \frac{AB}{AD}$?
[asy]unitsize(5mm);
defaultpen(linewidth(0.8pt)+fontsize(10pt));
pair A=(0,3), B=(8,3), C=(8,2), D=(0,2), Ep=(0,4), F=(4,4), G=(4,0), H=(0,0);
fill(shift(0,2)*xscale(4)*unitsquare,grey);
draw(Ep--F--G--H--cycle);
draw(A--B--C--D);
label("$A$",A,W);
label("$B$",B,E);
label("$C$",C,E);
label("$D$",D,W);
label("$E$",Ep,NW);
label("$F$",F,NE);
label("$G$",G,SE);
label("$H$",H,SW);[/asy]$ \textbf{(A)}\ 4\qquad \textbf{(B)}\ 5\qquad \textbf{(C)}\ 6\qquad \textbf{(D)}\ 8\qquad \textbf{(E)}\ 10$
Let $n$ and $t$ be positive integers. What is the number of ways to place $t$ dominoes $(1\times 2$ or $2\times 1$ rectangles) in a $2\times n$ table so that there is no $2\times 2$ square formed by $2$ dominoes and each $2\times 3$ rectangle either does not have a horizontal domino in the middle and last cell in the first row or does not have a horizontal domino in the first and middle cell in the second row (or both)?
A $(2^n - 1) \times (2^n +1)$ board is to be divided into rectangles with sides parallel to the sides of the board and integer side lengths such that the area of each rectangle is a power of 2. Find the minimum number of rectangles that the board may be divided into.
Let $ABCD$ be a rectangle and $P$ a point outside of it such that $\angle{BPC} = 90^{\circ}$ and the area of the pentagon $ABPCD$ is equal to $AB^{2}$.
Show that $ABPCD$ can be divided in 3 pieces with straight cuts in such a way that a square can be built using those 3 pieces, without leaving any holes or placing pieces on top of each other.
Note: the pieces can be rotated and flipped over.
Several knights are arranged on an infinite chessboard. No square is attacked by more than one knight (in particular, a square occupied by a knight can be attacked by one knight but not by two). Sasha outlined a $ 14\times 16$ rectangle. What maximum number of knights can this rectangle contain?
Congruent rectangles with sides $m(cm)$ and $n(cm)$ are given ($m, n$ positive integers). Characterize the rectangles that can be constructed from these rectangles (in the fashion of a jigsaw puzzle). (The number of rectangles is unbounded.)
Let $ABCDA'B'C'D'$ be a rectangular prism with $|AB|=2|BC|$. $E$ is a point on the edge $[BB']$ satisfying $|EB'|=6|EB|$. Let $F$ and $F'$ be the feet of the perpendiculars from $E$ at $\triangle AEC$ and $\triangle A'EC'$, respectively. If $m(\widehat{FEF'})=60^{\circ}$, then $|BC|/|BE| = ? $
$ \textbf{(A)}\ \sqrt\frac53 \qquad \textbf{(B)}\ \sqrt\frac{15}2 \qquad \textbf{(C)}\ \frac32\sqrt{15} \qquad \textbf{(D)}\ 5\sqrt\frac53 \qquad \textbf{(E)}\ \text{None}$
Ali Barber, the carpet merchant, has a rectangular piece of carpet whose dimensions are unknown. Unfortunately, his tape measure is broken and he has no other measuring instruments. However, he finds that if he lays it flat on the floor of either of his storerooms, then each corner of the carpet touches a different wall of that room. If the two rooms have dimensions of 38 feet by 55 feet and 50 feet by 55 feet, what are the carpet dimensions?
A square $ (n \minus{} 1) \times (n \minus{} 1)$ is divided into $ (n \minus{} 1)^2$ unit squares in the usual manner. Each of the $ n^2$ vertices of these squares is to be coloured red or blue. Find the number of different colourings such that each unit square has exactly two red vertices. (Two colouring schemse are regarded as different if at least one vertex is coloured differently in the two schemes.)
Does the bellow depicted figure fit into a square $5\times5$.
Two circles touch in $M$, and lie inside a rectangle $ABCD$. One of them touches the sides $AB$ and $AD$, and the other one touches $AD,BC,CD$. The radius of the second circle is four times that of the first circle. Find the ratio in which the common tangent of the circles in $M$ divides $AB$ and $CD$.
A point $K$ is chosen on the side $CD$ of a rectangle $ABCD$. From the vertex $B$, the perpendicular $BH$ is dropped to the segment $AK$. The segments $AK$ and $BH$ divide the rectangle into three parts such that each of them has the inscribed circle (see figure). Prove that if the circles tangent to $CD$ are equal then the third circle is also equal to them.
A rectangle has been divided into 8 smaller rectangles as shown below. Given the area of seven of these rectangles, find the area of the shaded rectangle.
[i]Lightning 2.1[/i]
In the figure, point $P, Q,R,S$ lies on the side of the rectangle $ABCD$.
[img]https://1.bp.blogspot.com/-Ff9rMibTuHA/X9PRPbGVy-I/AAAAAAAAMzA/2ytG0aqe-k0fPL3hbSp_zHrMYAfU-1Y_ACLcBGAsYHQ/s426/2020%2BIndonedia%2BMO%2BProvince%2BP2%2Bq1.png[/img]
If it is known that the area of the small square is $1$ unit, determine the area of the rectangle $ABCD$.
A regular octagon $ ABCDEFGH$ has an area of one square unit. What is the area of the rectangle $ ABEF$?
[asy]unitsize(8mm);
defaultpen(linewidth(.8pt)+fontsize(6pt));
pair C=dir(22.5), B=dir(67.5), A=dir(112.5), H=dir(157.5), G=dir(202.5), F=dir(247.5), E=dir(292.5), D=dir(337.5);
draw(A--B--C--D--E--F--G--H--cycle);
label("$A$",A,NNW);
label("$B$",B,NNE);
label("$C$",C,ENE);
label("$D$",D,ESE);
label("$E$",E,SSE);
label("$F$",F,SSW);
label("$G$",G,WSW);
label("$H$",H,WNW);[/asy]$ \textbf{(A)}\ 1\minus{}\frac{\sqrt2}{2} \qquad
\textbf{(B)}\ \frac{\sqrt2}{4} \qquad
\textbf{(C)}\ \sqrt2\minus{}1 \qquad
\textbf{(D)}\ \frac12 \qquad
\textbf{(E)}\ \frac{1\plus{}\sqrt2}{4}$
A bug travels in the coordinate plane, moving only along the lines that are parallel to the $x$-axis or $y$-axis. Let $A=(-3, 2)$ and $B=(3, -2)$. Consider all possible paths of the bug from $A$ to $B$ of length at most $20$. How many points with integer coordinates lie on at least one of these paths?
$ \textbf{(A)}\ 161 \qquad
\textbf{(B)}\ 185 \qquad
\textbf{(C)}\ 195 \qquad
\textbf{(D)}\ 227 \qquad
\textbf{(E)}\ 255 $
[asy]
real t=pi/12;real u=8*t;
real cu=cos(u);real su=sin(u);
draw(unitcircle);
draw((cos(-t),sin(-t))--(cos(13*t),sin(13*t)));
draw((cu,su)--(cu,-su));
label("A",(cos(13*t),sin(13*t)),W);
label("B",(cos(-t),sin(-t)),E);
label("C",(cu,su),N);
label("D",(cu,-su),S);
label("E",(cu,sin(-t)),NE);
label("2",((cu-1)/2,sin(-t)),N);
label("6",((cu+1)/2,sin(-t)),N);
label("3",(cu,(sin(-t)-su)/2),E);
//Credit to Zimbalono for the diagram[/asy]
Chords $AB$ and $CD$ in the circle above intersect at $E$ and are perpendicular to each other. If segments $AE$, $EB$, and $ED$ have measures $2$, $3$, and $6$ respectively, then the length of the diameter of the circle is
$\textbf{(A) }4\sqrt{5}\qquad\textbf{(B) }\sqrt{65}\qquad\textbf{(C) }2\sqrt{17}\qquad\textbf{(D) }3\sqrt{7}\qquad \textbf{(E) }6\sqrt{2}$
The width of rectangle $ABCD$ is twice its height, and the height of rectangle $EFCG$ is twice its width. The point $E$ lies on the diagonal $BD$. Which fraction of the area of the big rectangle is that of the small one?
[img]https://1.bp.blogspot.com/-aeqefhbBh5E/XzcBjhgg7sI/AAAAAAAAMXM/B0qSgWDBuqc3ysd-mOitP1LarOtBdJJ3gCLcBGAsYHQ/s0/2004%2BMohr%2Bp1.png[/img]
$\text{ABCD}$ is a rectangle, $\text{D}$ is the center of the circle, and $\text{B}$ is on the circle. If $\text{AD}=4$ and $\text{CD}=3$, then the area of the shaded region is between
[asy]
pair A,B,C,D;
A=(0,4); B=(3,4); C=(3,0); D=origin;
draw(circle(D,5));
fill((0,5)..(1.5,4.7697)..B--A--cycle,black);
fill(B..(4,3)..(5,0)--C--cycle,black);
draw((0,5)--D--(5,0));
label("A",A,NW);
label("B",B,NE);
label("C",C,S);
label("D",D,SW);
[/asy]
$\text{(A)}\ 4\text{ and }5 \qquad \text{(B)}\ 5\text{ and }6 \qquad \text{(C)}\ 6\text{ and }7 \qquad \text{(D)}\ 7\text{ and }8 \qquad \text{(E)}\ 8\text{ and }9$
In the picture below, the circles are tangent to each other and to the edges of the rectangle. The larger circle's radius equals 1. Determine the area of the rectangle.
[img]https://i.imgur.com/g3GUg4Z.png[/img]
Several identical square sheets of paper are laid out on a rectangular table so that their sides are parallel to the edges of the table (sheets may overlap). Prove that you can stick a few pins in such a way that each sheet will be attached to the table exactly by one pin.
Divide a $ 2\times 4 $ rectangle into $ 8 $ unit squares to obtain a set of $ 15 $ vertices denoted by $ \mathcal{M} . $ Find the points $ A\in\mathcal{M} $ that have the property that the set $ \mathcal{M}\setminus \{ A\} $ can form $ 7 $ pairs $ \left( A_1,B_1\right) ,\left( A_2,B_2\right) ,\ldots ,\left( A_7,B_7\right)\in\mathcal{M}\times\mathcal{M} $ such that
$$ \overrightarrow{A_1B_1} +\overrightarrow{A_2B_2} +\cdots +\overrightarrow{A_7B_7} =\overrightarrow{O} . $$