Found problems: 1342
The diagram below shows a rectangle with side lengths $36$ and $48$. Each of the sides is trisected and edges are added between the trisection points as shown. Then the shaded corner regions are removed, leaving the octagon which is not shaded in the diagram. Find the perimeter of this octagon.
[asy]
size(4cm);
dotfactor=3.5;
pair A,B,C,D,E,F,G,H,W,X,Y,Z;
A=(0,12);
B=(0,24);
C=(16,36);
D=(32,36);
E=(48,24);
F=(48,12);
G=(32,0);
H=(16,0);
W=origin;
X=(0,36);
Y=(48,36);
Z=(48,0);
filldraw(W--A--H--cycle^^B--X--C--cycle^^D--Y--E--cycle^^F--Z--G--cycle,rgb(.76,.76,.76));
draw(W--X--Y--Z--cycle,linewidth(1.2));
dot(A);
dot(B);
dot(C);
dot(D);
dot(E);
dot(F);
dot(G);
dot(H);
[/asy]
In a $12\times 12$ grid, colour each unit square with either black or white, such that there is at least one black unit square in any $3\times 4$ and $4\times 3$ rectangle bounded by the grid lines. Determine, with proof, the minimum number of black unit squares.
Vertex $ E$ of equilateral $ \triangle{ABE}$ is in the interior of unit square $ ABCD$. Let $ R$ be the region consisting of all points inside $ ABCD$ and outside $ \triangle{ABE}$ whose distance from $ \overline{AD}$ is between $ \frac{1}{3}$ and $ \frac{2}{3}$. What is the area of $ R$?
$ \textbf{(A)}\ \frac{12\minus{}5\sqrt3}{72} \qquad
\textbf{(B)}\ \frac{12\minus{}5\sqrt3}{36} \qquad
\textbf{(C)}\ \frac{\sqrt3}{18} \qquad
\textbf{(D)}\ \frac{3\minus{}\sqrt3}{9} \qquad
\textbf{(E)}\ \frac{\sqrt3}{12}$
Let $T$ be an acute triangle. Inscribe a rectangle $R$ in $T$ with one side along a side of $T.$ Then inscribe a rectangle $S$ in the triangle formed by the side of $R$ opposite the side on the boundary of $T,$ and the other two sides of $T,$ with one side along the side of $R.$ For any polygon $X,$ let $A(X)$ denote the area of $X.$ Find the maximum value, or show that no maximum exists, of $\tfrac{A(R)+A(S)}{A(T)},$ where $T$ ranges over all triangles and $R,S$ over all rectangles as above.
We divide up the plane into disjoint regions using a circle, a rectangle and a triangle. What is the greatest number of regions that we can get?
Peter made a paper rectangle, put it on an identical rectangle and pasted both rectangles along their perimeters. Then he cut the upper rectangle along one of its diagonals and along the perpendiculars to this diagonal from two remaining
vertices. After this he turned back the obtained triangles in such a way that they, along with the lower rectangle form a new rectangle.
Let this new rectangle be given. Restore the original rectangle using compass and ruler.
Let $n \ge 2$ be a fixed integer. The cells of an $n \times n$ table are filled with the integers from $1$ to $n^2$ with each number appearing exactly once. Let $N$ be the number of unordered quadruples of cells on this board which form an axis-aligned rectangle, with the two smaller integers being on opposite vertices of this rectangle. Find the largest possible value of $N$.
[i]Anonymous[/i]
A dart board is a regular octagon divided into regions as shown. Suppose that a dart thrown at the board is equally likely to land anywhere on the board. What is probability that the dart lands within the center square?
[asy]
unitsize(10mm);
defaultpen(linewidth(.8pt)+fontsize(10pt));
dotfactor=4;
pair A=(0,1), B=(1,0), C=(1+sqrt(2),0), D=(2+sqrt(2),1), E=(2+sqrt(2),1+sqrt(2)), F=(1+sqrt(2),2+sqrt(2)), G=(1,2+sqrt(2)), H=(0,1+sqrt(2));
draw(A--B--C--D--E--F--G--H--cycle);
draw(A--D);
draw(B--G);
draw(C--F);
draw(E--H);
[/asy]
$ \textbf{(A)}\ \frac{\sqrt{2} - 1}{2} \qquad\textbf{(B)}\ \frac{1}{4} \qquad\textbf{(C)}\ \frac{2 - \sqrt{2}}{2} \qquad\textbf{(D)}\ \frac{\sqrt{2}}{4} \qquad\textbf{(E)}\ 2 - \sqrt{2}$
Determine all integers $m$, for which it is possible to dissect the square $m\times m$ into five rectangles, with the side lengths being the integers $1, 2, … ,10$ in some order.
A rectangle with sides $ n$ and $p$ is divided into $np$ unit squares. Initially there are m unitary squares painted black and the remaining painted white. The following processoccurs repeatedly: if a unit square painted white has at minus two sides in common with squares painted black then Its color also turns black. Find the smallest integer $m$ that satisfies the property: there exists an initial position of $m$ black unit squares such that the entire $ n \times p$ rectangle is painted black when repeat the process a finite number of times.
In the arrow-shaped polygon [see figure], the angles at vertices $A$, $C$, $D$, $E$ and $F$ are right angles, $BC = FG = 5$, $CD = FE = 20$, $DE = 10$, and $AB = AG$. The area of the polygon is closest to
[asy]
size(200);
defaultpen(linewidth(0.7)+fontsize(10));
pair A=origin, B=(10,10), C=(10,5), D=(30,5), E=(30,-5), F=(10,-5), G=(10,-10);
draw(A--B--C--D--E--F--G--A);
label("$A$", A, W);
label("$B$", B, NE);
label("$C$", C, S);
label("$D$", D, NE);
label("$E$", E, SE);
label("$F$", F, N);
label("$G$", G, SE);
label("$5$", (11,7.5));
label("$5$", (11,-7.5));
label("$20$", (C+D)/2, N);
label("$20$", (F+E)/2, S);
label("$10$", (31,0));
[/asy]
$ \textbf{(A)}\ 288\qquad\textbf{(B)}\ 291\qquad\textbf{(C)}\ 294\qquad\textbf{(D)}\ 297\qquad\textbf{(E)}\ 300 $
A rectangular piece of paper has the side lengths $12$ and $15$. A corner is bent about as shown in the figure. Determine the area of the gray triangle.
[img]https://1.bp.blogspot.com/-HCfqWF0p_eA/XzcIhnHS1rI/AAAAAAAAMYg/KfY14frGPXUvF-H6ZVpV4RymlhD_kMs-ACLcBGAsYHQ/s0/1993%2BMohr%2Bp2.png[/img]
Let $ABCD$ be a rectangle with sides $AB=a$ and $BC=b$. Let $O$ be the intersection point of it's diagonals. Extent side $BA$ towards $A$ at a segment $AE=AO$, and diagonal $DB$ towards $B$ at a segment $BZ=BO$. If the triangle $EZC$ is an equilateral, then prove that:
i) $b=a\sqrt3$
ii) $AZ=EO$
iii) $EO \perp ZD$
$ABCD$ is a rectangle. The segment $MA$ is perpendicular to plane $ABC$ . $MB= 15$ , $MC=24$ , $MD=20$. Find the length of $MA$ .
A rectangle of dimensions $2024 \times 2023$ is filled with pieces of the following types:
[asy]
size(200);
// Figure (A)
draw((0,0)--(4,0)--(4,1)--(0,1)--cycle);
draw((1,0)--(1,1));
draw((2,0)--(2,1));
draw((3,0)--(3,1));
// Figure (B)
draw((6,0)--(8,0)--(8,2)--(6,2)--cycle);
draw((7,0)--(7,2));
draw((6,1)--(8,1));
// Figure (C)
draw((10,0)--(12,0)--(12,1)--(11,1)--(11,2)--(9,2)--(9,1)--(10,1)--cycle);
draw((10,0)--(10,1));
draw((11,0)--(11,1));
draw((10,1)--(11,1));
draw((9,1)--(9,2));
draw((10,1)--(10,2));
draw((11,0)--(12,0));
draw((10,1)--(12,1));
// Labeling
label("(A)", (2, -0.5));
label("(B)", (7, -0.5));
label("(C)", (10.5, -0.5));
[/asy]
Each piece can be turned arround, and each square has side length $1$.
Is it possible to use exactly 2023 pieces of type $(A)$?
Let $ABCS$ be an isosceles trapezoid. Denote $A',B',C',D'$ the incenters of triangles $BCD,CDA,$ $DAB,ABC,$ respectively. Show that $A',B',C',D'$ are vertices of a rectangle.
Triangle $ABC$ has side lengths $AB=12$, $BC=25$, and $CA=17$. Rectangle $PQRS$ has vertex $P$ on $\overline{AB}$, vertex $Q$ on $\overline{AC}$, and vertices $R$ and $S$ on $\overline{BC}$. In terms of the side length $PQ=w$, the area of $PQRS$ can be expressed as the quadratic polynomial
\[\text{Area}(PQRS)=\alpha w-\beta\cdot w^2\]
Then the coefficient $\beta=\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
The base of the quadrilateral pyramid $SABCD$ lies the $ABCD$ rectangle with the sides $AB = 1$ and $AD =
10$. The edge $SA$ of the pyramid is perpendicular to the base, $SA = 4$. On the edge of $AD$, find a point $M$ such that the perimeter of the triangle of $SMC$ was minimal.
Let $ABC$ be a triangle with sides 3, 4, and 5, and $DEFG$ be a 6-by-7 rectangle. A segment is drawn to divide triangle $ABC$ into a triangle $U_1$ and a trapezoid $V_1$ and another segment is drawn to divide rectangle $DEFG$ into a triangle $U_2$ and a trapezoid $V_2$ such that $U_1$ is similar to $U_2$ and $V_1$ is similar to $V_2$. The minimum value of the area of $U_1$ can be written in the form $m/n$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
A "letter $ T$" erected at point $ A$ of the $ x$-axis in the $ xy$-plane is the union of a segment $ AB$ in the upper half-plane perpendicular to the $ x$-axis and a segment $ CD$ containing $ B$ in its interior and parallel to the $ x$-axis. Show that it is impossible to erect a letter $ T$ at every point of the $ x$-axis so that the union of those erected at rational points is disjoint from the union of those erected at irrational points.
[i]A.Csaszar[/i]
Rectangle $ ABCD$ and a semicircle with diameter $ AB$ are coplanar and have nonoverlapping interiors. Let $ \mathcal{R}$ denote the region enclosed by the semicircle and the rectangle. Line $ \ell$ meets the semicircle, segment $ AB$, and segment $ CD$ at distinct points $ N$, $ U$, and $ T$, respectively. Line $ \ell$ divides region $ \mathcal{R}$ into two regions with areas in the ratio $ 1: 2$. Suppose that $ AU \equal{} 84$, $ AN \equal{} 126$, and $ UB \equal{} 168$. Then $ DA$ can be represented as $ m\sqrt {n}$, where $ m$ and $ n$ are positive integers and $ n$ is not divisible by the square of any prime. Find $ m \plus{} n$.
We are given $101$ rectangles with sides of integer lengths not exceeding $100$ . Prove that among these $101$ rectangles there are $3$ rectangles, say $A , B$ and $C$ such that $A$ will fit inside $B$ and $B$ inside $C$.
( N . Sedrakyan, Yerevan)
Two decompositions of a square into three rectangles are called substantially different, if reordering the rectangles does not change one into the other.
How many substantially different decompositions of a $2010 \times 2010$ square in three rectangles with integer side lengths are there such that the area of one rectangle is equal to the arithmetic mean of the areas of the other rectangles?
There is a square sheet of paper. How to get a rectangular sheet of paper with an aspect ratio equal to $\sqrt2$? (There are no tools, the sheet can only be bent.)
Consider a convex $n$-gon $A_1A_2 \dots A_n$. (Note: In a convex polygon, all interior angles are less than $180 \circ$.) Let $h$ be a positive number. Using the sides of the polygon as bases, we draw $n$ rectangles, each of height $h$, so that each rectangle is either entirely inside the $n$-gon or partially overlaps the inside of the $n$-gon.
As an example, the left figure below shows a pentagon with a correct configuration of rectangles, while the right figure shows an incorrect configuration of rectangles (since some of the rectangles do not overlap with the pentagon):