Found problems: 1342
In the coordinate plane a rectangle with vertices $ (0, 0),$ $ (m, 0),$ $ (0, n),$ $ (m, n)$ is given where both $ m$ and $ n$ are odd integers. The rectangle is partitioned into triangles in such a way that
[i](i)[/i] each triangle in the partition has at least one side (to be called a “good” side) that lies on a line of the form $ x \equal{} j$ or $ y \equal{} k,$ where $ j$ and $ k$ are integers, and the altitude on this side has length 1;
[i](ii)[/i] each “bad” side (i.e., a side of any triangle in the partition that is not a “good” one) is a common side of two triangles in the partition.
Prove that there exist at least two triangles in the partition each of which has two good sides.
Let $ R$ be a rectangle that is the union of a finite number of rectangles $ R_i,$ $ 1 \leq i \leq n,$ satisfying the following conditions:
[b](i)[/b] The sides of every rectangle $ R_i$ are parallel to the sides of $ R.$
[b](ii)[/b] The interiors of any two different rectangles $ R_i$ are disjoint.
[b](iii)[/b] Each rectangle $ R_i$ has at least one side of integral length.
Prove that $ R$ has at least one side of integral length.
[i]Variant:[/i] Same problem but with rectangular parallelepipeds having at least one integral side.
Is it possible to write a positive integer in every cell of an infinite chessboard, in such a manner that, for all positive integers $m, n$, the sum of numbers in every $m\times n$ rectangle is divisible by $m + n$ ?
In the diagram below, square $ABCD$ with side length 23 is cut into nine rectangles by two lines parallel to $\overline{AB}$ and two lines parallel to $\overline{BC}$. The areas of four of these rectangles are indicated in the diagram. Compute the largest possible value for the area of the central rectangle.
[asy]
size(250);
defaultpen (linewidth (0.7) + fontsize (10));
draw ((0,0)--(23,0)--(23,23)--(0,23)--cycle);
label("$A$", (0,23), NW);
label("$B$", (23, 23), NE);
label("$C$", (23,0), SE);
label("$D$", (0,0), SW);
draw((0,6)--(23,6));
draw((0,19)--(23,19));
draw((5,0)--(5,23));
draw((12,0)--(12,23));
label("13", (17/2, 21));
label("111",(35/2,25/2));
label("37",(17/2,3));
label("123",(2.5,12.5));[/asy]
[i]Proposed by Lewis Chen[/i]
Let A and B be two rectangles such that it is possible to get rectangle similar to A by putting together rectangles equal to B. Show that it is possible to get rectangle similar to B by putting together rectangles equal to A.
At a store, when a length is reported as $x$ inches that means the length is at least $x-0.5$ inches and at most $x+0.5$ inches. Suppose the dimensions of a rectangular tile are reported as $2$ inches by $3$ inches. In square inches, what is the minimum area for the rectangle?
$ \textbf{(A)}\ 3.75 \qquad
\textbf{(B)}\ 4.5 \qquad
\textbf{(C)}\ 5 \qquad
\textbf{(D)}\ 6 \qquad
\textbf{(E)}\ 8.75 $
A rectangle of paper of $3$ cm by $9$ cm is folded along a straight line, making two opposite vertices coincide. In this way a pentagon is formed. Calculate it's area.
Consider two circles, tangent at $T$, both inscribed in a rectangle of height $2$ and width $4$. A point $E$ moves counterclockwise around the circle on the left, and a point $D$ moves clockwise around the circle on the right. $E$ and $D$ start moving at the same time; $E$ starts at $T$, and $D$ starts at $A$, where $A$ is the point where the circle on the right intersects the top side of the rectangle. Both points move with the same speed. Find the locus of the midpoints of the segments joining $E$ and $D$.
Consider the shape formed from taking equilateral triangle $ABC$ with side length $6$ and tracing out the arc $BC$ with center $A$. Set the shape down on line $l$ so that segment $AB$ is perpendicular to $l$, and $B$ touches $l$. Beginning from arc $BC$ touching $l$, we roll $ABC$ along $l$ until both points $A$ and $C$ are on the line. The area traced out by the roll can be written in the form $n\pi$, where $n$ is an integer. Find $n$.
Prove that it is not possible to cover a $6\times 6$ square board with eighteen $2\times 1$ rectangles, in such a way that each of the lines going along the interior gridlines cuts at least one of the rectangles. Show also that it is possible to cover a $6\times 5$ rectangle with fifteen $2\times 1 $ rectangles so that the above condition is fulfilled.
The kid cut out of grid paper with the side of the cell $1$ rectangle along the grid lines and calculated its area and perimeter. Carlson snatched his scissors and cut out of this rectangle along the lines of the grid a square adjacent to the boundary of the rectangle.
- My rectangle ... - kid sobbed. - There is something strange about this figure!
- Nonsense, do not mention it - Carlson said - waving his hand carelessly. - Here you see, in this figure the perimeter is the same as the area of the rectangle was, and the area is the same as was the perimeter!
What size square did Carlson cut out?
The figure shows a rectangle, its circumscribed circle and four semicircles, which have the rectangle’s sides as diameters. Prove that the combined area of the four dark gray crescentshaped regions is equal to the area of the light gray rectangle.
[img]https://1.bp.blogspot.com/-gojv6KfBC9I/XzT9ZMKrIeI/AAAAAAAAMVU/NB-vUldjULI7jvqiFWmBC_Sd8QFtwrc7wCLcBGAsYHQ/s0/2013%2BMohr%2Bp3.png[/img]
Let $N_{14}$ be the answer to problem 14.
Rectangle $ABCD$ has area $\sqrt{2N_{14}}$. Points $E$, $F$, $G$, and $H$ lie on the rays $\overrightarrow{AB}$, $\overrightarrow{BC}$, $\overrightarrow{CD}$, and $\overrightarrow{DA}$, respectively, such that $EFGH$ is a rectangle with area $2\sqrt{2N_{14}}$ that contains all of $ABCD$ in its interior. If
\[ \tan\angle AEH = \tan\angle BFE = \tan\angle CGF = \tan\angle DHG = \sqrt{\frac{1}{48}}, \]
then $EG=\tfrac{m\sqrt{n}}{p}$, where $m$, $n$, and $p$ are positive integers, $m$ and $p$ are relatively prime, and $n$ is not divisible by the square of any prime. Compute $m + n + p$.
Which of the following has the greatest number of line of symmetry?
$ \textbf{(A)}\ \text{ Equilateral Triangle}$
$\textbf{(B)}\ \text{Non-square rhombus} $
$\textbf{(C)}\ \text{Non-square rectangle}$
$\textbf{(D)}\ \text{Isosceles Triangle}$
$\textbf{(E)}\ \text{Square} $
Let us consider all rectangles with sides of length $a,b$ both of which are whole numbers. Do more of these rectangles have perimeter $2000$ or perimeter $2002$?
The base of a triangle is of length $b$, and the latitude is of length $h$. A rectangle of height $x$ is inscribed in the triangle with the base of the rectangle in the base of the triangle. The area of the rectangle is:
$ \textbf{(A)}\ \frac{bx}{h}(h-x)\qquad\textbf{(B)}\ \frac{hx}{b}(b-x)\qquad\textbf{(C)}\ \frac{bx}{h}(h-2x)\qquad$
$\textbf{(D)}\ x(b-x)\qquad\textbf{(E)}\ x(h-x) $
Let $ABCD$ be a rectangle of center $O$, such that $\angle DAC = 60^o$. The angle bisector of $\angle DAC$ meets $DC$ at $S$. Lines $OS$ and $AD$ meet at $L$ and lines $BL$ and $AC$ meet at $M$. Prove that lines $SM$ and $CL$ are parallel.
The diagram below shows rectangle $ABDE$ where $C$ is the midpoint of side $\overline{BD}$, and $F$ is the midpoint of side $\overline{AE}$. If $AB=10$ and $BD=24$, find the area of the shaded region.
[asy]
size(300);
defaultpen(linewidth(0.8));
pair A = (0,10),B=origin,C=(12,0),D=(24,0),E=(24,10),F=(12,10),G=extension(C,E,D,F);
filldraw(A--C--G--F--cycle,gray(0.7));
draw(A--B--D--E--F^^E--G--D);
label("$A$",A,NW);
label("$B$",B,SW);
label("$C$",C,S);
label("$D$",D,SE);
label("$E$",E,NE);
label("$F$",F,N);
[/asy]
A circle with area $40$ is tangent to a circle with area $10$. Let R be the smallest rectangle containing both circles. The area of $R$ is $\frac{n}{\pi}$. Find $n$.
[asy]
defaultpen(linewidth(0.7)); size(120);
real R = sqrt(40/pi), r = sqrt(10/pi);
draw(circle((0,0), R)); draw(circle((R+r,0), r));
draw((-R,-R)--(-R,R)--(R+2*r,R)--(R+2*r,-R)--cycle);[/asy]
In rectangle $ABCD$, points $E$ and $F$ lie on sides $AB$ and $CD$ respectively such that both $AF$ and $CE$ are perpendicular to diagonal $BD$. Given that $BF$ and $DE$ separate $ABCD$ into three polygons with equal area, and that $EF = 1$, find the length of $BD$.
Let be $ABCD$ a rectangle with $|AB|=a\ge b=|BC|$. Find points $P,Q$ on the line $BD$ such that $|AP|=|PQ|=|QC|$. Discuss the solvability with respect to the lengths $a,b$.
The bisectors of trapezoid's angles form a quadrilateral with perpendicular diagonals. Prove that this trapezoid is isosceles.
Prove that circles constructed on the sides of a convex quadrilateral as diameters completely cover this quadrilateral.
Given is an acute angled triangle $ \triangle ABC$ with side lengths $ a$, $ b$ and $ c$ (in an usual way) and circumcenter $ O$. Angle bisector of angle $ \angle BAC$ intersects circumcircle at points $ A$ and $ A_{1}$. Let $ D$ be projection of point $ A_{1}$ onto line $ AB$, $ L$ and $ M$ be midpoints of $ AC$ and $ AB$ , respectively.
(i) Prove that $ AD\equal{}\frac{1}{2}(b\plus{}c)$
(ii) If triangle $ \triangle ABC$ is an acute angled prove that $ A_{1}D\equal{}OM\plus{}OL$
In the plane we consider rectangles whose sides are parallel to the coordinate axes and have positive length. Such a rectangle will be called a [i]box[/i]. Two boxes [i]intersect[/i] if they have a common point in their interior or on their boundary. Find the largest $ n$ for which there exist $ n$ boxes $ B_1$, $ \ldots$, $ B_n$ such that $ B_i$ and $ B_j$ intersect if and only if $ i\not\equiv j\pm 1\pmod n$.
[i]Proposed by Gerhard Woeginger, Netherlands[/i]