A rectangle has sides of length $a$ and $36$. A hinge is installed at each vertex of the rectangle and at the midpoint of each side of length $36$. The sides of length $a$ can be pressed toward each other keeping those two sides parallel so the rectangle becomes a convex hexagon as shown. When the figure is a hexagon with the sides of length $a$ parallel and separated by a distance of $24,$ the hexagon has the same area as the original rectangle. Find $a^2$. [asy] pair A,B,C,D,E,F,R,S,T,X,Y,Z; dotfactor = 2; unitsize(.1cm); A = (0,0); B = (0,18); C = (0,36); // don't look here D = (12*2.236, 36); E = (12*2.236, 18); F = (12*2.236, 0); draw(A--B--C--D--E--F--cycle); dot(" ",A,NW); dot(" ",B,NW); dot(" ",C,NW); dot(" ",D,NW); dot(" ",E,NW); dot(" ",F,NW); //don't look here R = (12*2.236 +22,0); S = (12*2.236 + 22 - 13.4164,12); T = (12*2.236 + 22,24); X = (12*4.472+ 22,24); Y = (12*4.472+ 22 + 13.4164,12); Z = (12*4.472+ 22,0); draw(R--S--T--X--Y--Z--cycle); dot(" ",R,NW); dot(" ",S,NW); dot(" ",T,NW); dot(" ",X,NW); dot(" ",Y,NW); dot(" ",Z,NW); // sqrt180 = 13.4164 // sqrt5 = 2.236 [/asy]

Let $C (O)$ be a circle (with center $O$ ) and $A, B$ points on the circle with $\angle AOB = 90^o$. Circles $C_1 (O_1)$ and $C_2 (O_2)$ are tangent internally with circle $C$ at $A$ and $B$, respectively, and, also, are tangent to each other. Consider another circle $C_3 (O_3)$ tangent externally to the circles $C_1, C_2$ and tangent internally to circle $C$, located inside angle $\angle AOB$. Show that the points $O, O_1, O_2, O_3$ are the vertices of a rectangle.

Let $ n$ be natural number. Find the limit value of ${ \lim_{n\to\infty} \frac{1}{n}(\frac{1}{\sqrt{2}}+\frac{2}{\sqrt{5}}}+\cdots\cdots +\frac{n}{\sqrt{n^2+1}}).$

$ABCD$ is a quadrilateral with a circumcircle centre $O$ and an inscribed circle centre $I$. The diagonals intersect at $S$. Show that if two of $O,I,S$ coincide, then it must be a square.

Let $m$ and $n$ be integers greater than 1. Prove that $\left\lfloor \dfrac{mn}{6} \right\rfloor$ non-overlapping 2-by-3 rectangles can be placed in an $m$-by-$n$ rectangle. Note: $\lfloor x \rfloor$ means the greatest integer that is less than or equal to $x$.

A square is divided by horizontal and vertical lines that form $n^2$ squares each with side $1$. What is the greatest possible value of $n$ such that it is possible to select $n$ squares such that any rectangle with area $n$ formed by the horizontal and vertical lines would contain at least one of the selected $n$ squares.

Let $S$ be the square one of whose diagonals has endpoints $(0.1,0.7)$ and $(-0.1,-0.7)$. A point $v=(x,y)$ is chosen uniformly at random over all pairs of real numbers $x$ and $y$ such that $0\le x \le 2012$ and $0 \le y \le 2012$. Let $T(v)$ be a translated copy of $S$ centered at $v$. What is the probability that the square region determined by $T(v)$ contains exactly two points with integer coordinates in its interior? $ \textbf{(A)}\ 0.125\qquad\textbf{(B)}\ 0.14\qquad\textbf{(C)}\ 0.16\qquad\textbf{(D)}\ 0.25\qquad\textbf{(E)}\ 0.32 $

Given a square $ABCD$ with side $10$. On sides BC and $AD$ of this square are selected respectively points $E$ and $F$ such that formed a rectangle $ABEF$. Rectangle $KLMN$ is located so that its the vertices $K, L, M$ and $N$ lie one on each segments $CD, DF, FE$ and $EC$, respectively. It turned out that the rectangles $ABEF$ and $KLMN$ are equal with $AB = MN$. Find the length of segment $AL$.

A square sheet of paper $ABCD$ is so folded that $B$ falls on the mid point of $M$ of $CD$. Prove that the crease will divide $BC$ in the ration $5 : 3$.

Let $ABCD$ be a rectangle inscribed in circle $\omega$ with center $O$. Line $l$ passes trough $O$ and intersects lines $BC$ and $AD$ at points $E$ and $F$ respectively. Points $K$ and $L$ are the intersection points of $l$ and $\omega$ and points $K, E, F, L$ lie in this order on the line $l$. Lines tangent to $w$ in $K$ and $L$ intersect $CD$ at $M$ and $N$ respectively. Prove that $E, F, M, N$ lie on a common circle.

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.

Is it possible to cover a rectangle of dimensions $m \times n$ with bricks that have the trimino angular shape (an arrangement of three unit squares forming the letter $\text L$) if: [b](a)[/b] $m \times n = 1985 \times 1987;$ [b](b)[/b] $m \times n = 1987 \times 1989 \quad ?$

In the rectangle $ABCD, BC = 5, EC = 1/3 CD$ and $F$ is the point where $AE$ and $BD$ are cut. The triangle $DFE$ has area $12$ and the triangle $ABF$ has area $27$. Find the area of the quadrilateral $BCEF$ . [img]https://1.bp.blogspot.com/-4w6e729AF9o/XNY9hqHaBaI/AAAAAAAAKL0/eCaNnWmgc7Yj9uV4z29JAvTcWCe21NIMgCK4BGAYYCw/s400/may%2B2011%2Bl1.png[/img]

Turki has divided a square into finitely many white and green rectangles, each with sides parallel to the sides of the square. Within each white rectangle, he writes down its width divided by its height. Within each green rectangle, he writes down its height divided by its width. Finally, he calculates $S$, the sum of these numbers. If the total area of white rectangles equals the total area of green rectangles, determine the minimum possible value of $S$.

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]

Given a finite string $S$ of symbols $X$ and $O$, we write $@(S)$ for the number of $X$'s in $S$ minus the number of $O$'s. (For example, $@(XOOXOOX) =-1$.) We call a string $S$ [b]balanced[/b] if every substring $T$ of (consecutive symbols) $S$ has the property $-2 \leq @(T) \leq 2$. (Thus $XOOXOOX$ is not balanced since it contains the sub-string $OOXOO$ whose $@$-value is $-3$.) Find, with proof, the number of balanced strings of length $n$.

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]

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}$

Let $ ABCD $ be a rectangle that has $ M $ on its $ BD $ diagonal. If $ N,P $ are the projections of $ M $ on $ AB, $ respectively, $ AD, $ what's the locus of the intersection between $ CP $ and $ DN? $

Let $ABC$ be an acute triangle. Let $D$ be a point on the line parallel to $AC$ that passes through $B$, such that $\angle BDC = 2\angle BAC$ as well as such that $ABDC$ is a convex quadrilateral. Show that $BD + DC = AC$.

In the rectangle $ABCD$, $AB = 2BC$. An equilateral triangle $ABE$ is constructed on the side $AB$ of the rectangle so that its sides $AE$ and $BE$ intersect the segment $CD$. Point $M$ is the midpoint of $BE$. Find the $\angle MCD$.

A square board is divided by lines parallel to the board sides ($7$ lines in each direction, not necessarily equidistant ) into $64$ rectangles. Rectangles are colored into white and black in alternating order. Assume that for any pair of white and black rectangles the ratio between area of white rectangle and area of black rectangle does not exceed $2.$ Determine the maximal ratio between area of white and black part of the board. White (black) part of the board is the total sum of area of all white (black) rectangles.

Each point of a plane is colored in one of three colors: red, black and blue. Prove that there exists a rectangle in this plane whose vertices all have the same color.

Circles with centers $ O$ and $ P$ have radii 2 and 4, respectively, and are externally tangent. Points $ A$ and $ B$ are on the circle centered at $ O$, and points $ C$ and $ D$ are on the circle centered at $ P$, such that $ \overline{AD}$ and $ \overline{BC}$ are common external tangents to the circles. What is the area of hexagon $ AOBCPD$? [asy] size(250);defaultpen(linewidth(0.8)); pair X=(-6,0), O=origin, P=(6,0), B=tangent(X, O, 2, 1), A=tangent(X, O, 2, 2), C=tangent(X, P, 4, 1), D=tangent(X, P, 4, 2); pair top=X+15*dir(X--A), bottom=X+15*dir(X--B); draw(Circle(O, 2)^^Circle(P, 4)); draw(bottom--X--top); draw(A--O--B^^O--P^^D--P--C); pair point=X; label("$2$", midpoint(O--A), dir(point--midpoint(O--A))); label("$4$", midpoint(P--D), dir(point--midpoint(P--D))); label("$O$", O, SE); label("$P$", P, dir(point--P)); pair point=O; label("$A$", A, dir(point--A)); label("$B$", B, dir(point--B)); pair point=P; label("$C$", C, dir(point--C)); label("$D$", D, dir(point--D)); fill((-3,7)--(-3,-7)--(-7,-7)--(-7,7)--cycle, white);[/asy] $ \textbf{(A) } 18\sqrt {3} \qquad \textbf{(B) } 24\sqrt {2} \qquad \textbf{(C) } 36 \qquad \textbf{(D) } 24\sqrt {3} \qquad \textbf{(E) } 32\sqrt {2}$

A rectangle $ABCD$ and a point $P$ are given. Lines passing through $A$ and $B$ and perpendicular to $PC$ and $PD$ respectively, meet at a point $Q$. Prove that $PQ \perp AB$.