A square with sides of length $6$ has the same area as a rectangle with a length of $9$. What is the width of the rectangle? $\textbf{(A) } 2\qquad\textbf{(B) } \frac73\qquad\textbf{(C) } 3\qquad\textbf{(D) } \frac{10}{3}\qquad\textbf{(E) } 4$

Let $S > 1$ be a real number. The Cartesian plane is partitioned into rectangles whose sides are parallel to the axes of the coordinate system. and whose vertices have integer coordinates. Prove that if the area of each triangle if at most $S$, then for any positive integer $k$ there exist $k$ vertices of these rectangles which lie on a line.

Rectangle $ ABCD$ has $ AB \equal{} 4$ and $ BC \equal{} 3$. Segment $ EF$ is constructed through $ B$ so that $ EF$ is perpendicular to $ DB$, and $ A$ and $ C$ lie on $ DE$ and $ DF$, respectively. What is $ EF$? $ \textbf{(A)}\ 9\qquad \textbf{(B)}\ 10\qquad \textbf{(C)}\ \frac {125}{12}\qquad \textbf{(D)}\ \frac {103}{9}\qquad \textbf{(E)}\ 12$

Let $ABCD$ be a rectangle and $E$ the reflection of $A$ with respect to the diagonal $BD$. If $EB = EC$, what is the ratio $\frac{AD}{AB}$ ?

In a regular pyramid with top point $ T$ and equilateral base $ ABC$, let $ P$, $ Q$, $ R$, $ S$ be the midpoints of $ \left[AB\right]$, $ \left[BC\right]$, $ \left[CT\right]$ and $ \left[TA\right]$, respectively. If $ \left|AB\right| \equal{} 6$ and the altitude of pyramid is equal to $ 2\sqrt {15}$, then area of $ PQRS$ will be $\textbf{(A)}\ 4\sqrt {15} \qquad\textbf{(B)}\ 8\sqrt {2} \qquad\textbf{(C)}\ 8\sqrt {3} \qquad\textbf{(D)}\ 6\sqrt {5} \qquad\textbf{(E)}\ 9\sqrt {2}$

$A,B,C$ are points in the plane such that $\angle ABC=90^\circ$. Circles with diameters $BA$ and $BC$ meet at $D$. If $BA=20$ and $BC=21$, then the length of segment $BD$ can be expressed in the form $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. What is $m+n$? [i]Ray Li[/i]

Rectangle $ABCD$ is divided into four parts of equal area by five segments as shown in the figure, where $XY = YB + BC + CZ = ZW = WD + DA + AX$, and $PQ$ is parallel to $AB$. Find the length of $AB$ (in cm) if $BC = 19$ cm and $PQ = 87$ cm. [asy] size(250); pair A=origin, B=(96,0), C=(96,22), D=(0,22), W=(16,22), X=(20,0), Y=(80,0), Z=(76,22), P=(24,11), Q=(72,11); draw(P--X--A--D--W--P--Q--Y--B--C--Z--Q^^W--Z^^X--Y); dot(A^^B^^C^^D^^P^^Q^^W^^X^^Y^^Z); pair point=(48,11); label("$A$", A, SW); label("$B$", B, SE); label("$C$", C, NE); label("$D$", D, NW); label("$P$", P, dir(point--P)); label("$Q$", Q, dir(point--Q)); label("$W$", W, N); label("$X$", X, S); label("$Y$", Y, S); label("$Z$", Z, N);[/asy]

Six rectangles each with a common base width of $2$ have lengths of $1, 4, 9, 16, 25,$ and $36$. What is the sum of the areas of the six rectangles? $\textbf{(A) }91\qquad\textbf{(B) }93\qquad\textbf{(C) }162\qquad\textbf{(D) }182\qquad \textbf{(E) }202$

We have a $ (n+2)\times n $ rectangle and we’ve divided it into $ n(n+2) \ \ 1\times1 $ squares. $ n(n+2) $ soldiers are standing on the intersection points ($ n+2 $ rows and $ n $ columns). The commander shouts and each soldier stands on its own location or gaits one step to north, west, east or south so that he stands on an adjacent intersection point. After the shout, we see that the soldiers are standing on the intersection points of a $ n\times(n+2) $ rectangle ($ n $ rows and $ n+2 $ columns) such that the first and last row are deleted and 2 columns are added to the right and left (To the left $1$ and $1$ to the right). Prove that $ n $ is even.

In a rectangle $ABCD, E$ is the midpoint of $AB, F$ is a point on $AC$ such that $BF$ is perpendicular to $AC$, and $FE$ perpendicular to $BD$. Suppose $BC = 8\sqrt3$. Find $AB$.

Let $P$ be a point in the plane and let $\gamma$ be a circle which does not contain $P$. Two distinct variable lines $\ell$ and $\ell'$ through $P$ meet the circle $\gamma$ at points $X$ and $Y$, and $X'$ and $Y'$, respectively. Let $M$ and $N$ be the antipodes of $P$ in the circles $PXX'$ and $PYY'$, respectively. Prove that the line $MN$ passes through a fixed point. [i]Mihai Chis[/i]

A rectangle $\mathcal{R}$ with odd integer side lengths is divided into small rectangles with integer side lengths. Prove that there is at least one among the small rectangles whose distances from the four sides of $\mathcal{R}$ are either all odd or all even. [i]Proposed by Jeck Lim, Singapore[/i]

Is it possible to cut a square with side $\sqrt{2015}$ into no more than five pieces so that these pieces can be rearranged into a rectangle with sides of integer length? (The cuts should be made using straight lines, and flipping of the pieces is disallowed.)

Let $n$ be an integer greater than $1$. For a positive integer $m$, let $S_{m}= \{ 1,2,\ldots, mn\}$. Suppose that there exists a $2n$-element set $T$ such that (a) each element of $T$ is an $m$-element subset of $S_{m}$; (b) each pair of elements of $T$ shares at most one common element; and (c) each element of $S_{m}$ is contained in exactly two elements of $T$. Determine the maximum possible value of $m$ in terms of $n$.

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.)