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 right rectangular prism $P$ (i.e., a rectangular parallelpiped) has sides of integral length $a, b, c,$ with $a\le b\le c.$ A plane parallel to one of the faces of $P$ cuts $P$ into two prisms, one of which is similar to $P,$ and both of which have nonzero volume. Given that $b=1995,$ for how many ordered triples $(a, b, c)$ does such a plane exist?

Rectangle $ABCD$ has $AB=6$ and $BC=3$. Point $M$ is chosen on side $AB$ so that $\angle AMD = \angle CMD$. What is the degree measure of $\angle AMD$? $ \textbf{(A)}\ 15 \qquad \textbf{(B)}\ 30 \qquad \textbf{(C)}\ 45 \qquad \textbf{(D)}\ 60 \qquad \textbf{(E)}\ 75 $

Let $u$ and $v$ be integers satisfying $0<v<u.$ Let $A=(u,v),$ let $B$ be the reflection of $A$ across the line $y=x,$ let $C$ be the reflection of $B$ across the y-axis, let $D$ be the reflection of $C$ across the x-axis, and let $E$ be the reflection of $D$ across the y-axis. The area of pentagon $ABCDE$ is 451. Find $u+v.$

What is the perimeter of a rectangle of area $32$ inscribed in a circle of radius $4$?

In a square $10^{2019} \times 10^{2019}, 10^{4038}$ points are marked. Prove that there is such a rectangle with sides parallel to the sides of a square whose area differs from the number of points located in it by at least $6$.

A collection of cardboard circles, each with a diameter of at most $1$, lie on a $5\times 8$ table without overlapping or overhanging the edge of the table. A cardboard circle of diameter $2$ is added to the collection. Prove that this new collection of cardboard circles can be placed on a $7\times 7$ table without overlapping or overhanging the edge.

Consider a chess board, with the numbers $1$ through $64$ placed in the squares as in the diagram below. \[\begin{tabular}{| c | c | c | c | c | c | c | c |} \hline 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\ \hline 9 & 10 & 11 & 12 & 13 & 14 & 15 & 16 \\ \hline 17 & 18 & 19 & 20 & 21 & 22 & 23 & 24 \\ \hline 25 & 26 & 27 & 28 & 29 & 30 & 31 & 32 \\ \hline 33 & 34 & 35 & 36 & 37 & 38 & 39 & 40 \\ \hline 41 & 42 & 43 & 44 & 45 & 46 & 47 & 48 \\ \hline 49 & 50 & 51 & 52 & 53 & 54 & 55 & 56 \\ \hline 57 & 58 & 59 & 60 & 61 & 62 & 63 & 64 \\ \hline \end{tabular}\] Assume we have an infinite supply of knights. We place knights in the chess board squares such that no two knights attack one another and compute the sum of the numbers of the cells on which the knights are placed. What is the maximum sum that we can attain? Note. For any $2\times3$ or $3\times2$ rectangle that has the knight in its corner square, the knight can attack the square in the opposite corner.

In rectangle $ABCD$, $AB=1$, $BC=2$, and points $E$, $F$, and $G$ are midpoints of $\overline{BC}$, $\overline{CD}$, and $\overline{AD}$, respectively. Point $H$ is the midpoint of $\overline{GE}$. What is the area of the shaded region? [asy] import graph; size(9cm); pen dps = fontsize(10); defaultpen(dps); pair D = (0,0); pair F = (1/2,0); pair C = (1,0); pair G = (0,1); pair E = (1,1); pair A = (0,2); pair B = (1,2); pair H = (1/2,1); // do not look pair X = (1/3,2/3); pair Y = (2/3,2/3); draw(A--B--C--D--cycle); draw(G--E); draw(A--F--B); draw(D--H--C); filldraw(H--X--F--Y--cycle,grey); label("$A$",A,NW); label("$B$",B,NE); label("$C$",C,SE); label("$D$",D,SW); label("$E$",E,E); label("$F$",F,S); label("$G$",G,W); label("$H$",H,N); label("$\displaystyle\frac12$",(0.25,0),S); label("$\displaystyle\frac12$",(0.75,0),S); label("$1$",(1,0.5),E); label("$1$",(1,1.5),E); [/asy] $ \textbf{(A)}\ \dfrac1{12}\qquad\textbf{(B)}\ \dfrac{\sqrt3}{18}\qquad\textbf{(C)}\ \dfrac{\sqrt2}{12}\qquad\textbf{(D)}\ \dfrac{\sqrt3}{12}\qquad\textbf{(E)}\ \dfrac16 $

Consider an 8x8 board divided in 64 unit squares. We call [i]diagonal[/i] in this board a set of 8 squares with the property that on each of the rows and the columns of the board there is exactly one square of the [i]diagonal[/i]. Some of the squares of this board are coloured such that in every [i]diagonal[/i] there are exactly two coloured squares. Prove that there exist two rows or two columns whose squares are all coloured.

Two equal rectangles of area $10$ are arranged as follows. Find the area of the gray rectangle. [img]https://cdn.artofproblemsolving.com/attachments/7/1/112b07530a2ef42e5b2cf83a2cb9fb11dfc9e6.png[/img]

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$

In the parallelogram $ABCD$, the diagonal $AC$ touches the incircles of triangles $ABC$ and $ADC$ at $W$ and $Y$ respectively, and the diagonal $BD$ touches the incircles of triangles $BAD$ and $BCD$ at $X$ and $Z$ respectively. Prove that either $W,X, Y$ and $Z$ coincide, or $WXYZ$ is a rectangle.

A rectangular table with 2001 rows and 2002 columns is partitioned into $1\times 2$ rectangles. It is known that any other partition of the table into $1\times 2$ rectangles contains a rectangle belonging to the original partition. Prove that the original partition contains two successive columns covered by 2001 horizontal rectangles. [i]Proposed by S. Volchenkov[/i]

Consider all possible rectangles that can be drawn on a $8 \times 8$ chessboard, covering only whole cells. Calculate the sum of their areas. What formula is obtained if “$8 \times 8$” is replaced with “$a \times b$”, where $a, b$ are positive integers?

The area of rectangle $ABCD$ is $72$. If point $A$ and the midpoints of $\overline{BC}$ and $\overline{CD}$ are joined to form a triangle, the area of that triangle is [asy] pair A,B,C,D; A = (0,8); B = (9,8); C = (9,0); D = (0,0); draw(A--B--C--D--A--(9,4)--(4.5,0)--cycle); label("$A$",A,NW); label("$B$",B,NE); label("$C$",C,SE); label("$D$",D,SW); [/asy] $\text{(A)}\ 21 \qquad \text{(B)}\ 27 \qquad \text{(C)}\ 30 \qquad \text{(D)}\ 36 \qquad \text{(E)}\ 40$

Two circles of equal radius can tightly fit inside right triangle $ABC$, which has $AB=13$, $BC=12$, and $CA=5$, in the three positions illustrated below. Determine the radii of the circles in each case. [asy] size(400); defaultpen(linewidth(0.7)+fontsize(12)); picture p = new picture; pair s1 = (20,0), s2 = (40,0); real r1 = 1.5, r2 = 10/9, r3 = 26/7; pair A=(12,5), B=(0,0), C=(12,0); draw(p,A--B--C--cycle); label(p,"$B$",B,SW); label(p,"$A$",A,NE); label(p,"$C$",C,SE); add(p); add(shift(s1)*p); add(shift(s2)*p); draw(circle(C+(-r1,r1),r1)); draw(circle(C+(-3*r1,r1),r1)); draw(circle(s1+C+(-r2,r2),r2)); draw(circle(s1+C+(-r2,3*r2),r2)); pair D=s2+156/17*(A-B)/abs(A-B), E=s2+(169/17,0), F=extension(D,E,s2+A,s2+C); draw(incircle(s2+B,D,E)); draw(incircle(s2+A,D,F)); label("Case (i)",(6,-3)); label("Case (ii)",s1+(6,-3)); label("Case (iii)",s2+(6,-3));[/asy]

In an acute triangle $ABC,$ let $O,G,H$ be its circumcentre, centroid and orthocenter. Let $D\in BC, E\in CA$ and $OD\perp BC, HE\perp CA.$ Let $F$ be the midpoint of $AB.$ If the triangles $ODC, HEA, GFB$ have the same area, find all the possible values of $\angle C.$

A unit square is divided into two rectangles in such a way that the smaller rectangle can be put on the greater rectangle with every vertex of the smaller on exactly one of the edges of the greater. Calculate the dimensions of the smaller rectangle.

Let $C_1$ and $C_2$ be two circles intersecting at $A $ and $B$. Let $S$ and $T $ be the centres of $C_1 $ and $C_2$, respectively. Let $P$ be a point on the segment $AB$ such that $ |AP|\ne |BP|$ and $P\ne A, P \ne B$. We draw a line perpendicular to $SP$ through $P$ and denote by $C$ and $D$ the points at which this line intersects $C_1$. We likewise draw a line perpendicular to $TP$ through $P$ and denote by $E$ and F the points at which this line intersects $C_2$. Show that $C, D, E,$ and $F$ are the vertices of a rectangle.

In rectangle $ABCD$, diagonal $AC$ is met by the angle bisector from $B$ at $B'$ and the angle bisector from $D$ at $D'$. Diagonal $BD$ is met by the angle bisector from $A$ at $A'$ and the angle bisector from $C$ at $C'$. The area of quadrilateral $A'B'C'D'$ is $\frac{9}{16}$ the area of rectangle $ABCD$. What is the ratio of the longer side and shorter side of rectangle $ABCD$?

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.)

You have some white one-by-one tiles and some black and white two-bye-one tiles as shown below. There are four different color patterns that can be generated when using these tiles to cover a three-by-one rectangoe by laying these tiles side by side (WWW, BWW, WBW, WWB). How many different color patterns can be generated when using these tiles to cover a ten-by-one rectangle? [asy] import graph; size(5cm); real labelscalefactor = 0.5; pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); draw((12,0)--(12,1)--(11,1)--(11,0)--cycle); fill((13.49,0)--(13.49,1)--(12.49,1)--(12.49,0)--cycle, black); draw((13.49,0)--(13.49,1)--(14.49,1)--(14.49,0)--cycle); draw((15,0)--(15,1)--(16,1)--(16,0)--cycle); fill((17,0)--(17,1)--(16,1)--(16,0)--cycle, black); [/asy]

Let $L$ be the length of the altitude to the hypotenuse of a right triangle with legs $5$ and $12$. Find the least integer greater than $L$.

A rectangular sheet of paper (white on one side and gray on the other) was folded three times, as shown in the figure: Rectangle $1$, which was white after the first fold, has $20$ cm more perimeter than rectangle $2$, which was white after the second fold, and this in turn has $16$ cm more perimeter than rectangle $3$, which was white after the third fold. Determine the area of the sheet. [img]https://cdn.artofproblemsolving.com/attachments/d/f/8e363b40654ad0d8e100eac38319ee3784a7a7.png[/img]