$p$ parallel lines are drawn in the plane and $q$ lines perpendicular to them are also drawn. How many rectangles are bounded by the lines?

Let $S$ be a square of side length $1$. Two points are chosen independently at random on the sides of $S$. The probability that the straight-line distance between the points is at least $\tfrac12$ is $\tfrac{a-b\pi}c$, where $a$, $b$, and $c$ are positive integers and $\gcd(a,b,c)=1$. What is $a+b+c$? $\textbf{(A) }59\qquad\textbf{(B) }60\qquad\textbf{(C) }61\qquad\textbf{(D) }62\qquad\textbf{(E) }63$

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 $D$.

A rectangle is divided into $200\times 3$ unit squares. Prove that the number of ways of splitting this rectangle into rectangles of size $1\times 2$ is divisible by $3$.

A square with side length $8$ is cut in half, creating two congruent rectangles. What are the dimensions of one of these rectangles? $ \textbf{(A)}\ 2\text{ by }4 \qquad\textbf{(B)}\ 2\text{ by }6 \qquad\textbf{(C)}\ 2\text{ by }8 \qquad\textbf{(D)}\ 4\text{ by }4 \qquad\textbf{(E)}\ 4\text{ by }8 $

Circle $\omega_1$ with radius $6$ centered at point $A$ is internally tangent at point $B$ to circle $\omega_2$ with radius $15$. Points $C$ and $D$ lie on $\omega_2$ such that $\overline{BC}$ is a diameter of $\omega_2$ and $\overline{BC} \perp \overline{AD}$. The rectangle $EFGH$ is inscribed in $\omega_1$ such that $\overline{EF} \perp \overline{BC}$, $C$ is closer to $\overline{GH}$ than to $\overline{EF}$, and $D$ is closer to $\overline{FG}$ than to $\overline{EH}$, as shown. Triangles $\triangle DGF$ and $\triangle CHG$ have equal areas. The area of rectangle $EFGH$ is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$. [asy] size(5cm); defaultpen(fontsize(10pt)); pair A = (9, 0), B = (15, 0), C = (-15, 0), D = (9, 12), E = (9+12/sqrt(5), -6/sqrt(5)), F = (9+12/sqrt(5), 6/sqrt(5)), G = (9-12/sqrt(5), 6/sqrt(5)), H = (9-12/sqrt(5), -6/sqrt(5)); filldraw(G--H--C--cycle, lightgray); filldraw(D--G--F--cycle, lightgray); draw(B--C); draw(A--D); draw(E--F--G--H--cycle); draw(circle(origin, 15)); draw(circle(A, 6)); dot(A); dot(B); dot(C); dot(D); dot(E); dot(F); dot(G); dot(H); label("$A$", A, (.8, -.8)); label("$B$", B, (.8, 0)); label("$C$", C, (-.8, 0)); label("$D$", D, (.4, .8)); label("$E$", E, (.8, -.8)); label("$F$", F, (.8, .8)); label("$G$", G, (-.8, .8)); label("$H$", H, (-.8, -.8)); label("$\omega_1$", (9, -5)); label("$\omega_2$", (-1, -13.5)); [/asy]

$ABCD$ 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.

We say that a rectangle with side lengths $a$ and $b$ [i]fits inside[/i] a rectangle with side lengths $c$ and $d$ if either ($a \le c$ and $b \le d$) or ($a \le d$ and $b \le c$). For instance, a rectangle with side lengths $1$ and $5$ [i]fits inside[/i] another rectangle with side lengths $1$ and $5$, and also [i]fits inside[/i] a rectangle with side lengths $6$ and $2$. Suppose $S$ is a set of $2019$ rectangles, all with integer side lengths between $1$ and $2018$ inclusive. Show that there are three rectangles $A$, $B$, and $C$ in $S$ such that $A$ fits inside $B$, and $B$ [i]fits inside [/i]$C$.

Diagonal $ DB$ of rectangle $ ABCD$ is divided into $ 3$ segments of length $ 1$ by parallel lines $ L$ and $ L'$ that pass through $ A$ and $ C$ and are perpendicular to $ DB$. The area of $ ABCD$, rounded to the nearest tenth, is [asy]size(200); defaultpen(linewidth(0.7)+fontsize(10)); real x=sqrt(6), y=sqrt(3), a=0.4; pair D=origin, A=(0,y), B=(x,y), C=(x,0), E=foot(C,B,D), F=foot(A,B,D); real r=degrees(B); pair M1=F+3*dir(r)*dir(90), M2=F+3*dir(r)*dir(-90), N1=E+3*dir(r)*dir(90), N2=E+3*dir(r)*dir(-90); markscalefactor=0.02; draw(B--C--D--A--B--D^^M1--M2^^N1--N2^^rightanglemark(A,F,B,6)^^rightanglemark(N1,E,B,6)); pair W=A+a*dir(135), X=B+a*dir(45), Y=C+a*dir(-45), Z=D+a*dir(-135); label("A", A, NE); label("B", B, NE); label("C", C, dir(0)); label("D", D, dir(180)); label("$L$", (x/2,0), SW); label("$L^\prime$", C, SW); label("1", D--F, NW); label("1", F--E, SE); label("1", E--B, SE); clip(W--X--Y--Z--cycle); [/asy] $ \textbf{(A)}\ 4.1 \qquad \textbf{(B)}\ 4.2 \qquad \textbf{(C)}\ 4.3 \qquad \textbf{(D)}\ 4.4 \qquad \textbf{(E)}\ 4.5$

A game is played on a $23\times 23$ board. The first player controls two white chips which start in the bottom left and top right corners. The second player controls two black ones which start in bottom right and top left corners. The players move alternately. In each move, a player moves one of the chips under control to a square which shares a side with the square the chip is currently in. The first player wins if he can bring the white chips to squares which share a side with each other. Can the second player prevent the first player from winning?

The $ 8\times 18$ rectangle $ ABCD$ is cut into two congruent hexagons, as shown, in such a way that the two hexagons can be repositioned without overlap to form a square. What is $ y$? [asy] unitsize(2mm); defaultpen(fontsize(10pt)+linewidth(.8pt)); dotfactor=4; draw((0,4)--(18,4)--(18,-4)--(0,-4)--cycle); draw((6,4)--(6,0)--(12,0)--(12,-4)); label("$D$",(0,4),NW); label("$C$",(18,4),NE); label("$B$",(18,-4),SE); label("$A$",(0,-4),SW); label("$y$",(9,1)); [/asy]$ \textbf{(A) } 6\qquad \textbf{(B) } 7\qquad \textbf{(C) } 8\qquad \textbf{(D) } 9\qquad \textbf{(E) } 10$

The keystone arch is an ancient architectural feature. It is composed of congruent isosceles trapezoids fitted together along the non-parallel sides, as shown. The bottom sides of the two end trapezoids are horizontal. In an arch made with $ 9$ trapezoids, let $ x$ be the angle measure in degrees of the larger interior angle of the trapezoid. What is $ x$? [asy]unitsize(4mm); defaultpen(linewidth(.8pt)); int i; real r=5, R=6; path t=r*dir(0)--r*dir(20)--R*dir(20)--R*dir(0); for(i=0; i<9; ++i) { draw(rotate(20*i)*t); } draw((-r,0)--(R+1,0)); draw((-R,0)--(-R-1,0));[/asy]$ \textbf{(A)}\ 100 \qquad \textbf{(B)}\ 102 \qquad \textbf{(C)}\ 104 \qquad \textbf{(D)}\ 106 \qquad \textbf{(E)}\ 108$

A paper rectangle $ABCD $ of area $1$ is folded along a straight line so that $C$ coincides with $A$. Prove that the area of the pentagon thus obtained is less than $3/4$.

The diagram shows a rectangle that has been dissected into nine non-overlapping squares. Given that the width and the height of the rectangle are relatively prime positive integers, find the perimeter of the rectangle. [asy] defaultpen(linewidth(0.7)); draw((0,0)--(69,0)--(69,61)--(0,61)--(0,0));draw((36,0)--(36,36)--(0,36)); draw((36,33)--(69,33));draw((41,33)--(41,61));draw((25,36)--(25,61)); draw((34,36)--(34,45)--(25,45)); draw((36,36)--(36,38)--(34,38)); draw((36,38)--(41,38)); draw((34,45)--(41,45));[/asy]

Given a rectangle $ABCD$, points $M$ -- the midpoint of $[AD]$ side, $N$ -- the midpoint of $[BC]$ side. Let us take a point $P$ on the extension of the $[DC]$ segment over the point $D$. Let us denote the intersection point of lines $(PM)$ and $(AC)$ as $Q$. Prove that the $\angle QNM= \angle MNP$

A circle of radius $ r$ is concentric with and outside a regular hexagon of side length 2. The probability that three entire sides of hexagon are visible from a randomly chosen point on the circle is 1/2. What is $ r$? $ \textbf{(A) } 2\sqrt {2} \plus{} 2\sqrt {3} \qquad \textbf{(B) } 3\sqrt {3} \plus{} \sqrt {2} \qquad \textbf{(C) } 2\sqrt {6} \plus{} \sqrt {3} \qquad \textbf{(D) } 3\sqrt {2} \plus{} \sqrt {6}\\ \textbf{(E) } 6\sqrt {2} \minus{} \sqrt {3}$

Let $ABCD$ be a rectangle of length $m$ and width $n$, with $m, n$ positive integers. Consider a ray of light that starts from $A$, reflects with an angle of $45^o$ on an opposite side and continues reflecting away at the same angle. $\bullet$ For any pair $(m,n)$, show that the ray meets a vertex at some point. $\bullet$ Suppose $m$ and $n$ are coprime. Determine the number of reflections made by the ray of light before encountering a vertex for the first time.

The area of trapezoid $ ABCD$ is $ 164 \text{cm}^2$. The altitude is $ 8 \text{cm}$, $ AB$ is $ 10 \text{cm}$, and $ CD$ is $ 17 \text{cm}$. What is $ BC$, in centimeters? [asy]/* AMC8 2003 #21 Problem */ size(4inch,2inch); draw((0,0)--(31,0)--(16,8)--(6,8)--cycle); draw((11,8)--(11,0), linetype("8 4")); draw((11,1)--(12,1)--(12,0)); label("$A$", (0,0), SW); label("$D$", (31,0), SE); label("$B$", (6,8), NW); label("$C$", (16,8), NE); label("10", (3,5), W); label("8", (11,4), E); label("17", (22.5,5), E);[/asy] $ \textbf{(A)}\ 9\qquad\textbf{(B)}\ 10\qquad\textbf{(C)}\ 12\qquad\textbf{(D)}\ 15\qquad\textbf{(E)}\ 20$

One container of paint is exactly enough to cover the inside of an old rectangle which is three times as long as it is wide. If we make a new rectangle by shortening the old rectangle by $18$ feet and widening it by $8$ feet as shown below, one container of paint is also exactly enough to cover the inside of the new rectangle. Find the length in feet of the perimeter of the new rectangle. [asy] size(250); defaultpen(linewidth(0.8)); draw((-2,0)--(-2,5)--(13,5)--(13,0)--cycle^^(16,-1)--(16,6)--(27,6)--(27,-1)--cycle^^(9,5)--(9,0)^^(16,4)--(27,4)); path rect1=(13,5)--(13,0)--(9,0)--(9,5)--cycle,rect2=(16,6)--(16,4)--(27,4)--(27,6)--cycle; fill(rect1,lightgray); fill(rect2,lightgray); draw(rect1^^rect2); [/asy]

Jackson's paintbrush makes a narrow strip that is $6.5$ mm wide. Jackson has enough paint to make a strip of 25 meters. How much can he paint, in $\text{cm}^2$? $\textbf{(A) }162{,}500\qquad\textbf{(B) }162.5\qquad\textbf{(C) }1{,}625\qquad\textbf{(D) }1{,}625{,}000\qquad\textbf{(E) }16{,}250$

At the vertices $A, B, C, D, E, F, G, H$ of a cube, $2001, 2002, 2003, 2004, 2005, 2008, 2007$ and $2006$ stones respectively are placed. It is allowed to move a stone from a vertex to each of its three neighbours, or to move a stone to a vertex from each of its three neighbours. Which of the following arrangements of stones at $A, B, \ldots , H$ can be obtained? $(\text{a})\quad 2001, 2002, 2003, 2004, 2006, 2007, 2008, 2005;$ $(\text{b})\quad 2002, 2003, 2004, 2001, 2006, 2005, 2008, 2007;$ $(\text{c})\quad 2004, 2002, 2003, 2001, 2005, 2008, 2007, 2006.$

We are given sufficiently many stones of the forms of a rectangle $2\times 1$ and square $1\times 1$. Let $n > 3$ be a natural number. In how many ways can one tile a rectangle $3 \times n$ using these stones, so that no two $2 \times 1$ rectangles have a common point, and each of them has the longer side parallel to the shorter side of the big rectangle?

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]

Do there exist a rectangle that can be partitioned into a regular hexagon with side length $1$, and several right triangles with side lengths $1, \sqrt3 , 2$?

Two tangents $AT$ and $BT$ touch a circle at $A$ and $B$, respectively, and meet perpendicularly at $T$. $Q$ is on $AT$, $S$ is on $BT$, and $R$ is on the circle, so that $QRST$ is a rectangle with $QT = 8$ and $ST = 9$. Determine the radius of the circle.