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 $\angle MCD$.

The polygon ABCDEFG (shown on the right) is a regular octagon. Prove that the area of the rectangle $ADEH$ is one half the area of the whole polygon $ABCDEFGH$. [asy] draw((0,1.414)--(1.414,0)--(3.414,0)--(4.828,1.414)--(4.828,3.414)--(3.414,4.828)--(1.414,4.828)--(0,3.414)--(0,1.414)); fill((0,1.414)--(0,3.414)--(4.828,3.414)--(4.828,1.414)--cycle, mediumgrey); label("$B$",(1.414,0),SW); label("$C$",(3.414,0),SE); label("$D$",(4.828,1.414),SE); label("$E$",(4.828,3.414),NE); label("$F$",(3.414,4.828),NE); label("$G$",(1.414,4.828),NW); label("$H$",(0,3.414),NW); label("$A$",(0,1.414),SW); [/asy]

Let $A$ and $B$ be points on a circle $\mathcal{C}$ with center $O$ such that $\angle AOB = \dfrac {\pi}2$. Circles $\mathcal{C}_1$ and $\mathcal{C}_2$ are internally tangent to $\mathcal{C}$ at $A$ and $B$ respectively and are also externally tangent to one another. The circle $\mathcal{C}_3$ lies in the interior of $\angle AOB$ and it is tangent externally to $\mathcal{C}_1$, $\mathcal{C}_2$ at $P$ and $R$ and internally tangent to $\mathcal{C}$ at $S$. Evaluate the value of $\angle PSR$.

A rectangle of dimensions $2024 \times 2023$ is filled with pieces of the following types: [asy] size(200); // Figure (A) draw((0,0)--(4,0)--(4,1)--(0,1)--cycle); draw((1,0)--(1,1)); draw((2,0)--(2,1)); draw((3,0)--(3,1)); // Figure (B) draw((6,0)--(8,0)--(8,2)--(6,2)--cycle); draw((7,0)--(7,2)); draw((6,1)--(8,1)); // Figure (C) draw((10,0)--(12,0)--(12,1)--(11,1)--(11,2)--(9,2)--(9,1)--(10,1)--cycle); draw((10,0)--(10,1)); draw((11,0)--(11,1)); draw((10,1)--(11,1)); draw((9,1)--(9,2)); draw((10,1)--(10,2)); draw((11,0)--(12,0)); draw((10,1)--(12,1)); // Labeling label("(A)", (2, -0.5)); label("(B)", (7, -0.5)); label("(C)", (10.5, -0.5)); [/asy] Each piece can be turned arround, and each square has side length $1$. Is it possible to use exactly 2023 pieces of type $(A)$?

Holding a rectangular sheet of paper $ABCD$, Prair folds triangle $ABD$ over diagonal $BD$, so that the new location of point $A$ is $A'$. She notices that $A'C =\frac13 BD$. If the area of $ABCD$ is $27\sqrt2$, find $BD$.

A rectangular field has dimensions $120$ meters and $192$ meters. You want to divide it into equal square plots. The measure of the sides of these squares must be an integer number . In addition, you want to place a post in each corner of plot. Determine the smallest number of plots in which you can divide the land and the number of posts needed. [hide=Original wording]Un terreno de forma rectangular de 120 metros por 192 metros se quiere dividir en parcelas cuadradas iguales sin que sobre terreno. La medida de los lados de estos cuadrados debe ser un nu´mero entero. Adem´as se desea colocar un poste en cada esquina de parcela. Determinar el menor nu´mero de parcelas en que se puede dividir el terreno y el nu´mero de postes que se necesitan.[/hide]

A polygon can be dissected into $100$ rectangles, but it cannot be dissected into $99$ rectangles. Prove that this polygon cannot be dissected into $100$ triangles.

There are $1998$ rectangular pieces $2$ cm wide and $3$ cm long and with them squares are assembled (without overlapping or gaps). What is the greatest number of different squares that can be had at the same time?

We are given $n$ distinct rectangles in the plane. Prove that between the $4n$ interior angles formed by these rectangles at least $4\sqrt n$ are distinct.

Three points inside a rectangle determine a triangle. A fourth point is taken inside the triangle. Prove that at least one of the three concave quadrilaterals formed by these four points has perimeter lesser than that of the rectangle. [i](Dan Schwarz)[/i]

Circles $A$, $B$, and $C$ each have radius $1$. Circles $A$ and $B$ share one point of tangency. Circle $C$ has a point of tangency with the midpoint of $\overline{AB}$. What is the area inside circle $C$ but outside circle $A$ and circle $B$? [asy] size(170); defaultpen(linewidth(.8pt)+fontsize(8pt)); dotfactor=3; filldraw(arc((1,0),1,90,180)--arc((-1,0),1,0,90)--arc((0,1), 1, 180, 0)--cycle,gray); draw(circle((0,1),1)); draw(circle((1,0),1)); draw(circle((-1,0),1)); dot((-1,0)); dot((1,0)); dot((0,1)); label("$A$",(-1,0),SW); label("$B$",(1,0),SE); label("$C$",(0,1),N);[/asy] $ \textbf{(A)}\ 3-\frac{\pi}{2} \qquad \textbf{(B)}\ \frac{\pi}{2} \qquad \textbf{(C)}\ 2 \qquad \textbf{(D)}\ \frac{3\pi}{4} \qquad \textbf{(E)}\ 1+\frac{\pi}{2}$

Let $ABC$ be a right triangle with a right angle at $C.$ Two lines, one parallel to $AC$ and the other parallel to $BC,$ intersect on the hypotenuse $AB.$ The lines split the triangle into two triangles and a rectangle. The two triangles have areas $512$ and $32.$ What is the area of the rectangle? [i]Author: Ray Li[/i]

Compute the number of ways to divide a $20 \times 24 $ rectangle into $4 \times 5$ rectangles. (Rotations and reflections are considered distinct.)

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.

Let $ABCD$ be a rectangle of centre $O$, such that $\angle DAC=60^{\circ}$. 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.

Prove that the parallelogram $ABCD$ with relation $\angle ABD + \angle DAC = 90^o$, is either a rectangle or a rhombus.

Consider the set of all rectangles with a given area $S$. Find the largest value o $ M = \frac{S}{2S+p + 2}$ where $p$ is the perimeter of the rectangle.

A rectangle $3 \times 5$ is divided into $15$ $1 \times 1$ cells. The middle $3$ cells that have no common points with the border of the rectangle are deleted. Is it possible to put in the remaining $12$ cells numbers $1, 2, \ldots, 12$ in some order, so that the sums of the numbers in the cells along each of the four sides of the rectangle are equal? [i]Proposed by Mariia Rozhkova[/i]

Let $C_1, C_2$ be circles of radius $1/2$ tangent to each other and both tangent internally to a circle $C$ of radius $1$. The circles $C_1$ and $C_2$ are the first two terms of an infinite sequence of distinct circles $C_n$ defined as follows: $C_{n+2}$ is tangent externally to $C_n$ and $C_{n+1}$ and internally to $C$. Show that the radius of each $C_n$ is the reciprocal of an integer.

Let $P$ be the intersection of lines $l_1$ and $l_2$. Let $S_1$ and $S_2$ be two circles externally tangent at $P$ and both tangent to $l_1$, and let $T_1$ and $T_2$ be two circles externally tangent at $P$ and both tangent to $l_2$. Let $A$ be the second intersection of $S_1$ and $T_1, B$ that of $S_1$ and $T_2, C$ that of $S_2$ and $T_1$, and $D$ that of $S_2$ and $T_2$. Show that the points $A,B,C,D$ are concyclic if and only if $l_1$ and $l_2$ are perpendicular.

Let $ABC$ be a triangle such that $CA \neq CB$, the points $A_{1}$ and $B_{1}$ are tangency points for the ex-circles relative to sides $CB$ and $CA$, respectively, and $I$ the incircle. The line $CI$ intersects the cincumcircle of the triangle $ABC$ in the point $P$. The line that trough $P$ that is perpendicular to $CP$, intersects the line $AB$ in $Q$. Prove that the lines $QI$ and $A_{1}B_{1}$ are parallels.

The projection of the vertex $C$ of the rectangle $ABCD$ on the diagonal $BD$ is $E$. The projections of $E$ on $AB$ and $AD$ are $F$ and $G$ respectively. Prove that $$AF^{2/3} + AG^{2/3} = AC^{2/3}$$ .

Let $ABCD$ be a cyclic quadrilateral whose diagonals $AC$ and $BD$ are perpendicular. Let $O$ be the circumcenter of $ABCD$, $K$ the intersection of the diagonals, $ L\neq O $ the intersection of the circles circumscribed to $OAC$ and $OBD$, and $G$ the intersection of the diagonals of the quadrilateral whose vertices are the midpoints of the sides of $ABCD$. Prove that $O, K, L$ and $G$ are collinear

We have an $n \times n$ board, and in every square there is an integer. The sum of all integers on the board is $0$. We define an action on a square where the integer in the square is decreased by the number of neighbouring squares, and the number inside each of the neighbouring squares is increased by 1. Determine if there exists $n\geq 2$ such that we can turn all the integers into zeros in a finite number of actions.

Can we find $N$ such that all $m\times n$ rectangles with $m,n>N$ can be tiled with $4\times6$ and $5\times7$ rectangles?