In the triangle $ABC$, the medians $AA_1$, $BB_1$, and $CC_1$ are concurrent at a point $T$ such that $BA_1=TA_1$. The points $C_2$ and $B_2$ are chosen on the extensions of $CC_1$ and $BB_2$, respectively, such that $$C_1C_2 = \frac{CC_1}{3} \quad \text{and} \quad B_1B_2 = \frac{BB_1}{3}.$$ Show that $TB_2AC_2$ is a rectangle.

Let be $ABCD$ a rectangle with $|AB|=a\ge b=|BC|$. Find points $P,Q$ on the line $BD$ such that $|AP|=|PQ|=|QC|$. Discuss the solvability with respect to the lengths $a,b$.

Three congruent equilateral triangles $T_1$, $T_2$, and $T_3$ are stacked from left to right inside rectangle $JHMT$ such that the bottom left vertex of $T_1$ is $T$, the bottom side of $T_1$ lies on $\overline{MT}$, the bottom left vertex of $T_2$ is the midpoint of a side of $T_1$, the bottom left vertex of $T_3$ is the midpoint of a side of $T_2$, and the other two vertices of $T_3$ lie on $\overline{JH}$ and $\overline{HM}$, as shown below. Given that rectangle $JHMT$ has area $2022$, find the area of any one of the triangles $T_1$, $T_2$, or $T_3$. [asy] unitsize(0.111111111111111111cm); real s = sqrt(4044/sqrt(75)); real l = 5s/2; real w = s * sqrt(3); pair J,H,M,T,V1,V2,V3,V4,V5,V6,V7,V8,C1,C2,C3; J = (0,w); H = (l,w); M = (l,0); T = (0,0); V1 = (s,0); V2 = (s/2,s * sqrt(3)/2); V3 = (V1+V2)/2; V4 = (3 * s/4+s,s * sqrt(3)/4); V5 = (3 * s/4+s/2,s * sqrt(3)/4+s * sqrt(3)/2); V6 = (V4+V5)/2; V7 = (l,s * sqrt(3)/4+s * sqrt(3)/4); V8 = (l-s/2,w); C1 = (T+V1+V2)/3; C2 = (V3+V4+V5)/3; C3 = (V6+V7+V8)/3; draw(J--H--M--T--cycle); draw(V1--V2--T); draw(V3--V4--V5--cycle); draw(V6--V7--V8--cycle); label("$J$", J, NW); label("$H$", H, NE); label("$M$", M, SE); label("$T$", T, SW); label("$T_1$", C1); label("$T_2$", C2); label("$T_3$", C3); [/asy]

$\text{ABCD}$ is a rectangle, $\text{D}$ is the center of the circle, and $\text{B}$ is on the circle. If $\text{AD}=4$ and $\text{CD}=3$, then the area of the shaded region is between [asy] pair A,B,C,D; A=(0,4); B=(3,4); C=(3,0); D=origin; draw(circle(D,5)); fill((0,5)..(1.5,4.7697)..B--A--cycle,black); fill(B..(4,3)..(5,0)--C--cycle,black); draw((0,5)--D--(5,0)); label("A",A,NW); label("B",B,NE); label("C",C,S); label("D",D,SW); [/asy] $\text{(A)}\ 4\text{ and }5 \qquad \text{(B)}\ 5\text{ and }6 \qquad \text{(C)}\ 6\text{ and }7 \qquad \text{(D)}\ 7\text{ and }8 \qquad \text{(E)}\ 8\text{ and }9$

Determine the maximum possible number of points you can place in a rectangle with lengths $14$ and $28$ such that any two of those points are more than $10$ apart from each other.

Let $ABCD$ be a rectangle with $AB = 30$ and $BC = 28$. Point $P$ and $Q$ lie on $\overline{BC}$ and $\overline{CD}$ respectively so that all sides of $\triangle{ABP}, \triangle{PCQ},$ and $\triangle{QDA}$ have integer lengths. What is the perimeter of $\triangle{APQ}$? (A) 84 (B) 86 (C) 88 (D)90 (E)92

Given a rectangle $\mathcal{R}$ of area $100000 $, Pancho must completely cover the rectangle $\mathcal{R}$ with a finite number of rectangles with sides parallel to the sides of $\mathcal{R}$ . Next, Martín colors some rectangles of Pancho's cover red so that no two red rectangles have interior points in common. If the red area is greater than $0.00001$, Martin wins. Otherwise, Pancho wins. Prove that Pancho can cover to ensure victory,

A rectangle $ABCD$ has side lengths $AB = m$, $AD = n$, with $m$ and $n$ relatively prime and both odd. It is divided into unit squares and the diagonal AC intersects the sides of the unit squares at the points $A_1 = A, A_2, A_3, \ldots , A_k = C$. Show that \[ A_1A_2 - A_2A_3 + A_3A_4 - \cdots + A_{k-1}A_k = {\sqrt{m^2+n^2}\over mn}. \]

A square of side $1$ is divided into rectangles . We choose one of the two smaller sides of each rectangle (if the rectangle is a square, then we choose any of the four sides) . Prove that the sum of the lengths of all the chosen sides is at least $1$ . (Folklore)

Seven squares are arranged to form a rectangle as shown below. The side length of the smallest square is $3$ cm. What is the perimeter in centimeters of the rectangle formed by the $7$ squares? [asy] draw((0,0)--(57,0)--(57,63)--(0,63)--cycle); draw((12,27)--(12,39)); draw((24,27)--(24,63)); draw((27,0)--(27,30)); draw((0,27)--(27,27)); draw((24,30)--(57,30)); draw((0,39)--(24,39)); [/asy]

In quadrilateral $ ABCD$, it is given that $ \angle A \equal{} 120^\circ$, angles $ B$ and $ D$ are right angles, $ AB \equal{} 13$, and $ AD \equal{} 46$. Then $ AC \equal{}$ $ \textbf{(A)}\ 60 \qquad \textbf{(B)}\ 62 \qquad \textbf{(C)}\ 64 \qquad \textbf{(D)}\ 65 \qquad \textbf{(E)}\ 72$

Let $AXYZB$ be a convex pentagon inscribed in a semicircle of diameter $AB$. Denote by $P$, $Q$, $R$, $S$ the feet of the perpendiculars from $Y$ onto lines $AX$, $BX$, $AZ$, $BZ$, respectively. Prove that the acute angle formed by lines $PQ$ and $RS$ is half the size of $\angle XOZ$, where $O$ is the midpoint of segment $AB$.

Inside an $11\times 11$ square, Pablo drew a rectangle and extending its sides divided the square into $5$ rectangles, as shown in the figure. [img]https://cdn.artofproblemsolving.com/attachments/5/a/7774da7085f283b3aae74fb5ff472572571827.gif[/img] Sofía did the same, but she also managed to make the lengths of the sides of the $5$ rectangles be whole numbers between $1$ and $10$, all different. Show a figure like the one Sofia made.

The perimeter of a rectangle is $100$ and its diagonal has length $x$. What is the area of this rectangle? $\textbf{(A) }625-x^2\qquad\textbf{(B) }625-\dfrac{x^2}2\qquad\textbf{(C) }1250-x^2\qquad\textbf{(D) }1250-\dfrac{x^2}2\qquad\textbf{(E) }2500-\dfrac{x^2}2$

A $ 6\times 6$ square is dissected in to 9 rectangles by lines parallel to its sides such that all these rectangles have integer sides. Prove that there are always [b]two[/b] congruent rectangles.

At a store, when a length is reported as $x$ inches that means the length is at least $x-0.5$ inches and at most $x+0.5$ inches. Suppose the dimensions of a rectangular tile are reported as $2$ inches by $3$ inches. In square inches, what is the minimum area for the rectangle? $ \textbf{(A)}\ 3.75 \qquad \textbf{(B)}\ 4.5 \qquad \textbf{(C)}\ 5 \qquad \textbf{(D)}\ 6 \qquad \textbf{(E)}\ 8.75 $

How many rectangles are there in the diagram below such that the sum of the numbers within the rectangle is a multiple of 7? [asy] int n; n=0; for (int i=0; i<=7;++i) { draw((i,0)--(i,7)); draw((0,i)--(7,i)); for (int a=0; a<=7;++a) { if ((a != 7)&&(i != 7)) { n=n+1; label((string) n,(a,i),(2,2)); } } } [/asy]

The bisectors of trapezoid's angles form a quadrilateral with perpendicular diagonals. Prove that this trapezoid is isosceles.

A rectangle is decomposed by $6$ vertical lines and $6$ horizontal lines in the $49$ small rectangles (see figure). The perimeter of each small rectangle is known to be a whole number of meters. In this case, will the perimeter of the large rectangle be a whole number of meters? [asy] unitsize(0.8 cm); draw((0,0)--(8.6,0)--(8.6,4.2)--(0,4.2)--cycle, linewidth(1.5*bp)); draw((0.7,0)--(0.7,4.2)); draw((1.2,0)--(1.2,4.2)); draw((4.1,0)--(4.1,4.2)); draw((5.6,0)--(5.6,4.2)); draw((6.0,0)--(6.0,4.2)); draw((7.4,0)--(7.4,4.2)); draw((0,0.4)--(8.6,0.4)); draw((0,1.0)--(8.6,1.0)); draw((0,1.5)--(8.6,1.5)); draw((0,2.5)--(8.6,2.5)); draw((0,3.1)--(8.6,3.1)); draw((0,3.5)--(8.6,3.5)); [/asy]

The area of the rectangular region is [asy] draw((0,0)--(4,0)--(4,2.2)--(0,2.2)--cycle,linewidth(.5 mm)); label(".22 m",(4,1.1),E); label(".4 m",(2,0),S); [/asy] $\text{(A)}\ \text{.088 m}^2 \qquad \text{(B)}\ \text{.62 m}^2 \qquad \text{(C)}\ \text{.88 m}^2 \qquad \text{(D)}\ \text{1.24 m}^2 \qquad \text{(E)}\ \text{4.22 m}^2$

The diagram below shows the regular hexagon $BCEGHJ$ surrounded by the rectangle $ADFI$. Let $\theta$ be the measure of the acute angle between the side $\overline{EG}$ of the hexagon and the diagonal of the rectangle $\overline{AF}$. There are relatively prime positive integers $m$ and $n$ so that $\sin^2\theta  = \tfrac{m}{n}$. Find $m + n$. [asy] import graph; size(3.2cm); real labelscalefactor = 0.5; pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); draw((-1,3)--(-1,2)--(-0.13,1.5)--(0.73,2)--(0.73,3)--(-0.13,3.5)--cycle); draw((-1,3)--(-1,2)); draw((-1,2)--(-0.13,1.5)); draw((-0.13,1.5)--(0.73,2)); draw((0.73,2)--(0.73,3)); draw((0.73,3)--(-0.13,3.5)); draw((-0.13,3.5)--(-1,3)); draw((-1,3.5)--(0.73,3.5)); draw((0.73,3.5)--(0.73,1.5)); draw((-1,1.5)--(0.73,1.5)); draw((-1,3.5)--(-1,1.5)); label("$ A $",(-1.4,3.9),SE*labelscalefactor); label("$ B $",(-1.4,3.28),SE*labelscalefactor); label("$ C $",(-1.4,2.29),SE*labelscalefactor); label("$ D $",(-1.4,1.45),SE*labelscalefactor); label("$ E $",(-0.3,1.4),SE*labelscalefactor); label("$ F $",(0.8,1.45),SE*labelscalefactor); label("$ G $",(0.8,2.24),SE*labelscalefactor); label("$ H $",(0.8,3.26),SE*labelscalefactor); label("$ I $",(0.8,3.9),SE*labelscalefactor); label("$ J $",(-0.25,3.9),SE*labelscalefactor); [/asy]

Let $AXYZB$ be a convex pentagon inscribed in a semicircle of diameter $AB$. Denote by $P$, $Q$, $R$, $S$ the feet of the perpendiculars from $Y$ onto lines $AX$, $BX$, $AZ$, $BZ$, respectively. Prove that the acute angle formed by lines $PQ$ and $RS$ is half the size of $\angle XOZ$, where $O$ is the midpoint of segment $AB$.

There is a $11\times 11$ square grid. We divided this in $5$ rectangles along unit squares. How many ways that one of the rectangles doesn’t have a edge on basic circumference. Note that we count as different ways that one way coincides with another way by rotating or reversing.

A ball was launched on a rectangular billiard table at an angle of $45^o$ to one of the sides. Reflected from all sides (the angle of incidence is equal to the angle of reflection), he returned to his original position . It is known that one of the sides of the table has a length of one meter. Find the length of the second side. [img]https://cdn.artofproblemsolving.com/attachments/3/d/e0310ea910c7e3272396cd034421d1f3e88228.png[/img]

On the cartesian plane are drawn several rectangles with the sides parallel to the coordinate axes. Assume that any two rectangles can be cut by a vertical or a horizontal line. Show that it's possible to draw one horizontal and one vertical line such that each rectangle is cut by at least one of these two lines.