Rectangle $ ABCD$ has $ AB\equal{}8$ and $ BC\equal{}6$. Point $ M$ is the midpoint of diagonal $ \overline{AC}$, and E is on $ \overline{AB}$ with $ \overline{ME}\perp\overline{AC}$. What is the area of $ \triangle AME$? $ \textbf{(A)}\ \frac{65}{8} \qquad \textbf{(B)}\ \frac{25}{3} \qquad \textbf{(C)}\ 9 \qquad \textbf{(D)}\ \frac{75}{8} \qquad \textbf{(E)}\ \frac{85}{8}$

Find the area of inscribed convex octagon, if the length of four sides is $2$, and length of other four sides is $ 6\sqrt {2}$. $\textbf{(A)}\ 120 \qquad\textbf{(B)}\ 24 \plus{} 68\sqrt {2} \qquad\textbf{(C)}\ 88\sqrt {2} \qquad\textbf{(D)}\ 124 \qquad\textbf{(E)}\ 72\sqrt {3}$

A square is dissected into rectangles with sides parallel to the sides of the square. For each of these rectangles, the ratio of its shorter side to its longer side is considered. Show that the sum of all these ratios is at least $1$.

A square board is divided by lines parallel to the board sides ($7$ lines in each direction, not necessarily equidistant ) into $64$ rectangles. Rectangles are colored into white and black in alternating order. Assume that for any pair of white and black rectangles the ratio between area of white rectangle and area of black rectangle does not exceed $2.$ Determine the maximal ratio between area of white and black part of the board. White (black) part of the board is the total sum of area of all white (black) rectangles.

A cylinder radius $12$ and a cylinder radius $36$ are held tangent to each other with a tight band. The length of the band is $m\sqrt{k}+n\pi$ where $m$, $k$, and $n$ are positive integers, and $k$ is not divisible by the square of any prime. Find $m + k + n$. [asy] size(150); real t=0.3; void cyl(pair x, real r, real h) { pair xx=(x.x,t*x.y); path B=ellipse(xx,r,t*r), T=ellipse((x.x,t*x.y+h),r,t*r), S=xx+(r,0)--xx+(r,h)--(xx+(-r,h))--xx-(r,0); unfill(S--cycle); draw(S); unfill(B); draw(B); unfill(T); draw(T); } real h=8, R=3,r=1.2; pair X=(0,0), Y=(R+r)*dir(-50); cyl(X,R,h); draw(shift((0,5))*yscale(t)*arc(X,R,180,360)); cyl(Y,r,h); void str (pair x, pair y, real R, real r, real h, real w) { real u=(angle(y-x)+asin((R-r)/(R+r)))*180/pi+270; path P=yscale(t)*(arc(x,R,180,u)--arc(y,r,u,360)); path Q=shift((0,h))*P--shift((0,h+w))*reverse(P)--cycle; fill(Q,grey);draw(Q); } str(X,Y,R,r,3.5,1.5);[/asy]

A rectangle is divided by $10$ horizontal and $10$ vertical lines into $121$ rectangular cells. If $111$ of them have integer perimeters, prove that they all have integer perimeters.

An acute-angled triangle $AKL$ is given on a plane. Consider all rectangles $ABCD$ circumscribed to triangle $AKL$ such that point $K$ lies on side $BC$ and point $L$ lieson side $CD$. Find the locus of the intersection $S$ of the diagonals $AC$ and $BD$.

$ ABCD$ is a rectangle of area 2. $ P$ is a point on side $ CD$ and $ Q$ is the point where the incircle of $ \triangle PAB$ touches the side $ AB$. The product $ PA \cdot PB$ varies as $ ABCD$ and $ P$ vary. When $ PA \cdot PB$ attains its minimum value, a) Prove that $ AB \geq 2BC$, b) Find the value of $ AQ \cdot BQ$.

Let R be a rectangle. How many circles in the plane of R have a diameter both of whose endpoints are vertices of R? $ \textbf{(A)}\ 1\qquad \textbf{(B)}\ 2\qquad \textbf{(C)}\ 4\qquad \textbf{(D)}\ 5\qquad \textbf{(E)}\ 6$

The seats in the Parliament of some country are arranged in a rectangle of $10$ rows of $10$ seats each. All the $100$ $MP$s have different salaries. Each of them asks all his neighbours (sitting next to, in front of, or behind him, i.e. $4$ members at most) how much they earn. They feel a lot of envy towards each other: an $MP$ is content with his salary only if he has at most one neighbour who earns more than himself. What is the maximum possible number of $MP$s who are satisfied with their salaries?

A circular piece of metal of maximum size is cut out of a square piece and then a square piece of maximum size is cut out of the circular piece. The total amount of metal wasted is: $ \textbf{(A)}\ \frac{1}{4} \text{ the area of the original square}\\ \textbf{(B)}\ \frac{1}{2} \text{ the area of the original square}\\ \textbf{(C)}\ \frac{1}{2} \text{ the area of the circular piece}\\ \textbf{(D)}\ \frac{1}{4} \text{ the area of the circular piece}\\ \textbf{(E)}\ \text{none of these}$

Let $ABC$ be a triangle. Construct rectangles $BA_1A_2C$, $CB_1B_2A$, and $AC_1C_2B$ outside $ABC$ such that $\angle BCA_1=\angle CAB_1=\angle ABC_1$. Let $A_1B_2$ and $A_2C_1$ intersect at $A'$ and define $B',C'$ similarly. Prove that line $AA'$ bisects $B'C'$. [i]Linus Tang[/i]

A $100\times \sqrt{3}$ rectangular table is given. What is the minimum number of disk-shaped napkins of radius $1$ required to cover the table completely? [i]Remark:[/i] The napkins are allowed to overlap and protrude the table's edges.

Inside a rectangular piece of paper $n$ rectangular holes with sides parallel to the sides of the paper have been cut out. Into what minimal number of rectangular pieces (without holes) is it always possible to cut this piece of paper? (A Shapovalov)

Determine all pairs $(m,n)$ of positive integers such that a $m\times n$ rectangle can be tiled with L-trominoes.

In rectangle $ABCD$, points $E$ and $F$ lie on sides $AB$ and $CD$ respectively such that both $AF$ and $CE$ are perpendicular to diagonal $BD$. Given that $BF$ and $DE$ separate $ABCD$ into three polygons with equal area, and that $EF = 1$, find the length of $BD$.

Treasure was buried in a single cell of an $M\times N$ ($2\le M$, $N$) grid. Detectors were brought to find the cell with the treasure. For each detector, you can set it up to scan a specific subgrid $[a,b]\times[c,d]$ with $1\le a\le b\le M$ and $1\le c\le d\le N$. Running the detector will tell you whether the treasure is in the region or not, though it cannot say where in the region the treasure was detected. You plan on setting up $Q$ detectors, which may only be run simultaneously after all $Q$ detectors are ready. In terms of $M$ and $N$, what is the minimum $Q$ required to gaurantee to determine the location of the treasure?

Sebastian has a certain number of rectangles with areas that sum up to 3 and with side lengths all less than or equal to $1$. Demonstrate that with each of these rectangles it is possible to cover a square with side $1$ in such a way that the sides of the rectangles are parallel to the sides of the square. [b]Note:[/b] The rectangles can overlap and they can protrude over the sides of the square.

Helen has a wooden rectangle of unknown dimensions, a straightedge, and a pencil (no compass). Is it possible for her to construct a line segment on the rectangle connecting the midpoints of two opposite sides, where she cannot draw any lines or points outside the rectangle? Note: Helen is allowed to draw lines between two points she has already marked, and mark the intersection of any two lines she has already drawn, if the intersection lies on the rectangle. Further, Helen is allowed to mark arbitrary points either on the rectangle or on a segment she has previously drawn. Assume that only the four vertices of the rectangle have been marked prior to the beginning of this process.

Let $f:[0,1]\to\mathbb R$ be a function satisfying $$|f(x)-f(y)|\le|x-y|$$for every $x,y\in[0,1]$. Show that for every $\varepsilon>0$ there exists a countable family of rectangles $(R_i)$ of dimensions $a_i\times b_i$, $a_i\le b_i$ in the plane such that $$\{(x,f(x)):x\in[0,1]\}\subset\bigcup_iR_i\text{ and }\sum_ia_i<\varepsilon.$$(The edges of the rectangles are not necessarily parallel to the coordinate axes.)

A point $M (x, y)$ of the Cartesian plane of $xOy$ coordinates is called [i]lattice [/i] if it has integer coordinates. Each lattice point is colored red or blue. Prove that in the plan there is at least one rectangle with lattice vertices of the same color.

Given the rectangle $ABCD$. The points $E ,F$ lie on the segments $(BC) , (DC)$ respectively, such that $\angle DAF = \angle FAE$. Proce that if $DF + BE = AE$ then $ABCD$ is square.

Matías has a rectangular sheet of paper $ABCD$, with $AB<AD$.Initially, he folds the sheet along a straight line $AE$, where $E$ is a point on the side $DC$ , so that vertex $D$ is located on side $BC$, as shown in the figure. Then folds the sheet again along a straight line $AF$, where $F$ is a point on side $BC$, so that vertex $B$ lies on the line $AE$; and finally folds the sheet along the line $EF$. Matías observed that the vertices $B$ and $C$ were located on the same point of segment $AE$ after making the folds. Calculate the measure of the angle $\angle DAE$. [img]https://cdn.artofproblemsolving.com/attachments/0/9/b9ab717e1806c6503a9310ee923f20109da31a.png[/img]

Let $ R$ be a rectangle that is the union of a finite number of rectangles $ R_i,$ $ 1 \leq i \leq n,$ satisfying the following conditions: [b](i)[/b] The sides of every rectangle $ R_i$ are parallel to the sides of $ R.$ [b](ii)[/b] The interiors of any two different rectangles $ R_i$ are disjoint. [b](iii)[/b] Each rectangle $ R_i$ has at least one side of integral length. Prove that $ R$ has at least one side of integral length. [i]Variant:[/i] Same problem but with rectangular parallelepipeds having at least one integral side.

Let $A_1,B_1,C_1,D_1$ be the midpoints of the sides of a convex quadrilateral $ABCD$ and let $A_2, B_2, C_2, D_2$ be the midpoints of the sides of the quadrilateral $A_1B_1C_1D_1$. If $A_2B_2C_2D_2$ is a rectangle with sides $4$ and $6$, then what is the product of the lengths of the diagonals of $ABCD$ ?