Inside the rectangle $ABCD$ is taken a point $M$ such that $\angle BMC + \angle AMD = 180^o$. Determine the sum of the angles $BCM$ and $DAM$.

Suppose from $(m+2)\times(n+2)$ rectangle we cut $4$, $1\times1$ corners. Now on first and last row first and last columns we write $2(m+n)$ real numbers. Prove we can fill the interior $m\times n$ rectangle with real numbers that every number is average of it's $4$ neighbors.

We have an 8x8 chessboard with 64 squares. Then we have 3x1 dominoes which cover exactly 3 squares. Such dominoes can only be moved parallel to the borders of the chessboard and also only if the passing squares are free. If no dominoes can be moved, then the position is called stable. a. Find the smalles number of covered squares neccessary for a stable position. b. Prove: There exist a stable position with only one square uncovered. c. Find all Squares which are uncoverd in at least one position of b).

The rectangle is cut into 6 squares, as shown on the figure below. The gray square in the middle has a side equal to 1. What is the area of the rectangle? [img]https://i.ibb.co/gg1tBTN/Kyiv-MO-2023-7-1.png[/img]

Let $ABCD$ be a rectangle and the circle of center $D$ and radius $DA$, which cuts the extension of the side $AD$ at point $P$. Line $PC$ cuts the circle at point $Q$ and the extension of the side $AB$ at point $R$. Show that $QB = BR$.

Compute the maximum area of a rectangle which can be inscribed in a triangle of area $M$.

Consider a $m\times n$ rectangular board consisting of $mn$ unit squares. Two of its unit squares are called [i]adjacent[/i] if they have a common edge, and a [i]path[/i] is a sequence of unit squares in which any two consecutive squares are adjacent. Two parths are called [i]non-intersecting[/i] if they don't share any common squares. Each unit square of the rectangular board can be colored black or white. We speak of a [i]coloring[/i] of the board if all its $mn$ unit squares are colored. Let $N$ be the number of colorings of the board such that there exists at least one black path from the left edge of the board to its right edge. Let $M$ be the number of colorings of the board for which there exist at least two non-intersecting black paths from the left edge of the board to its right edge. Prove that $N^{2}\geq M\cdot 2^{mn}$.

$A$ is a fixed point inside a given circle. Determine the locus of points $C$ such that $ABCD$ is a rectangle with $B$ and $D$ on the circumference of the given circle. (M Panov)

Suppose that a rectangle with sides $ a$ and $ b$ is arbitrarily cut into $ n$ squares with sides $ x_{1},\ldots,x_{n}$. Show that $ \frac{x_{i}}{a}\in\mathbb{Q}$ and $ \frac{x_{i}}{b}\in\mathbb{Q}$ for all $ i\in\{1,\cdots, n\}$.

The set $ G$ is defined by the points $ (x,y)$ with integer coordinates, $ 3\le|x|\le7$, $ 3\le|y|\le7$. How many squares of side at least $ 6$ have their four vertices in $ G$? [asy]defaultpen(black+0.75bp+fontsize(8pt)); size(5cm); path p = scale(.15)*unitcircle; draw((-8,0)--(8.5,0),Arrow(HookHead,1mm)); draw((0,-8)--(0,8.5),Arrow(HookHead,1mm)); int i,j; for (i=-7;i<8;++i) { for (j=-7;j<8;++j) { if (((-7 <= i) && (i <= -3)) || ((3 <= i) && (i<= 7))) { if (((-7 <= j) && (j <= -3)) || ((3 <= j) && (j<= 7))) { fill(shift(i,j)*p,black); }}}} draw((-7,-.2)--(-7,.2),black+0.5bp); draw((-3,-.2)--(-3,.2),black+0.5bp); draw((3,-.2)--(3,.2),black+0.5bp); draw((7,-.2)--(7,.2),black+0.5bp); draw((-.2,-7)--(.2,-7),black+0.5bp); draw((-.2,-3)--(.2,-3),black+0.5bp); draw((-.2,3)--(.2,3),black+0.5bp); draw((-.2,7)--(.2,7),black+0.5bp); label("$-7$",(-7,0),S); label("$-3$",(-3,0),S); label("$3$",(3,0),S); label("$7$",(7,0),S); label("$-7$",(0,-7),W); label("$-3$",(0,-3),W); label("$3$",(0,3),W); label("$7$",(0,7),W);[/asy]$ \textbf{(A)}\ 125\qquad \textbf{(B)}\ 150\qquad \textbf{(C)}\ 175\qquad \textbf{(D)}\ 200\qquad \textbf{(E)}\ 225$

Given a rectangle $ABCD$ with $AB = a$ and $BC = b$. Point $O$ is the intersection of the two diagonals. Extend the side of the $BA$ so that $AE = AO$, also extend the diagonal of $BD$ so that $BZ = BO.$ Assume that triangle $EZC$ is equilateral. Prove that (i) $b = a\sqrt3$ (ii) $EO$ is perpendicular to $ZD$

The points $K$ and $M$ are the centres of the $AB$ and $CD$ sides of the convex quadrangle $ABCD$. The points $L$ and $M$ belong to two other sides and $KLMN$ is a rectangle. Prove that $KLMN$ area is a half of $ABCD$ area.

In rectangle $ ABCD$, we have $ A \equal{} (6, \minus{} 22)$, $ B \equal{} (2006,178)$, and $ D \equal{} (8,y)$, for some integer $ y$. What is the area of rectangle $ ABCD$? $ \textbf{(A) } 4000 \qquad \textbf{(B) } 4040 \qquad \textbf{(C) } 4400 \qquad \textbf{(D) } 40,000 \qquad \textbf{(E) } 40,400$

In the rectangle there is a broken line, the neighboring links of which are perpendicular and equal to the smaller side of the rectangle (see the figure). Find the ratio of the sides of the rectangle. [img]https://2.bp.blogspot.com/-QYj53KiPTJ8/XT_mVIw876I/AAAAAAAAKbE/gJ1roU4Bx-kfGVfJxYMAuLE0Ax0glRbegCK4BGAYYCw/s1600/oral%2Bmoscow%2B2016%2B8.9%2Bp2.png[/img]

Let $f(x)=2x^3-2x$. For what positive values of $a$ do there exist distinct $b,c,d$ such that $(a,f(a)),(b,f(b)),(c,f(c)),(d,f(d))$ is a rectangle?

The letter T is formed by placing two $ 2\times 4$ inch rectangles next to each other, as shown. What is the perimeter of the T, in inches? [asy]size(150); draw((0,6)--(4,6)--(4,4)--(3,4)--(3,0)--(1,0)--(1,4)--(0,4)--cycle, linewidth(1));[/asy] $ \textbf{(A)}\ 12 \qquad \textbf{(B)}\ 16 \qquad \textbf{(C)}\ 20 \qquad \textbf{(D)}\ 22 \qquad \textbf{(E)}\ 24$

An infinite set of rectangles in the Cartesian coordinate plane is given. The vertices of each of these rectangles have coordinates $(0, 0), (p, 0), (p, q), (0, q)$ for some positive integers $p, q$. Show that there must exist two among them one of which is entirely contained in the other.

From a given triangle, cut out the rectangle with the largest area.

In a square of sidelength $7$, $51$ points are given. Show that there's a disk of radius $1$ covering at least $3$ of these points.

Two $4\times 4$ squares intersect at right angles, bisecting their intersecting sides, as shown. The circle's diameter is the segment between the two points of intersection. What is the area of the shaded region created by removing the circle from the squares? [asy] filldraw((0,1)--(1,2)--(3,0)--(1,-2)--(0,-1)--(-1,-2)--(-3,0)--(-1,2)--cycle, gray, black+linewidth(0.8)); filldraw(Circle(origin, 1.01), white, black+linewidth(0.8)); [/asy] $ \textbf{(A)}\ 16-4\pi\qquad\textbf{(B)}\ 16-2\pi\qquad\textbf{(C)}\ 28-4\pi\qquad\textbf{(D)}\ 28-2\pi\qquad\textbf{(E)}\ 32-2\pi $

Consider the shape formed from taking equilateral triangle $ABC$ with side length $6$ and tracing out the arc $BC$ with center $A$. Set the shape down on line $l$ so that segment $AB$ is perpendicular to $l$, and $B$ touches $l$. Beginning from arc $BC$ touching $l$, we roll $ABC$ along $l$ until both points $A$ and $C$ are on the line. The area traced out by the roll can be written in the form $n\pi$, where $n$ is an integer. Find $n$.

Let $n$ be an integer greater than $3$. Show that it is possible to divide a square into $n^2 + 1$ or more disjointed rectangles and with sides parallel to those of the square so that any line parallel to one of the sides intersects at most the interior of $n$ rectangles. Note: We say that two rectangles are [i]disjointed [/i] if they do not intersect or only intersect at their perimeters.

Given an $ a \times b$ rectangle with $ a > b > 0,$ determine the minimum side of a square that covers the rectangle. (A square covers the rectangle if each point in the rectangle lies inside the square.)

An infinite rectangular stripe of width $3$ cm is folded along a line. What is the minimum possible area of the region of overlapping?

Consider the rectangle $ ABCD $ and the points $ M,N,P,Q $ on the segments $ AB,BC,CD, $ respectively, $ DA, $ excluding its extremities. Denote with $ p_{\square} , A_{\square} $ the perimeter, respectively, the area of $ \square. $ Prove that: [b]a)[/b] $ p_{MNPQ}\ge AC+BD. $ [b]b)[/b] $ p_{MNPQ} =AC+BD\implies A_{MNPQ}\le \frac{A_{ABCD}}{2} . $ [b]c)[/b] $ p_{MNPQ} =AC+BD\implies MP^2 +NQ^2\ge AC^2. $ [i]Dan Brânzei[/i] and [i]Gheorghe Iurea[/i]