Found problems: 1342
1980 All Soviet Union Mathematical Olympiad, 295
Some squares of the infinite sheet of cross-lined paper are red. Each $2\times 3$ rectangle (of $6$ squares) contains exactly two red squares. How many red squares can be in the $9\times 11$ rectangle of $99$ squares?
I Soros Olympiad 1994-95 (Rus + Ukr), 11.2
Given a rectangle $ABCD$ with $AB> BC$. On the side $CD$, take a point $L$ such that $BL$ and $AC$ are perpendicular. Let $K$ be the intersection point of segments $BL$ and $AC$. It is known that segments $AL$. and $DK$ are perpendicular. Find $\angle ACB.$
2013 Costa Rica - Final Round, G3
Let $ABCD$ be a rectangle with center $O$ such that $\angle DAC = 60^o$. Bisector of $\angle DAC$ cuts a $DC$ at $S$, $OS$ and $AD$ intersect at $L$, $BL$ and $AC$ intersect at $M$. Prove that $SM \parallel CL$.
2007 Stanford Mathematics Tournament, 15
A number $ x$ is uniformly chosen on the interval $ [0,1]$, and $ y$ is uniformly randomly chosen on $ [\minus{}1,1]$. Find the probability that $ x>y$.
Novosibirsk Oral Geo Oly IX, 2021.1
Cut the $19 \times 20$ grid rectangle along the grid lines into several squares so that there are exactly four of them with odd sidelengths.
2010 AMC 8, 6
Which of the following has the greatest number of line of symmetry?
$ \textbf{(A)}\ \text{ Equilateral Triangle}$
$\textbf{(B)}\ \text{Non-square rhombus} $
$\textbf{(C)}\ \text{Non-square rectangle}$
$\textbf{(D)}\ \text{Isosceles Triangle}$
$\textbf{(E)}\ \text{Square} $
2018 Harvard-MIT Mathematics Tournament, 1
In an $n \times n$ square array of $1\times1$ cells, at least one cell is colored pink. Show that you can always divide the square into rectangles along cell borders such that each rectangle contains exactly one pink cell.
2009 Purple Comet Problems, 1
The pentagon below has three right angles. Find its area.
[asy]
size(150);
defaultpen(linewidth(1));
draw((4,10)--(0,10)--origin--(10,0)--(10,2)--cycle);
label("4",(2,10),N);
label("10",(0,5),W);
label("10",(5,0),S);
label("2",(10,1),E);
label("10",(7,6),NE);
[/asy]
1988 Greece National Olympiad, 3
Bisectors of $\angle BAC$, $\angle CAD$ in a rectangle $ABCD$ , intersect the sides $BC$, $CD$ at points $M$ and $N$ resp. Prove that $\frac{(MB)}{(MC)}+\frac{(ND)}{(NC)}>1$
Kvant 2022, M2718
$m\times n$ grid is tiled by mosaics $2\times2$ and $1\times3$ (horizontal and vertical). Prove that the number of ways to choose a $1\times2$ rectangle (horizontal and vertical) such that one of its cells is tiled by $2\times2$ mosaic and the other cell is tiled by $1\times3$ mosaic [horizontal and vertical] is an even number.
Kyiv City MO Juniors Round2 2010+ geometry, 2017.7.4
On the sides $AD$ and $BC$ of a rectangle $ABCD$ select points $M, N$ and $P, Q$ respectively such that $AM = MN = ND = BP = PQ = QC$. On segment $QC$ selected point $X$, different from the ends of the segment. Prove that the perimeter of $\vartriangle ANX$ is more than the perimeter of $\vartriangle MDX$.
1971 AMC 12/AHSME, 9
An uncrossed belt is fitted without slack around two circular pulleys with radii of $14$ inches and $4$ inches. If the distance between the points of contact of the belt with the pulleys is $24$ inches, then the distance between the centers of the pulleys in inches is
$\textbf{(A) }24\qquad\textbf{(B) }2\sqrt{119}\qquad\textbf{(C) }25\qquad\textbf{(D) }26\qquad \textbf{(E) }4\sqrt{35}$
2005 MOP Homework, 3
Squares of an $n \times n$ table ($n \ge 3$) are painted black and white as in a chessboard. A move allows one to choose any $2 \times 2$ square and change all of its squares to the opposite color. Find all such n that there is a finite number of the moves described after which all squares are the same color.
2003 Tournament Of Towns, 5
Prove that one can cut $a \times b$ rectangle, $\frac{b}{2} < a < b$, into three pieces and rearrange them into a square (without overlaps and holes).
2000 AMC 8, 25
The area of rectangle $ABCD$ is $72$. If point $A$ and the midpoints of $\overline{BC}$ and $\overline{CD}$ are joined to form a triangle, the area of that triangle is
[asy]
pair A,B,C,D;
A = (0,8); B = (9,8); C = (9,0); D = (0,0);
draw(A--B--C--D--A--(9,4)--(4.5,0)--cycle);
label("$A$",A,NW);
label("$B$",B,NE);
label("$C$",C,SE);
label("$D$",D,SW);
[/asy]
$\text{(A)}\ 21 \qquad \text{(B)}\ 27 \qquad \text{(C)}\ 30 \qquad \text{(D)}\ 36 \qquad \text{(E)}\ 40$
2002 Moldova National Olympiad, 1
Several pupils wrote a solution of a math problem on the blackboard on the break. When the teacher came in, a pupil was just clearing the blackboard, so the teacher could only observe that there was a rectangle with the sides of integer lenghts and a diagonal of lenght $ 2002$. Then the teacher pointed out that there was a computation error in pupils' solution. Why did he conclude that?
2014 Cono Sur Olympiad, 3
Let $ABCD$ be a rectangle and $P$ a point outside of it such that $\angle{BPC} = 90^{\circ}$ and the area of the pentagon $ABPCD$ is equal to $AB^{2}$.
Show that $ABPCD$ can be divided in 3 pieces with straight cuts in such a way that a square can be built using those 3 pieces, without leaving any holes or placing pieces on top of each other.
Note: the pieces can be rotated and flipped over.
1992 All Soviet Union Mathematical Olympiad, 561
Given an infinite sheet of square ruled paper. Some of the squares contain a piece. A move consists of a piece jumping over a piece on a neighbouring square (which shares a side) onto an empty square and removing the piece jumped over. Initially, there are no pieces except in an $m x n$ rectangle ($m, n > 1$) which has a piece on each square. What is the smallest number of pieces that can be left after a series of moves?
2005 Iran MO (3rd Round), 3
$f(n)$ is the least number that there exist a $f(n)-$mino that contains every $n-$mino.
Prove that $10000\leq f(1384)\leq960000$.
Find some bound for $f(n)$
2004 Flanders Junior Olympiad, 1
Two $5\times1$ rectangles have 2 vertices in common as on the picture.
(a) Determine the area of overlap
(b) Determine the length of the segment between the other 2 points of intersection, $A$ and $B$.
[img]https://cdn.artofproblemsolving.com/attachments/9/0/4f1721c7ccdecdfe4d9cc05a17a553a0e9f670.png[/img]
2006 All-Russian Olympiad, 8
A $3000\times 3000$ square is tiled by dominoes (i. e. $1\times 2$ rectangles) in an arbitrary way. Show that one can color the dominoes in three colors such that the number of the dominoes of each color is the same, and each dominoe $d$ has at most two neighbours of the same color as $d$. (Two dominoes are said to be [i]neighbours[/i] if a cell of one domino has a common edge with a cell of the other one.)
1972 IMO Longlists, 33
A rectangle $ABCD$ is given whose sides have lengths $3$ and $2n$, where $n$ is a natural number. Denote by $U(n)$ the number of ways in which one can cut the rectangle into rectangles of side lengths $1$ and $2$.
$(a)$ Prove that
\[U(n + 1)+U(n -1) = 4U(n);\]
$(b)$ Prove that
\[U(n) =\frac{1}{2\sqrt{3}}[(\sqrt{3} + 1)(2 +\sqrt{3})^n + (\sqrt{3} - 1)(2 -\sqrt{3})^n].\]
2001 AIME Problems, 15
Let $EFGH$, $EFDC$, and $EHBC$ be three adjacent square faces of a cube, for which $EC=8$, and let $A$ be the eighth vertex of the cube. Let $I$, $J$, and $K$, be the points on $\overline{EF}$, $\overline{EH}$, and $\overline{EC}$, respectively, so that $EI=EJ=EK=2$. A solid $S$ is obtained by drilling a tunnel through the cube. The sides of the tunnel are planes parallel to $\overline{AE}$, and containing the edges, $\overline{IJ}$, $\overline{JK}$, and $\overline{KI}$. The surface area of $S$, including the walls of the tunnel, is $m+n\sqrt{p}$, where $m$, $n$, and $p$ are positive integers and $p$ is not divisible by the square of any prime. Find $m+n+p$.
1987 IMO Longlists, 51
The function $F$ is a one-to-one transformation of the plane into itself that maps rectangles into rectangles (rectangles are closed; continuity is not assumed). Prove that $F$ maps squares into squares.
2012 AMC 12/AHSME, 2
A circle of radius $5$ is inscribed in a rectangle as shown. The ratio of the the length of the rectangle to its width is $2\ :\ 1$. What is the area of the rectangle?
[asy]
draw((0,0)--(0,10)--(20,10)--(20,0)--cycle);
draw(circle((10,5),5));
[/asy]
$ \textbf{(A)}\ 50\qquad\textbf{(B)}\ 100\qquad\textbf{(C)}\ 125\qquad\textbf{(D)}\ 150\qquad\textbf{(E)}\ 200 $