Found problems: 1342
Novosibirsk Oral Geo Oly VIII, 2021.7
Two congruent rectangles are located as shown in the figure. Find the area of the shaded part.
[img]https://cdn.artofproblemsolving.com/attachments/2/e/10b164535ab5b3a3b98ce1a0b84892cd11d76f.png[/img]
May Olympiad L2 - geometry, 2023.4
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]
1988 IMO Longlists, 50
Prove that the numbers $A,B$ and $C$ are equal, where:
- $A=$ number of ways that we can cover a $2 \times n$ rectangle with $2 \times 1$ retangles.
- $B=$ number of sequences of ones and twos that add up to $n$
- $C= \sum^m_{k=0} \binom{m + k}{2 \cdot k}$ if $n = 2 \cdot m,$ and
- $C= \sum^m_{k=0} \binom{m + k + 1}{2 \cdot k + 1}$ if $n = 2 \cdot m + 1.$
2010 AMC 12/AHSME, 3
Rectangle $ ABCD$, pictured below, shares $50\%$ of its area with square $ EFGH$. Square $ EFGH$ shares $20\%$ of its area with rectangle $ ABCD$. What is $ \frac{AB}{AD}$?
[asy]unitsize(5mm);
defaultpen(linewidth(0.8pt)+fontsize(10pt));
pair A=(0,3), B=(8,3), C=(8,2), D=(0,2), Ep=(0,4), F=(4,4), G=(4,0), H=(0,0);
fill(shift(0,2)*xscale(4)*unitsquare,grey);
draw(Ep--F--G--H--cycle);
draw(A--B--C--D);
label("$A$",A,W);
label("$B$",B,E);
label("$C$",C,E);
label("$D$",D,W);
label("$E$",Ep,NW);
label("$F$",F,NE);
label("$G$",G,SE);
label("$H$",H,SW);[/asy]$ \textbf{(A)}\ 4\qquad \textbf{(B)}\ 5\qquad \textbf{(C)}\ 6\qquad \textbf{(D)}\ 8\qquad \textbf{(E)}\ 10$
2006 AMC 12/AHSME, 6
The $ 8\times 18$ rectangle $ ABCD$ is cut into two congruent hexagons, as shown, in such a way that the two hexagons can be repositioned without overlap to form a square. What is $ y$?
[asy] unitsize(2mm); defaultpen(fontsize(10pt)+linewidth(.8pt)); dotfactor=4; draw((0,4)--(18,4)--(18,-4)--(0,-4)--cycle); draw((6,4)--(6,0)--(12,0)--(12,-4)); label("$D$",(0,4),NW); label("$C$",(18,4),NE); label("$B$",(18,-4),SE); label("$A$",(0,-4),SW); label("$y$",(9,1)); [/asy]$ \textbf{(A) } 6\qquad \textbf{(B) } 7\qquad \textbf{(C) } 8\qquad \textbf{(D) } 9\qquad \textbf{(E) } 10$
2013 Iran Team Selection Test, 14
we are given $n$ rectangles in the plane. Prove that between $4n$ right angles formed by these rectangles there are at least $[4\sqrt n]$ distinct right angles.
2017 Yasinsky Geometry Olympiad, 1
Rectangular sheet of paper $ABCD$ is folded as shown in the figure. Find the rato $DK: AB$, given that $C_1$ is the midpoint of $AD$.
[img]https://3.bp.blogspot.com/-9EkSdxpGnPU/W6dWD82CxwI/AAAAAAAAJHw/iTkEOejlm9U6Dbu427vUJwKMfEOOVn0WwCK4BGAYYCw/s400/Yasinsky%2B2017%2BVIII-IX%2Bp1.png[/img]
2012-2013 SDML (Middle School), 5
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]
2005 China National Olympiad, 5
There are 5 points in a rectangle (including its boundary) with area 1, no three of them are in the same line. Find the minimum number of triangles with the area not more than $\frac 1{4}$, vertex of which are three of the five points.
1995 APMO, 4
Let $C$ be a circle with radius $R$ and centre $O$, and $S$ a fixed point in the interior of $C$. Let $AA'$ and $BB'$ be perpendicular chords through $S$. Consider the rectangles $SAMB$, $SBN'A'$, $SA'M'B'$, and $SB'NA$. Find the set of all points $M$, $N'$, $M'$, and $N$ when $A$ moves around the whole circle.
2012 AMC 10, 2
A square with side length $8$ is cut in half, creating two congruent rectangles. What are the dimensions of one of these rectangles?
$ \textbf{(A)}\ 2\text{ by }4
\qquad\textbf{(B)}\ 2\text{ by }6
\qquad\textbf{(C)}\ 2\text{ by }8
\qquad\textbf{(D)}\ 4\text{ by }4
\qquad\textbf{(E)}\ 4\text{ by }8
$
1985 All Soviet Union Mathematical Olympiad, 416
Given big enough sheet of cross-lined paper with the side of the squares equal to $1$. We are allowed to cut it along the lines only. Prove that for every $m>12$ we can cut out a rectangle of the greater than $m$ area such, that it is impossible to cut out a rectangle of $m$ area from it.
2006 MOP Homework, 1
Determine if there is a way to tile a $5 \times 6$ unit square board by dominos such that one can not use a needle to peer through the tiling? Determine if there is a way to tile a $5 \times 6$ unit square board by dominos such that one can use a needle to through the tiling? What if it is a $6 \times 6$ board?
2016 Stars of Mathematics, 2
Let $ m,n\ge 2 $ and consider a rectangle formed by $ m\times n $ unit squares that are colored, either white, or either black. A [i]step[/i] is the action of selecting from it a rectangle of dimensions $ 1\times k, $ where $ k $ is an odd number smaller or equal to $ n, $ or a rectangle of dimensions $ l\times 1, $ where $ l $ is and odd number smaller than $ m, $ and coloring all the unit squares of this chosen rectangle with the color that appears the least in it.
[b]a)[/b] Show that, for any $ m,n\ge 5, $ there exists a succession of [i]steps[/i] that make the rectagle to be single-colored.
[b]b)[/b] What about $ m=n+1=5? $
2005 Sharygin Geometry Olympiad, 8
Around the convex quadrilateral $ABCD$, three rectangles are circumscribed .
It is known that two of these rectangles are squares. Is it true that the third one is necessarily a square?
(A rectangle is circumscribed around the quadrilateral $ABCD$ if there is one vertex $ABCD$ on each side of the rectangle).
1981 All Soviet Union Mathematical Olympiad, 312
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.
1981 All Soviet Union Mathematical Olympiad, 308
Given real $a$. Find the least possible area of the rectangle with the sides parallel to the coordinate axes and containing the figure determined by the system of inequalities $$y \le -x^2 \,\,\, and \,\,\, y \ge x^2 - 2x + a$$
2019 Taiwan TST Round 3, 1
For each lattice point on the Cartesian Coordinate Plane, one puts a positive integer at the lattice such that [b]for any given rectangle with sides parallel to coordinate axes, the sum of the number inside the rectangle is not a prime. [/b] Find the minimal possible value of the maximum of all numbers.
2022 Portugal MO, 3
The Proenc has a new $8\times 8$ chess board and requires composing it into rectangles that do not overlap, so that:
(i) each rectangle has as many white squares as black ones;
(ii) there are no two rectangles with the same number of squares.
Determines the maximum value of $n$ for which such a decomposition is possible. For this value of $n$, determine all possible sets ${A_1,... ,A_n}$, where $A_i$ is the number of rectangle $i$ in squares, for which a decomposition of the board under the conditions intended actions is possible.
1998 Junior Balkan Team Selection Tests - Romania, 2
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]
1953 AMC 12/AHSME, 32
Each angle of a rectangle is trisected. The intersections of the pairs of trisectors adjacent to the same side always form:
$ \textbf{(A)}\ \text{a square} \qquad\textbf{(B)}\ \text{a rectangle} \qquad\textbf{(C)}\ \text{a parallelogram with unequal sides} \\
\textbf{(D)}\ \text{a rhombus} \qquad\textbf{(E)}\ \text{a quadrilateral with no special properties}$
1984 AMC 12/AHSME, 4
A rectangle intersects a circle as shown: $AB=4$, $BC=5$, and $DE=3$. Then $EF$ equals:
[asy]size(200);
defaultpen(linewidth(0.7)+fontsize(10));
pair D=origin, E=(3,0), F=(10,0), G=(12,0), H=(12,1), A=(0,1), B=(4,1), C=(9,1), O=circumcenter(B,C,F);
draw(D--G--H--A--cycle);
draw(Circle(O, abs(O-C)));
label("$A$", A, NW);
label("$B$", B, NW);
label("$C$", C, NE);
label("$D$", D, SW);
label("$E$", E, SE);
label("$F$", F, SW);
label("4", (2,0.85), N);
label("3", D--E, S);
label("5", (6.5,0.85), N);
[/asy]
$\mathbf{(A)}\; 6\qquad \mathbf{(B)}\; 7\qquad \mathbf{(C)}\; \frac{20}3\qquad \mathbf{(D)}\; 8\qquad \mathbf{(E)}\; 9$
1999 Tournament Of Towns, 5
Is it possible to divide a $6 \times 6$ chessboard into $18$ rectangles, each either $1 \times 2$ or $2 \times 1$, and to draw exactly one diagonal on each rectangle such that no two of these diagonals have a common endpoint?
(A Shapovalov)
1990 Iran MO (2nd round), 3
We want to cover a rectangular $5 \times 137$ with the following figures, prove that this is impossible.
\[\text{Squars are the same and all are } \Huge{1 \times 1}\]
[asy]
import graph; size(400); real lsf = 0.5; pen dp = linewidth(0.7) + fontsize(10); defaultpen(dp); pen ds = black; pen xdxdff = rgb(0.49,0.49,1);
draw((2,4)--(0,4),linewidth(2pt)); draw((0,4)--(0,0),linewidth(2pt)); draw((0,0)--(2,0),linewidth(2pt)); draw((2,0)--(2,1),linewidth(2pt)); draw((2,1)--(0,1),linewidth(2pt)); draw((1,0)--(1,4),linewidth(2pt)); draw((2,4)--(2,3),linewidth(2pt)); draw((2,3)--(0,3),linewidth(2pt)); draw((0,2)--(1,2),linewidth(2pt));
label("(1)", (0.56,-1.54), SE*lsf); draw((4,2)--(4,1),linewidth(2pt)); draw((7,2)--(7,1),linewidth(2pt)); draw((4,2)--(7,2),linewidth(2pt)); draw((4,1)--(7,1),linewidth(2pt)); draw((6,0)--(6,3),linewidth(2pt)); draw((5,3)--(5,0),linewidth(2pt)); draw((5,0)--(6,0),linewidth(2pt)); draw((5,3)--(6,3),linewidth(2pt)); label("(2)", (5.13,-1.46), SE*lsf); draw((9,0)--(9,3),linewidth(2pt)); draw((10,3)--(10,0),linewidth(2pt)); draw((12,3)--(12,0),linewidth(2pt)); draw((11,0)--(11,3),linewidth(2pt)); draw((9,2)--(12,2),linewidth(2pt)); draw((12,1)--(9,1),linewidth(2pt)); draw((9,3)--(10,3),linewidth(2pt)); draw((11,3)--(12,3),linewidth(2pt)); draw((12,0)--(11,0),linewidth(2pt)); draw((9,0)--(10,0),linewidth(2pt)); label("(3)", (10.08,-1.48), SE*lsf); draw((14,1)--(17,1),linewidth(2pt)); draw((15,2)--(17,2),linewidth(2pt)); draw((15,2)--(15,0),linewidth(2pt)); draw((15,0)--(14,0)); draw((14,1)--(14,0),linewidth(2pt)); draw((16,2)--(16,0),linewidth(2pt)); label("(4)", (15.22,-1.5), SE*lsf); draw((14,0)--(16,0),linewidth(2pt)); draw((17,2)--(17,1),linewidth(2pt)); draw((19,3)--(19,0),linewidth(2pt)); draw((20,3)--(20,0),linewidth(2pt)); draw((20,3)--(19,3),linewidth(2pt)); draw((19,2)--(20,2),linewidth(2pt)); draw((19,1)--(20,1),linewidth(2pt)); draw((20,0)--(19,0),linewidth(2pt)); label("(5)", (19.11,-1.5), SE*lsf); dot((0,0),ds); dot((0,1),ds); dot((0,2),ds); dot((0,3),ds); dot((0,4),ds); dot((1,4),ds); dot((2,4),ds); dot((2,3),ds); dot((1,3),ds); dot((1,2),ds); dot((1,1),ds); dot((2,1),ds); dot((2,0),ds); dot((1,0),ds); dot((5,0),ds); dot((6,0),ds); dot((5,1),ds); dot((6,1),ds); dot((5,2),ds); dot((6,2),ds); dot((5,3),ds); dot((6,3),ds); dot((7,2),ds); dot((7,1),ds); dot((4,1),ds); dot((4,2),ds); dot((9,0),ds); dot((9,1),ds); dot((9,2),ds); dot((9,3),ds); dot((10,0),ds); dot((11,0),ds); dot((12,0),ds); dot((10,1),ds); dot((10,2),ds); dot((10,3),ds); dot((11,1),ds); dot((11,2),ds); dot((11,3),ds); dot((12,1),ds); dot((12,2),ds); dot((12,3),ds); dot((14,0),ds); dot((15,0),ds); dot((16,0),ds); dot((15,1),ds); dot((14,1),ds); dot((16,1),ds); dot((15,2),ds); dot((16,2),ds); dot((17,2),ds); dot((17,1),ds); dot((19,0),ds); dot((20,0),ds); dot((19,1),ds); dot((20,1),ds); dot((19,2),ds); dot((20,2),ds); dot((19,3),ds); dot((20,3),ds); clip((-0.41,-10.15)--(-0.41,8.08)--(21.25,8.08)--(21.25,-10.15)--cycle);
[/asy]
2000 South africa National Olympiad, 6
Let $A_n$ be the number of ways to tile a $4 \times n$ rectangle using $2 \times 1$ tiles. Prove that $A_n$ is divisible by 2 if and only if $A_n$ is divisible by 3.