Found problems: 1342
2009 AMC 8, 8
The length of a rectangle is increased by $ 10\%$ and the width is decreased by $ 10\%$. What percent of the old area is the new area?
$ \textbf{(A)}\ 90 \qquad
\textbf{(B)}\ 99 \qquad
\textbf{(C)}\ 100 \qquad
\textbf{(D)}\ 101 \qquad
\textbf{(E)}\ 110$
1993 All-Russian Olympiad, 2
Is it true that any two rectangles of equal area can be placed in the plane such that any horizontal line intersecting at least one of them will also intersect the other, and the segments of intersection will be equal?
1995 AIME Problems, 11
A right rectangular prism $P$ (i.e., a rectangular parallelpiped) has sides of integral length $a, b, c,$ with $a\le b\le c.$ A plane parallel to one of the faces of $P$ cuts $P$ into two prisms, one of which is similar to $P,$ and both of which have nonzero volume. Given that $b=1995,$ for how many ordered triples $(a, b, c)$ does such a plane exist?
2014 USAMTS Problems, 5:
A finite set $S$ of unit squares is chosen out of a large grid of unit squares. The squares of $S$ are tiled with isosceles right triangles of hypotenuse $2$ so that the triangles do not overlap each other, do not extend past $S$, and all of $S$ is fully covered by the triangles. Additionally, the hypotenuse of each triangle lies along a grid line, and the vertices of the triangles lie at the corners of the squares. Show that the number of triangles must be a multiple of $4$.
2011 AMC 12/AHSME, 10
Rectangle $ABCD$ has $AB=6$ and $BC=3$. Point $M$ is chosen on side $AB$ so that $\angle AMD = \angle CMD$. What is the degree measure of $\angle AMD$?
$ \textbf{(A)}\ 15 \qquad
\textbf{(B)}\ 30 \qquad
\textbf{(C)}\ 45 \qquad
\textbf{(D)}\ 60 \qquad
\textbf{(E)}\ 75 $
1998 Akdeniz University MO, 4
A floor has $2 \times 11$ dimension, and this floor covering with $1 \times 2$ rectangles. (No two rectangles overlap). How many cases we done this job?
Indonesia Regional MO OSP SMA - geometry, 2011.4
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 $ZD$
2010 Czech-Polish-Slovak Match, 3
Let $p$ be a prime number. Prove that from a $p^2\times p^2$ array of squares, we can select $p^3$ of the squares such that the centers of any four of the selected squares are not the vertices of a rectangle with sides parallel to the edges of the array.
2003 Tournament Of Towns, 5
What is the largest number of squares on $9 \times 9$ square board that can be cut along their both diagonals so that the board does not fall apart into several pieces?
1966 IMO Longlists, 20
Given three congruent rectangles in the space. Their centers coincide, but the planes they lie in are mutually perpendicular. For any two of the three rectangles, the line of intersection of the planes of these two rectangles contains one midparallel of one rectangle and one midparallel of the other rectangle, and these two midparallels have different lengths. Consider the convex polyhedron whose vertices are the vertices of the rectangles.
[b]a.)[/b] What is the volume of this polyhedron ?
[b]b.)[/b] Can this polyhedron turn out to be a regular polyhedron ? If yes, what is the condition for this polyhedron to be regular ?
2007 AMC 8, 8
In trapezoid $ABCD$, $AD$ is perpendicular to $DC$, $AD=AB=3$, and $DC=6$. In addition, E is on $DC$, and $BE$ is parallel to $AD$. Find the area of $\Delta BEC$.
[asy]
defaultpen(linewidth(0.7));
pair A=(0,3), B=(3,3), C=(6,0), D=origin, E=(3,0);
draw(E--B--C--D--A--B);
draw(rightanglemark(A, D, C));
label("$A$", A, NW);
label("$B$", B, NW);
label("$C$", C, SE);
label("$D$", D, SW);
label("$E$", E, NW);
label("$3$", A--D, W);
label("$3$", A--B, N);
label("$6$", E, S);[/asy]
$\textbf{(A)} \: 3\qquad \textbf{(B)} \: 4.5\qquad \textbf{(C)} \: 6\qquad \textbf{(D)} \: 9\qquad \textbf{(E)} \: 18\qquad $
2006 Australia National Olympiad, 1
In a square $ABCD$, $E$ is a point on diagonal $BD$. $P$ and $Q$ are the circumcentres of $\triangle ABE$ and $\triangle ADE$ respectively. Prove that $APEQ$ is a square.
1987 Mexico National Olympiad, 5
In a right triangle $ABC$, M is a point on the hypotenuse $BC$ and $P$ and $Q$ the projections of $M$ on $AB$ and $AC$ respectively. Prove that for no such point $M$ do the triangles $BPM, MQC$ and the rectangle $AQMP$ have the same area.
2010 Turkey Junior National Olympiad, 1
A circle that passes through the vertex $A$ of a rectangle $ABCD$ intersects the side $AB$ at a second point $E$ different from $B.$ A line passing through $B$ is tangent to this circle at a point $T,$ and the circle with center $B$ and passing through $T$ intersects the side $BC$ at the point $F.$ Show that if $\angle CDF= \angle BFE,$ then $\angle EDF=\angle CDF.$
2010 Romania Team Selection Test, 3
Two rectangles of unit area overlap to form a convex octagon. Show that the area of the octagon is at least $\dfrac {1} {2}$.
[i]Kvant Magazine [/i]
2010 Contests, 3
Let $p$ be a prime number. Prove that from a $p^2\times p^2$ array of squares, we can select $p^3$ of the squares such that the centers of any four of the selected squares are not the vertices of a rectangle with sides parallel to the edges of the array.
2011 District Olympiad, 2
[b]a)[/b] Show that if four distinct complex numbers have the same absolute value and their sum vanishes, then they represent a rectangle.
[b]b)[/b] Let $ x,y,z,t $ be four real numbers, and $ k $ be an integer. Prove the following implication:
$$ \sum_{j\in\{ x,y,z,t\}} \sin j = 0 = \sum_{j\in\{ x,y,z,t\}} \cos j\implies \sum_{j\in\{ x,y,z,t\}} \sin (1+2n)j. $$
2018 Peru IMO TST, 1
A rectangle $\mathcal{R}$ with odd integer side lengths is divided into small rectangles with integer side lengths. Prove that there is at least one among the small rectangles whose distances from the four sides of $\mathcal{R}$ are either all odd or all even.
[i]Proposed by Jeck Lim, Singapore[/i]
2003 Oral Moscow Geometry Olympiad, 5
Given triangle $ABC$. Point $O_1$ is the center of the $BCDE$ rectangle, constructed so that the side $DE$ of the rectangle contains the vertex $A$ of the triangle. Points $O_2$ and $O_3$ are the centers of rectangles constructed in the same way on the sides $AC$ and $AB$, respectively. Prove that lines $AO_1, BO_2$ and $CO_3$ meet at one point.
1989 Chile National Olympiad, 2
We have a rectangle with integer sides $m, n$ that is subdivided into $mn$ squares of side $1$. Find the number of little squares that are crossed by the diagonal (without counting those that are touched only in one vertex)
1990 AMC 12/AHSME, 20
$ABCD$ is a quadrilateral with right angles at $A$ and $C$. Points $E$ and $F$ are on $AC$, and $DE$ and $BF$ are perpendicular to $AC$. If $AE=3$, $DE=5$, and $CE=7$, then $BF=$
[asy]
draw((0,0)--(10,0)--(3,-5)--(0,0)--(6.5,3)--(10,0));
draw((6.5,0)--(6.5,3));
draw((3,0)--(3,-5));
dot((0,0));
dot((10,0));
dot((3,0));
dot((3,-5));
dot((6.5,0));
dot((6.5,3));
label("A", (0,0), W);
label("B", (6.5,3), N);
label("C", (10,0), E);
label("D", (3,-5), S);
label("E", (3,0), N);
label("F", (6.5,0), S);[/asy]
$\text{(A)} \ 3.6 \qquad \text{(B)} \ 4 \qquad \text{(C)} \ 4.2 \qquad \text{(D)} \ 4.5 \qquad \text{(E)} \ 5$
2015 Kazakhstan National Olympiad, 3
A rectangle is said to be $ inscribed$ in a triangle if all its vertices lie on the sides of the triangle. Prove that the locus of the centers (the meeting points of the diagonals) of all inscribed in an acute-angled triangle rectangles are three concurrent unclosed segments.
2013 Kazakhstan National Olympiad, 3
How many non-intersecting pairs of paths we have from (0,0) to (n,n) so that path can move two ways:top or right?
2012 National Olympiad First Round, 33
Let $ABCDA'B'C'D'$ be a rectangular prism with $|AB|=2|BC|$. $E$ is a point on the edge $[BB']$ satisfying $|EB'|=6|EB|$. Let $F$ and $F'$ be the feet of the perpendiculars from $E$ at $\triangle AEC$ and $\triangle A'EC'$, respectively. If $m(\widehat{FEF'})=60^{\circ}$, then $|BC|/|BE| = ? $
$ \textbf{(A)}\ \sqrt\frac53 \qquad \textbf{(B)}\ \sqrt\frac{15}2 \qquad \textbf{(C)}\ \frac32\sqrt{15} \qquad \textbf{(D)}\ 5\sqrt\frac53 \qquad \textbf{(E)}\ \text{None}$
1981 All Soviet Union Mathematical Olympiad, 324
Six points are marked inside the $3\times 4$ rectangle. Prove that there is a pair of marked points with the distance between them not greater than $\sqrt5$.