Found problems: 1342
2017 Yasinsky Geometry Olympiad, 5
$ABCD$ is a rectangle. The segment $MA$ is perpendicular to plane $ABC$ . $MB= 15$ , $MC=24$ , $MD=20$. Find the length of $MA$ .
2013 ELMO Shortlist, 9
Let $ABCD$ be a cyclic quadrilateral inscribed in circle $\omega$ whose diagonals meet at $F$. Lines $AB$ and $CD$ meet at $E$. Segment $EF$ intersects $\omega$ at $X$. Lines $BX$ and $CD$ meet at $M$, and lines $CX$ and $AB$ meet at $N$. Prove that $MN$ and $BC$ concur with the tangent to $\omega$ at $X$.
[i]Proposed by Allen Liu[/i]
2019 AMC 8, 2
Three identical rectangles are put together to form rectangle $ABCD$, as shown in the figure below. Given that the length of the shorter side of each of the smaller rectangles $5$ feet, what is the area in square feet of rectangle $ABCD$?
[asy]draw((0,0)--(0,10)--(15,10)--(15,0)--(0,0));
draw((0,5)--(10,5));
draw((10,0)--(10,10));
label("$A$",(0,0),SW);
label("$B$",(15,0),SE);
label("$C$",(15,10),NE);
label("$D$",(0,10),NW);
dot((0,10));
dot((15,0));
dot((15,10));
dot((0,0));
[/asy]
$\textbf{(A) }45\qquad
\textbf{(B) }75\qquad
\textbf{(C) }100\qquad
\textbf{(D) }125\qquad
\textbf{(E) }150\qquad$
2024 Kyiv City MO Round 1, Problem 1
Square $ABCD$ is cut by a line segment $EF$ into two rectangles $AEFD$ and $BCFE$. The lengths of the sides of each of these rectangles are positive integers. It is known that the area of the rectangle $AEFD$ is $30$ and it is larger than the area of the rectangle $BCFE$. Find the area of square $ABCD$.
[i]Proposed by Bogdan Rublov[/i]
2009 USA Team Selection Test, 1
Let $m$ and $n$ be positive integers. Mr. Fat has a set $S$ containing every rectangular tile with integer side lengths and area of a power of $2$. Mr. Fat also has a rectangle $R$ with dimensions $2^m \times 2^n$ and a $1 \times 1$ square removed from one of the corners. Mr. Fat wants to choose $m + n$ rectangles from $S$, with respective areas $2^0, 2^1, \ldots, 2^{m + n - 1}$, and then tile $R$ with the chosen rectangles. Prove that this can be done in at most $(m + n)!$ ways.
[i]Palmer Mebane.[/i]
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).
2006 Bosnia and Herzegovina Team Selection Test, 2
It is given a triangle $\triangle ABC$. Determine the locus of center of rectangle inscribed in triangle $ABC$ such that one side of rectangle lies on side $AB$.
1986 China Team Selection Test, 1
If $ABCD$ is a cyclic quadrilateral, then prove that the incenters of the triangles $ABC$, $BCD$, $CDA$, $DAB$ are the vertices of a rectangle.
1996 May Olympiad, 1
Let $ABCD$ be a rectangle. A line $r$ moves parallel to $AB$ and intersects diagonal $AC$ , forming two triangles opposite the vertex, inside the rectangle. Prove that the sum of the areas of these triangles is minimal when $r$ passes through the midpoint of segment $AD$ .
2003 Tournament Of Towns, 1
There is $3 \times 4 \times 5$ - box with its faces divided into $1 \times 1$ - squares. Is it possible to place numbers in these squares so that the sum of numbers in every stripe of squares (one square wide) circling the box, equals $120$?
2017 AMC 12/AHSME, 8
The ratio of the short side of a certain rectangle to the long side is equal to the ratio of the long side to the diagonal. What is the square of the ratio of the short side to the long side of this rectangle?
$\textbf{(A)} \text{ } \frac{\sqrt{3}-1}{2} \qquad \textbf{(B)} \text{ } \frac{1}{2} \qquad \textbf{(C)} \text{ } \frac{\sqrt{5}-1}{2} \qquad \textbf{(D)} \text{ } \frac{\sqrt{2}}{2} \qquad \textbf{(E)} \text{ } \frac{\sqrt{6}-1}{2}$
2011 Junior Balkan Team Selection Tests - Moldova, 7
In the rectangle $ABCD$ with $AB> BC$, the perpendicular bisecotr of $AC$ intersects the side $CD$ at point $E$. The circle with the center at point $E$ and the radius $AE$ intersects again the side $AB$ at point $F$. If point $O$ is the orthogonal projection of point $C$ on the line $EF$, prove that points $B, O$ and $D$ are collinear.
2019 India IMO Training Camp, P2
Determine all positive integers $m$ satisfying the condition that there exists a unique positive integer $n$ such that there exists a rectangle which can be decomposed into $n$ congruent squares and can also be decomposed into $m+n$ congruent squares.
2019 239 Open Mathematical Olympiad, 2
Several cells are marked in a $100 \times 100$ table. Vasya wants to split the square into several rectangles such that each rectangle does not contain more than two marked cells and there are at most $k$ rectangles containing less than two cells. What is the smallest $k$ such that Vasya will certainly be able to do this?
2003 Austrian-Polish Competition, 10
What is the smallest number of $5\times 1$ tiles which must be placed on a $31\times 5$ rectangle (each covering exactly $5$ unit squares) so that no further tiles can be placed? How many different ways are there of placing the minimal number (so that further tiles are blocked)? What are the answers for a $52\times 5$ rectangle?
2014 USAMTS Problems, 4:
A point $P$ in the interior of a convex polyhedron in Euclidean space is called a [i]pivot point[/i] of the polyhedron if every line through $P$ contains exactly $0$ or $2$ vertices of the polyhedron. Determine, with proof, the maximum number of pivot points that a polyhedron can contain.
1968 Dutch Mathematical Olympiad, 5
A square of side $n$ ($n$ natural) is divided into $n^2$ squares of side $1$. Each pair of "horizontal" boundary lines and each pair of "vertical" boundary lines enclose a rectangle (a square is also considered a rectangle). A rectangle has a length and a width; the width is less than or equal to the length.
(a) Prove that there are $8$ rectangles of width $n - 1$.
(b) Determine the number of rectangles with width $n -k$ ($0\le k \le n -1,k$ integer).
(c) Determine a formula for $1^3 + 2^3 +...+ n^3$.
2023/2024 Tournament of Towns, 6
6. The baker has baked a rectangular pancake. He then cut it into $n^{2}$ rectangles by making $n-1$ horizontal and $n-1$ vertical cuts. Being rounded to the closest integer, the areas of resulting rectangles equal to all positive integers from 1 to $n^{2}$ in some order. For which maximal $n$ could this happen? (Half-integers are rounded upwards.)
Georgy Karavaev
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.
2012 Indonesia TST, 3
Suppose $l(M, XYZ)$ is a Simson line of the triangle $XYZ$ that passes through $M$.
Suppose $ABCDEF$ is a cyclic hexagon such that $l(A, BDF), l(B, ACE), l(D, ABF), l(E, ABC)$ intersect at a single point. Prove that $CDEF$ is a rectangle.
[color=blue]Should the first sentence read:
Suppose $l(M, XYZ)$ is a Simson line of the triangle $XYZ$ [u]with respect to[/u] $M$.
? Since it appears weird that a Simson line that passes a point is to be constructed. However, this is Unsolved after all, so I'm not sure.[/color]
2009 India National Olympiad, 1
Let $ ABC$ be a tringle and let $ P$ be an interior point such that $ \angle BPC \equal{} 90 ,\angle BAP \equal{} \angle BCP$.Let $ M,N$ be the mid points of $ AC,BC$ respectively.Suppose $ BP \equal{} 2PM$.Prove that $ A,P,N$ are collinear.
2002 Junior Balkan Team Selection Tests - Romania, 3
Consider a $1 \times n$ rectangle and some tiles of size $1 \times 1$ of four different colours. The rectangle is tiled in such a way that no two neighboring square tiles have the same colour.
a) Find the number of distinct symmetrical tilings.
b) Find the number of tilings such that any consecutive square tiles have distinct colours.
2004 Bosnia and Herzegovina Junior BMO TST, 2
A rectangle is divided into $9$ smaller rectangles. The area of four of them is $5, 3, 9$ and $2$, as in the picture below.
(The picture is not at scale.)
[img]https://cdn.artofproblemsolving.com/attachments/8/e/0ccd6f41073f776b62e9ef4522df1f1639ee31.png[/img]
Determine the minimum area of the rectangle. Under what circumstances is it achieved?
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]
2025 PErA, P5
We have an $n \times n$ board, filled with $n$ rectangles aligned to the grid. The $n$ rectangles cover all the board and are never superposed. Find, in terms of $n$, the smallest value the sum of the $n$ diagonals of the rectangles can take.