Found problems: 1342
2009 Ukraine National Mathematical Olympiad, 3
Given $2009 \times 4018$ rectangular board. Frame is a rectangle $n \times n$ or $n \times(n + 2)$ for $ ( n \geq 3 )$ without all cells which don’t have any common points with boundary of rectangle. Rectangles $1\times1,1\times 2,1\times 3$ and $ 2\times 4$ are also frames. Two players by turn paint all cells of some frame that has no painted cells yet. Player that can't make such move loses. Who has a winning strategy?
1989 AMC 8, 24
Suppose a square piece of paper is folded in half vertically. The folded paper is then cut in half along the dashed line. Three rectangles are formed-a large one and two small ones. What is the ratio of the perimeter of one of the small rectangles to the perimeter of the large rectangle?
$\text{(A)}\ \frac{1}{2} \qquad \text{(B)}\ \frac{2}{3} \qquad \text{(C)}\ \frac{3}{4} \qquad \text{(D)}\ \frac{4}{5} \qquad \text{(E)}\ \frac{5}{6}$
[asy]
draw((0,0)--(0,8)--(6,8)--(6,0)--cycle);
draw((0,8)--(5,9)--(5,8));
draw((3,-1.5)--(3,10.3),dashed);
draw((0,5.5)..(-.75,4.75)..(0,4));
draw((0,4)--(1.5,4),EndArrow);
[/asy]
1990 IberoAmerican, 4
Let $\Gamma_{1}$ be a circle. $AB$ is a diameter, $\ell$ is the tangent at $B$, and $M$ is a point on $\Gamma_{1}$ other than $A$. $\Gamma_{2}$ is a circle tangent to $\ell$, and also to $\Gamma_{1}$ at $M$.
a) Determine the point of tangency $P$ of $\ell$ and $\Gamma_{2}$ and find the locus of the center of $\Gamma_{2}$ as $M$ varies.
b) Show that there exists a circle that is always orthogonal to $\Gamma_{2}$, regardless of the position of $M$.
2010 Tournament Of Towns, 3
An angle is given in a plane. Using only a compass, one must find out
$(a)$ if this angle is acute. Find the minimal number of circles one must draw to be sure.
$(b)$ if this angle equals $31^{\circ}$.(One may draw as many circles as one needs).
2013 JBMO Shortlist, 6
Let $P$ and $Q$ be the midpoints of the sides $BC$ and $CD$, respectively in a rectangle $ABCD$. Let $K$ and $M$ be the intersections of the line $PD$ with the lines $QB$ and $QA$, respectively, and let $N$ be the intersection of the lines $PA$ and $QB$. Let $X$, $Y$ and $Z$ be the midpoints of the segments $AN$, $KN$ and $AM$, respectively. Let $\ell_1$ be the line passing through $X$ and perpendicular to $MK$, $\ell_2$ be the line passing through $Y$ and perpendicular to $AM$ and $\ell_3$ the line passing through $Z$ and perpendicular to $KN$. Prove that the lines $\ell_1$, $\ell_2$ and $\ell_3$ are concurrent.
1988 IMO Longlists, 33
In a multiple choice test there were 4 questions and 3 possible answers for each question. A group of students was tested and it turned out that for any three of them there was a question which the three students answered differently. What is the maximum number of students tested?
2017 IMO Shortlist, C1
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]
2019 Purple Comet Problems, 1
The diagram shows a polygon made by removing six $2\times 2$ squares from the sides of an $8\times 12$ rectangle. Find the perimeter of this polygon.
[img]https://cdn.artofproblemsolving.com/attachments/6/3/c23510c821c159d31aff0e6688edebc81e2737.png[/img]
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 $AD$ .
1997 Pre-Preparation Course Examination, 2
Let $P$ be a variable point on arc $BC$ of the circumcircle of triangle $ABC$ not containing $A$. Let $I_1$ and $I_2$ be the incenters of the triangles $PAB$ and $PAC$, respectively. Prove that:
[b](a)[/b] The circumcircle of $?PI_1I_2$ passes through a fixed point.
[b](b)[/b] The circle with diameter $I_1I_2$ passes through a fixed point.
[b](c)[/b] The midpoint of $I_1I_2$ lies on a fixed circle.
1992 India Regional Mathematical Olympiad, 4
$ABCD$ is a cyclic quadrilateral with $AC \perp BD$; $AC$ meets $BD$ at $E$. Prove that \[ EA^2 + EB^2 + EC^2 + ED^2 = 4 R^2 \]
where $R$ is the radius of the circumscribing circle.
2001 Junior Balkan Team Selection Tests - Romania, 3
Let $ABCD$ be a quadrilateral inscribed in the circle $O$. For a point $E\in O$, its projections $K,L,M,N$ on the lines $DA,AB,BC,CD$, respectively, are considered. Prove that if $N$ is the orthocentre of the triangle $KLM$ for some point $E$, different from $A,B,C,D$, then this holds for every point $E$ of the circle.
2007 IMO Shortlist, 2
A rectangle $ D$ is partitioned in several ($ \ge2$) rectangles with sides parallel to those of $ D$. Given that any line parallel to one of the sides of $ D$, and having common points with the interior of $ D$, also has common interior points with the interior of at least one rectangle of the partition; prove that there is at least one rectangle of the partition having no common points with $ D$'s boundary.
[i]Author: Kei Irie, Japan[/i]
2008 AMC 12/AHSME, 13
Vertex $ E$ of equilateral $ \triangle{ABE}$ is in the interior of unit square $ ABCD$. Let $ R$ be the region consisting of all points inside $ ABCD$ and outside $ \triangle{ABE}$ whose distance from $ \overline{AD}$ is between $ \frac{1}{3}$ and $ \frac{2}{3}$. What is the area of $ R$?
$ \textbf{(A)}\ \frac{12\minus{}5\sqrt3}{72} \qquad
\textbf{(B)}\ \frac{12\minus{}5\sqrt3}{36} \qquad
\textbf{(C)}\ \frac{\sqrt3}{18} \qquad
\textbf{(D)}\ \frac{3\minus{}\sqrt3}{9} \qquad
\textbf{(E)}\ \frac{\sqrt3}{12}$
2011 Baltic Way, 9
Given a rectangular grid, split into $m\times n$ squares, a colouring of the squares in two colours (black and white) is called valid if it satisfies the following conditions:
[list]
[*]All squares touching the border of the grid are coloured black.
[*]No four squares forming a $2\times 2$ square are coloured in the same colour.
[*]No four squares forming a $2\times 2$ square are coloured in such a way that only diagonally touching
squares have the same colour.[/list]
Which grid sizes $m\times n$ (with $m,n\ge 3$) have a valid colouring?
2013 Argentina Cono Sur TST, 3
$1390$ ants are placed near a line, such that the distance between their heads and the line is less than $1\text{cm}$ and the distance between the heads of two ants is always larger than $2\text{cm}$. Show that there is at least one pair of ants such that the distance between their heads is at least $10$ meters (consider the head of an ant as point).
1993 Chile National Olympiad, 2
Given a rectangle, circumscribe a rectangle of maximum area.
2018 Hanoi Open Mathematics Competitions, 12
Let ABCD be a rectangle with $45^o < \angle ADB < 60^o$. The diagonals $AC$ and$ BD$ intersect at $O$. A line passing through $O$ and perpendicular to $BD$ meets $AD$ and $CD$ at $M$ and $N$ respectively. Let $K$ be a point on side $BC$ such that $MK \parallel AC$. Show that $\angle MKN = 90^o$.
[img]https://cdn.artofproblemsolving.com/attachments/4/1/1d37b96cebaea3409ade7ce6711ac2d3fc2ef9.png[/img]
Durer Math Competition CD Finals - geometry, 2015.D4
The projection of the vertex $C$ of the rectangle $ABCD$ on the diagonal $BD$ is $E$. The projections of $E$ on $AB$ and $AD$ are $F$ and $G$ respectively. Prove that $$AF^{2/3} + AG^{2/3} = AC^{2/3}$$
.
1993 Irish Math Olympiad, 5
$ (a)$ The rectangle $ PQRS$ with $ PQ\equal{}l$ and $ QR\equal{}m$ $ (l,m \in \mathbb{N})$ is divided into $ lm$ unit squares. Prove that the diagonal $ PR$ intersects exactly $ l\plus{}m\minus{}d$ of these squares, where $ d\equal{}(l,m)$.
$ (b)$ A box with edge lengths $ l,m,n \in \mathbb{N}$ is divided into $ lmn$ unit cubes. How many of the cubes does a main diagonal of the box intersect?
1998 IMO Shortlist, 7
A solitaire game is played on an $m\times n$ rectangular board, using $mn$ markers which are white on one side and black on the other. Initially, each square of the board contains a marker with its white side up, except for one corner square, which contains a marker with its black side up. In each move, one may take away one marker with its black side up, but must then turn over all markers which are in squares having an edge in common with the square of the removed marker. Determine all pairs $(m,n)$ of positive integers such that all markers can be removed from the board.
2007 Iran MO (3rd Round), 1
Consider two polygons $ P$ and $ Q$. We want to cut $ P$ into some smaller polygons and put them together in such a way to obtain $ Q$. We can translate the pieces but we can not rotate them or reflect them. We call $ P,Q$ equivalent if and only if we can obtain $ Q$ from $ P$(which is obviously an equivalence relation).
[img]http://i3.tinypic.com/4lrb43k.png[/img]
a) Let $ P,Q$ be two rectangles with the same area(their sides are not necessarily parallel). Prove that $ P$ and $ Q$ are equivalent.
b) Prove that if two triangles are not translation of each other, they are not equivalent.
c) Find a necessary and sufficient condition for polygons $ P,Q$ to be equivalent.
2016 Mathematical Talent Reward Programme, MCQ: P 11
In rectangle $ABCD$, $AD=1$, $P$ is on $AB$ and $DB$ and $DP$ trisect $\angle ADC$. What is the perimeter $\triangle BDP$
[list=1]
[*] $3+\frac{\sqrt{3}}{3}$
[*] $2+\frac{4\sqrt{3}}{3}$
[*] $2+2\sqrt{2}$
[*] $\frac{3+3\sqrt{5}}{2}$
[/list]
1989 AMC 12/AHSME, 3
A square is cut into three rectangles along two lines parallel to a side, as shown. If the perimeter of each of the three rectangles is 24, then the area of the original square is
[asy]
draw((0,0)--(9,0)--(9,9)--(0,9)--cycle);
draw((3,0)--(3,9), dashed);
draw((6,0)--(6,9), dashed);[/asy]
$\text{(A)} \ 24 \qquad \text{(B)} \ 36 \qquad \text{(C)} \ 64 \qquad \text{(D)} \ 81 \qquad \text{(E)} \ 96$
2015 Indonesia MO Shortlist, G1
Given a cyclic quadrilateral $ABCD$ so that $AB = AD$ and $AB + BC <CD$. Prove that the angle $ABC$ is more than $120$ degrees.