Olympiad problems

2017 NZMOC Camp Selection Problems, 5

Find all pairs $(m, n)$ of positive integers such that the $m \times n$ grid contains exactly $225$ rectangles whose side lengths are odd and whose edges lie on the lines of the grid.

2007 Hungary-Israel Binational, 1

Tags: geometry , rectangle , induction , combinatorics

A given rectangle $ R$ is divided into $mn$ small rectangles by straight lines parallel to its sides. (The distances between the parallel lines may not be equal.) What is the minimum number of appropriately selected rectangles’ areas that should be known in order to determine the area of $ R$?

1996 IberoAmerican, 3

Tags: geometry , rectangle , combinatorics

We have a grid of $k^2-k+1$ rows and $k^2-k+1$ columns, where $k=p+1$ and $p$ is prime. For each prime $p$, give a method to put the numbers 0 and 1, one number for each square in the grid, such that on each row there are exactly $k$ 0's, on each column there are exactly $k$ 0's, and there is no rectangle with sides parallel to the sides of the grid with 0s on each four vertices.

1977 Czech and Slovak Olympiad III A, 2

Tags: geometry , rectangle , construction , geometric construction

The numbers $p,q>0$ are given. Construct a rectangle $ABCD$ with $AE=p,AF=q$ where $E,F$ are midpoints of $BC,CD,$ respectively. Discuss conditions of solvability.

1986 IMO Longlists, 23

Tags: rectangle , geometry

Let $I$ and $J$ be the centers of the incircle and the excircle in the angle $BAC$ of the triangle $ABC$. For any point $M$ in the plane of the triangle, not on the line $BC$, denote by $I_M$ and $J_M$ the centers of the incircle and the excircle (touching $BC$) of the triangle $BCM$. Find the locus of points $M$ for which $II_MJJ_M$ is a rectangle.

2005 AMC 12/AHSME, 3

Tags: geometry , rectangle

A rectangle with a diagonal of length $ x$ is twice as long as it is wide. What is the area of the rectangle? $ \textbf{(A)}\ \frac14x^2 \qquad \textbf{(B)}\ \frac25x^2 \qquad \textbf{(C)}\ \frac12x^2 \qquad \textbf{(D)}\ x^2 \qquad \textbf{(E)}\ \frac32x^2$

MIPT Undergraduate Contest 2019, 2.3

Tags: function , geometry , rectangle , topology , mapping

Let $A$ and $B$ be rectangles in the plane and $f : A \rightarrow B$ be a mapping which is uniform on the interior of $A$, maps the boundary of $A$ homeomorphically to the boundary of $B$ by mapping the sides of $A$ to corresponding sides in $B$. Prove that $f$ is an affine transformation.

2013 International Zhautykov Olympiad, 3

Tags: square , rectangle , combinatorics , extremal combinatorics , covering , square grid

A $10 \times 10$ table consists of $100$ unit cells. A [i]block[/i] is a $2 \times 2$ square consisting of $4$ unit cells of the table. A set $C$ of $n$ blocks covers the table (i.e. each cell of the table is covered by some block of $C$ ) but no $n -1$ blocks of $C$ cover the table. Find the largest possible value of $n$.

2003 Costa Rica - Final Round, 4

Tags: geometry , rectangle , trapezoid

$S_{1}$ and $S_{2}$ are two circles that intersect at distinct points $P$ and $Q$. $\ell_{1}$ and $\ell_{2}$ are two parallel lines through $P$ and $Q$. $\ell_{1}$ intersects $S_{1}$ and $S_{2}$ at points $A_{1}$ and $A_{2}$, different from $P$, respectively. $\ell_{2}$ intersects $S_{1}$ and $S_{2}$ at points $B_{1}$ and $B_{2}$, different from $Q$, respectively. Show that the perimeters of the triangles $A_{1}QA_{2}$ and $B_{1}PB_{2}$ are equal.

1972 IMO Longlists, 43

Tags: parallelogram , rectangle , trapezoid , pythagorean theorem , perpendicular bisector , geometry

A fixed point $A$ inside a circle is given. Consider all chords $XY$ of the circle such that $\angle XAY$ is a right angle, and for all such chords construct the point $M$ symmetric to $A$ with respect to $XY$ . Find the locus of points $M$.

2008 District Round (Round II), 3

Tags: rectangle , geometry , combinatorics

For $n>2$, an $n\times n$ grid of squares is coloured black and white like a chessboard, with its upper left corner coloured black. Then we can recolour some of the white squares black in the following way: choose a $2\times 3$ (or $3\times 2$) rectangle which has exactly $3$ white squares and then colour all these $3$ white squares black. Find all $n$ such that after a series of such operations all squares will be black.

2024 Ecuador NMO (OMEC), 3

Tags: geometry , angle bisector , rectangle

Let $\triangle ABC$ with $\angle BAC=120 ^\circ$. Let $D, E, F$ points on sides $BC, CA, AB$, respectively, such that $AD, BE, CF$ are angle bisectors on $\triangle ABC$. \\ Prove that $\triangle ABC$ is isosceles if and only if $\triangle DEF$ is right-angled isosceles.

1998 Slovenia National Olympiad, Problem 3

Tags: geometry , rectangle

A rectangle $ABCD$ with $AB>AD$ is given. The circle with center $B$ and radius $AB$ intersects the line $CD$ at $E$ and $F$. (a) Prove that the circumcircle of triangle $EBF$ is tangent to the circle with diameter $AD$. Denote the tangency point by $G$. (b) Prove that the points $D,G,$ and $B$ are collinear.

Kyiv City MO Juniors 2003+ geometry, 2020.7.4

Tags: geometry , rectangle , square

Given a square $ABCD$ with side $10$. On sides BC and $AD$ of this square are selected respectively points $E$ and $F$ such that formed a rectangle $ABEF$. Rectangle $KLMN$ is located so that its the vertices $K, L, M$ and $N$ lie one on each segments $CD, DF, FE$ and $EC$, respectively. It turned out that the rectangles $ABEF$ and $KLMN$ are equal with $AB = MN$. Find the length of segment $AL$.

Novosibirsk Oral Geo Oly IX, 2020.1

Tags: geometry , rectangle , semicircle , tangent circle

Two semicircles touch the side of the rectangle, each other and the segment drawn in it as in the figure. What part of the whole rectangle is filled? [img]https://cdn.artofproblemsolving.com/attachments/3/e/70ca8b80240a282553294a58cb3ed807d016be.png[/img]

1996 IMO Shortlist, 2

Tags: geometry , rectangle , combinatorics , coloring , counting

A square $ (n \minus{} 1) \times (n \minus{} 1)$ is divided into $ (n \minus{} 1)^2$ unit squares in the usual manner. Each of the $ n^2$ vertices of these squares is to be coloured red or blue. Find the number of different colourings such that each unit square has exactly two red vertices. (Two colouring schemse are regarded as different if at least one vertex is coloured differently in the two schemes.)

2007 Croatia Team Selection Test, 4

Tags: geometry , rectangle , combinatorics

Given a finite string $S$ of symbols $X$ and $O$, we write $@(S)$ for the number of $X$'s in $S$ minus the number of $O$'s. (For example, $@(XOOXOOX) =-1$.) We call a string $S$ [b]balanced[/b] if every substring $T$ of (consecutive symbols) $S$ has the property $-2 \leq @(T) \leq 2$. (Thus $XOOXOOX$ is not balanced since it contains the sub-string $OOXOO$ whose $@$-value is $-3$.) Find, with proof, the number of balanced strings of length $n$.

1953 Polish MO Finals, 2

Tags: locus , rectangle , inscribed , geometry

Find the geometric locus of the center of a rectangle whose vertices lie on the perimeter of a given triangle.

2013 Stanford Mathematics Tournament, 6

Tags: geometry , rectangle

$ABCD$ is a rectangle with $AB = CD = 2$. A circle centered at $O$ is tangent to $BC$, $CD$, and $AD$ (and hence has radius $1$). Another circle, centered at $P$, is tangent to circle $O$ at point $T$ and is also tangent to $AB$ and $BC$. If line $AT$ is tangent to both circles at $T$, find the radius of circle $P$.

1972 IMO Longlists, 28

Tags: rectangle , geometry

The lengths of the sides of a rectangle are given to be odd integers. Prove that there does not exist a point within that rectangle that has integer distances to each of its four vertices.

2015 AMC 10, 11

Tags: rectangle , function , ratio , pythagorean theorem , geometry

The ratio of the length to the width of a rectangle is $4:3$. If the rectangle has diagonal of length $d$, then the area may be expressed as $kd^2$ for some constant $k$. What is $k$? $\textbf{(A) }\dfrac27\qquad\textbf{(B) }\dfrac37\qquad\textbf{(C) }\dfrac{12}{25}\qquad\textbf{(D) }\dfrac{16}{25}\qquad\textbf{(E) }\dfrac34$

1995 Portugal MO, 3

Tags: rectangle , geometry

Three ants are at three corners of a rectangle. It is assumed that each ant moves only when the other two are stopped and always parallel to the line defined by them. Will be is it possible that the three ants are simultaneously at midpoints on the sides of the rectangle?

Novosibirsk Oral Geo Oly IX, 2021.1

Tags: geometry , rectangle

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.

2008 China Team Selection Test, 1

Tags: rectangle , geometry

Given a rectangle $ ABCD,$ let $ AB \equal{} b, AD \equal{} a ( a\geq b),$ three points $ X,Y,Z$ are put inside or on the boundary of the rectangle, arbitrarily. Find the maximum of the minimum of the distances between any two points among the three points. (Denote it by $ a,b$)

2008 CentroAmerican, 2

Tags: parallelogram , rectangle , geometry

Let $ ABCD$ be a convex cuadrilateral inscribed in a circumference centered at $ O$ such that $ AC$ is a diameter. Pararellograms $ DAOE$ and $ BCOF$ are constructed. Show that if $ E$ and $ F$ lie on the circumference then $ ABCD$ is a rectangle.