Let $n \ge 3$ be an integer, and consider a $n \times n$-board, divided into $n^2$ unit squares. For all $m \ge 1$, arbitrarily many $1\times m$-rectangles (type I) and arbitrarily many $m\times 1$-rectangles (type II) are available. We cover the board with $N$ such rectangles, without overlaps, and such that every rectangle lies entirely inside the board. We require that the number of type I rectangles used is equal to the number of type II rectangles used.(Note that a $1 \times 1$-rectangle has both types.) What is the minimal value of $N$ for which this is possible?

A $ 9 \times 12$ rectangle is partitioned into unit squares. The centers of all the unit squares, except for the four corner squares and eight squares sharing a common side with one of them, are coloured red. Is it possible to label these red centres $ C_1,C_2,\ldots ,C_{96}$ in such way that the following to conditions are both fulfilled i) the distances $C_1C_2,\ldots ,C_{95}C_{96}, C_{96}C_{1}$ are all equal to $ \sqrt {13}$, ii) the closed broken line $ C_1C_2\ldots C_{96}C_1$ has a centre of symmetry? [i]Bulgaria[/i]

Two rooks on a chessboard are said to be attacking each other if they are placed in the same row or column of the board. (a) There are eight rooks on a chessboard, none of them attacks any other. Prove that there is an even number of rooks on black fields. (b) How many ways can eight mutually non-attacking rooks be placed on the 9 £ 9 chessboard so that all eight rooks are on squares of the same color.

Are the following statements true? a) if the four vertices of a rectangle lie on the four sides of a rhombus, then the sides of the rectangle are parallel to the diagonals of the rhombus; b) if the four vertices of a square lie on the four sides of a rhombus that is not a square, then the sides of the square are parallel to the diagonals of the rhombus.

Let $ ABC$ be an acute-angled triangle; $ AD$ be the bisector of $ \angle BAC$ with $ D$ on $ BC$; and $ BE$ be the altitude from $ B$ on $ AC$. Show that $ \angle CED > 45^\circ .$ [b][weightage 17/100][/b]

In the rectangle $ABCD, M, N, P$ and $Q$ are the midpoints of the sides. If the area of the shaded triangle is $1$, calculate the area of the rectangle $ABCD$. [img]https://2.bp.blogspot.com/-9iyKT7WP5fc/XNYuXirLXSI/AAAAAAAAKK4/10nQuSAYypoFBWGS0cZ5j4vn_hkYr8rcwCK4BGAYYCw/s400/may3.gif[/img]

Suppose $F$ is a polygon with lattice vertices and sides parralell to x-axis and y-axis.Suppose $S(F),P(F)$ are area and perimeter of $F$. Find the smallest k that: $S(F) \leq k.P(F)^2$

Cut the $10$ cm $x 20$ cm rectangle into two pieces with one straight cut so that they can be placed inside the $19.4$ cm diameter circle without intersecting.

Let $a,b,c,$ and $d$ be the lengths of sides $MN,NP,PQ,$ and $QM$, respectively, of quadrilateral $MNPQ$. If $A$ is the area of $MNPQ$, then $\textbf{(A) }A=\left(\frac{a+c}{2}\right)\left(\frac{b+d}{2}\right)\text{ if and only if }MNPQ\text{ is convex}$ $\textbf{(B) }A=\left(\frac{a+c}{2}\right)\left(\frac{b+d}{2}\right)\text{ if and only if }MNPQ\text{ is a rectangle}$ $\textbf{(C) }A\le\left(\frac{a+c}{2}\right)\left(\frac{b+d}{2}\right)\text{ if and only if }MNPQ\text{ is a rectangle}$ $\textbf{(D) }A\le\left(\frac{a+c}{2}\right)\left(\frac{b+d}{2}\right)\text{ if and only if }MNPQ\text{ is a parallelogram}$ $\textbf{(E) }A\ge\left(\frac{a+c}{2}\right)\left(\frac{b+d}{2}\right)\text{ if and only if }MNPQ\text{ is a parallelogram}$

Rectangle $p*q,$ where $p,q$ are relatively coprime positive integers with $p <q$ is divided into squares $1*1$.Diagonal which goes from lowest left vertice to highest right cuts triangles from some squares.Find sum of perimeters of all such triangles.

We attempt to cover the plane with an infinite sequence of rectangles, overlapping allowed. (a) Is the task always possible if the area of the $n$th rectangle is $n^2$ for each $n$? (b) Is the task always possible if each rectangle is a square, and for any number $N$, there exist squares with total area greater than $N$?

In a village $1998$ persons volunteered to clean up, for a fair, a rectangular field with integer sides and perimeter equla to $3996$ feet. For this purpose, the field was divided into $1998$ equal parts. If each part had an integer area, find the length and breadth of the field.

For a rectangle $ABCD$ which is not a square, there is $O$ such that $O$ is on the perpendicular bisector of $BD$ and $O$ is in the interior of $\triangle BCD$. Denote by $E$ and $F$ the second intersections of the circle centered at $O$ passing through $B, D$ and $AB, AD$. $BF$ and $DE$ meets at $G$, and $X, Y, Z$ are the foots of the perpendiculars from $G$ to $AB, BD, DA$. $L, M, N$ are the foots of the perpendiculars from $O$ to $CD, BD, BC$. $XY$ and $ML$ meets at $P$, $YZ$ and $MN$ meets at $Q$. Prove that $BP$ and $DQ$ are parallel.

Sebastian has a certain number of rectangles with areas that sum up to 3 and with side lengths all less than or equal to $1$. Demonstrate that with each of these rectangles it is possible to cover a square with side $1$ in such a way that the sides of the rectangles are parallel to the sides of the square. [b]Note:[/b] The rectangles can overlap and they can protrude over the sides of the square.

If a $5 \times n$ rectangle can be tiled using $n$ pieces like those shown in the diagram, prove that $n$ is even. Show that there are more than $2 \cdot 3^{k-1}$ ways to file a fixed $5 \times 2k$ rectangle $(k \geq 3)$ with $2k$ pieces. (symmetric constructions are supposed to be different.)

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 $n = 2 \cdot m + 1.$

A rectangle $ABCD$ is inscribed in a circle with centre $O$. The exterior bisectors of $\angle ABD$ and $\angle ADB$ intersect at $P$; those of $\angle DAB$ and $\angle DBA$ intersect at $Q$; those of $\angle ACD$ and $\angle ADC$ intersect at $R$; and those of $\angle DAC$ and $\angle DCA$ intersect at $S$. Prove that $P,Q,R$, and $S$ are concyclic.

Problem 4 Let $ABCD$ be a rectangle with diagonals $AC$ and $BD$. Let $E$ be the intersection of the bisector of $\angle CAD$ with segment $CD$, $F$ on $CD$ such that $E$ is midpoint of $DF$, and $G$ on $BC$ such that $BG = AC$ (with $C$ between $B$ and $G$). Prove that the circumference through $D$, $F$ and $G$ is tangent to $BG$.

We are given a positive integer $ r$ and a rectangular board $ ABCD$ with dimensions $ AB \equal{} 20, BC \equal{} 12$. The rectangle is divided into a grid of $ 20 \times 12$ unit squares. The following moves are permitted on the board: one can move from one square to another only if the distance between the centers of the two squares is $ \sqrt {r}$. The task is to find a sequence of moves leading from the square with $ A$ as a vertex to the square with $ B$ as a vertex. (a) Show that the task cannot be done if $ r$ is divisible by 2 or 3. (b) Prove that the task is possible when $ r \equal{} 73$. (c) Can the task be done when $ r \equal{} 97$?

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.

Consider a cuboid$ ABCDA'B'C'D'$ (where $ABCD$ is a rectangle and $AA' \parallel BB' \parallel CC' \parallel DD'$) with $AA' = d$, $\angle ABD' = \alpha, \angle A'D'B = \beta$. Express the lengths x = $AB$, $y = BC$ in terms of $d$ and (acute) angles $\alpha, \beta$. Discuss condition of solvability.

The length of a rectangle is $ 5$ inches and its width is less than $ 4$ inches. The rectangle is folded so that two diagonally opposite vertices coincide. If the length of the crease is $ \sqrt {6}$, then the width is: $ \textbf{(A)}\ \sqrt {2} \qquad \textbf{(B)}\ \sqrt {3} \qquad \textbf{(C)}\ 2 \qquad \textbf{(D)}\ \sqrt {5} \qquad \textbf{(E)}\ \sqrt {\frac {11}{2}}$

The function $F$ is a one-to-one transformation of the plane into itself that maps rectangles into rectangles (rectangles are closed; continuity is not assumed). Prove that $F$ maps squares into squares.

A rectangle $ABCD$ and its circumcircle are given. Let $E$ be an arbitrary point on the minor arc $BC$. The tangent to the circle at $B$ meets $CE$ at point $G$. The segments $AE$ and $BD$ meet at point $K$. Prove that $GK$ and $AD$ are perpendicular.

Let $T$ be the answer to question $18$. Rectangle $ZOMR$ has $ZO = 2T$ and $ZR = T$. Point $B$ lies on segment $ZO$, $O'$ lies on segment $OM$, and $E$ lies on segment $RM$ such that $BR = BE = EO'$, and $\angle BEO' = 90^o$. Compute $2(ZO + O'M + ER)$. PS. You had better calculate it in terms of $T$.