On a large chessboard $2n$ of its $1 \times 1$ squares have been marked such thar the rook (which moves only horizontally or vertically) can visit all the marked squares without jumpin over any unmarked ones. Prove that the figure consisting of all the marked squares can be cut into rectangles. (A Shapovalov)

In a matrix $2n \times 2n$, $n \in N$, are $4n^2$ real numbers with a sum equal zero. The absolute value of each of these numbers is not greater than $1$. Prove that the absolute value of a sum of all the numbers from one column or a row doesn't exceed $n$.

Define an integer $n \ge 0$ to be \textit{two-far} if there exist integers $a$ and $b$ such that $a$, $b$, and $n + a + b$ are all powers of two. If $N$ is the number of two-far integers less than 2048, find the remainder when $N$ is divided by 100. Let $T = TNYWR$. Let $CMU$ be a triangle with $CM=13$, $MU=14$, and $UC=15$. Rectangle $WEAN$ is inscribed in $\triangle CMU$ with points $W$ and $E$ on $\overline{MU}$, point $A$ on $\overline{CU}$, and point $N$ on $\overline{CM}$. If the area of $WEAN$ is $T$, what is its perimeter?

Let $ABCD$ be a fixed convex quadrilateral. Say a point $K$ is [i]pastanaga[/i] if there's a rectangle $PQRS$ centered at $K$ such that $A\in PQ, B\in QR, C\in RS, D\in SP$. Prove there exists a circle $\omega$ depending only on $ABCD$ that contains all pastanaga points.

Determine the maximum possible number of points you can place in a rectangle with lengths $14$ and $28$ such that any two of those points are more than $10$ apart from each other.

A convex, closed figure lies inside a given circle. The figure is seen from every point of the circumference at a right angle (that is, the two rays drawn from the point and supporting the convex figure are perpendicular). Prove that the center of the circle is a center of symmetry of the figure.

How many rectangles are there in the diagram below such that the sum of the numbers within the rectangle is a multiple of 7? [asy] int n; n=0; for (int i=0; i<=7;++i) { draw((i,0)--(i,7)); draw((0,i)--(7,i)); for (int a=0; a<=7;++a) { if ((a != 7)&&(i != 7)) { n=n+1; label((string) n,(a,i),(2,2)); } } } [/asy]

Given is trapezoid $ABCD$, $M$ and $N$ being the midpoints of the bases of $AD$ and $BC$, respectively. a) Prove that the trapezoid is isosceles if it is known that the intersection point of perpendicular bisectors of the lateral sides belongs to the segment $MN$. b) Does the statement of point a) remain true if it is only known that the intersection point of perpendicular bisectors of the lateral sides belongs to the line $MN$?

$D$ is an arbitrary point lying on side $BC$ of $\triangle{ABC}$. Circle $\omega_1$ is tangent to segments $AD$ , $BD$ and the circumcircle of $\triangle{ABC}$ and circle $\omega_2$ is tangent to segments $AD$ , $CD$ and the circumcircle of $\triangle{ABC}$. Let $X$ and $Y$ be the intersection points of $\omega_1$ and $\omega_2$ with $BC$ respectively and take $M$ as the midpoint of $XY$. Let $T$ be the midpoint of arc $BC$ which does not contain $A$. If $I$ is the incenter of $\triangle{ABC}$, prove that $TM$ goes through the midpoint of $ID$.

A natural number $k$ is said $n$-squared if by colouring the squares of a $2n \times k$ chessboard, in any manner, with $n$ different colours, we can find $4$ separate unit squares of the same colour, the centers of which are vertices of a rectangle having sides parallel to the sides of the board. Determine, in function of $n$, the smallest natural $k$ that is $n$-squared.

Let $ABCD$ be a rectangle of sides $AB = 4$ and $BC = 3$. The perpendicular on the diagonal $BD$ drawn from $A$ cuts $BD$ at point $H$. We call $M$ the midpoint of $BH$ and $N$ the midpoint of $CD$. Calculate the measure of the segment $MN$.

In a table with $n$ rows and $2n$ columns where $n$ is a fixed positive integer, we write either zero or one into each cell so that each row has $n$ zeros and $n$ ones. For $1 \le k \le n$ and $1 \le i \le n$, we define $a_{k,i}$ so that the $i^{\text{th}}$ zero in the $k^{\text{th}}$ row is the $a_{k,i}^{\text{th}}$ column. Let $\mathcal F$ be the set of such tables with $a_{1,i} \ge a_{2,i} \ge \dots \ge a_{n,i}$ for every $i$ with $1 \le i \le n$. We associate another $n \times 2n$ table $f(C)$ from $C \in \mathcal F$ as follows: for the $k^{\text{th}}$ row of $f(C)$, we write $n$ ones in the columns $a_{n,k}-k+1, a_{n-1,k}-k+2, \dots, a_{1,k}-k+n$ (and we write zeros in the other cells in the row). (a) Show that $f(C) \in \mathcal F$. (b) Show that $f(f(f(f(f(f(C)))))) = C$ for any $C \in \mathcal F$.

[size=130][b]Higher Secondary: 2011[/b] [/size] Time: 4 Hours [b]Problem 1:[/b] Prove that for any non-negative integer $n$ the numbers $1, 2, 3, ..., 4n$ can be divided in tow mutually exclusive classes with equal number of members so that the sum of numbers of each class is equal. http://matholympiad.org.bd/forum/viewtopic.php?f=13&t=709 [b]Problem 2:[/b] In the first round of a chess tournament, each player plays against every other player exactly once. A player gets $3, 1$ or $-1$ points respectively for winning, drawing or losing a match. After the end of the first round, it is found that the sum of the scores of all the players is $90$. How many players were there in the tournament? http://matholympiad.org.bd/forum/viewtopic.php?f=13&t=708 [b]Problem 3:[/b] $E$ is the midpoint of side $BC$ of rectangle $ABCD$. $A$ point $X$ is chosen on $BE$. $DX$ meets extended $AB$ at $P$. Find the position of $X$ so that the sum of the areas of $\triangle BPX$ and $\triangle DXC$ is maximum with proof. http://matholympiad.org.bd/forum/viewtopic.php?f=13&t=683 [b]Problem 4:[/b] Which one is larger 2011! or, $(1006)^{2011}$? Justify your answer. http://matholympiad.org.bd/forum/viewtopic.php?f=13&t=707 [b]Problem 5:[/b] In a scalene triangle $ABC$ with $\angle A = 90^{\circ}$, the tangent line at $A$ to its circumcircle meets line $BC$ at $M$ and the incircle touches $AC$ at $S$ and $AB$ at $R$. The lines $RS$ and $BC$ intersect at $N$ while the lines $AM$ and $SR$ intersect at $U$. Prove that the triangle $UMN$ is isosceles. http://matholympiad.org.bd/forum/viewtopic.php?f=13&t=706 [b]Problem 6:[/b] $p$ is a prime and sum of the numbers from $1$ to $p$ is divisible by all primes less or equal to $p$. Find the value of $p$ with proof. http://matholympiad.org.bd/forum/viewtopic.php?f=13&t=693 [b]Problem 7:[/b] Consider a group of $n > 1$ people. Any two people of this group are related by mutual friendship or mutual enmity. Any friend of a friend and any enemy of an enemy is a friend. If $A$ and $B$ are friends/enemies then we count it as $1$ [b]friendship/enmity[/b]. It is observed that the number of friendships and number of enmities are equal in the group. Find all possible values of $n$. http://matholympiad.org.bd/forum/viewtopic.php?f=13&t=694 [b]Problem 8:[/b] $ABC$ is a right angled triangle with $\angle A = 90^{\circ}$ and $D$ be the midpoint of $BC$. A point $F$ is chosen on $AB$. $CA$ and $DF$ meet at $G$ and $GB \parallel AD$. $CF$ and $AD$ meet at $O$ and $AF = FO$. $GO$ meets $BC$ at $R$. Find the sides of $ABC$ if the area of $GDR$ is $\dfrac{2}{\sqrt{15}}$ http://matholympiad.org.bd/forum/viewtopic.php?f=13&t=704 [b]Problem 9:[/b] The repeat of a natural number is obtained by writing it twice in a row (for example, the repeat of $123$ is $123123$). Find a positive integer (if any) whose repeat is a perfect square. http://matholympiad.org.bd/forum/viewtopic.php?f=13&t=703 [b]Problem 10:[/b] Consider a square grid with $n$ rows and $n$ columns, where $n$ is odd (similar to a chessboard). Among the $n^2$ squares of the grid, $p$ are black and the others are white. The number of black squares is maximized while their arrangement is such that horizontally, vertically or diagonally neighboring black squares are separated by at least one white square between them. Show that there are infinitely many triplets of integers $(p, q, n)$ so that the number of white squares is $q^2$. http://matholympiad.org.bd/forum/viewtopic.php?f=13&t=702 The problems of the Junior categories are available in [url=http://matholympiad.org.bd/forum/]BdMO Online forum[/url]: http://matholympiad.org.bd/forum/viewtopic.php?f=25&t=678

Let $ R$ be a rectangle that is the union of a finite number of rectangles $ R_i,$ $ 1 \leq i \leq n,$ satisfying the following conditions: [b](i)[/b] The sides of every rectangle $ R_i$ are parallel to the sides of $ R.$ [b](ii)[/b] The interiors of any two different rectangles $ R_i$ are disjoint. [b](iii)[/b] Each rectangle $ R_i$ has at least one side of integral length. Prove that $ R$ has at least one side of integral length. [i]Variant:[/i] Same problem but with rectangular parallelepipeds having at least one integral side.

A square with integer side length is cut into 10 squares, all of which have integer side length and at least 8 of which have area 1. What is the smallest possible value of the length of the side of the original square? $\textbf{(A)}\hspace{.05in}3 \qquad \textbf{(B)}\hspace{.05in}4 \qquad \textbf{(C)}\hspace{.05in}5 \qquad \textbf{(D)}\hspace{.05in}6 \qquad \textbf{(E)}\hspace{.05in}7 $

Four identical squares and one rectangle are placed together to form one large square as shown. The length of the rectangle is how many times as large as its width? [asy]unitsize(8mm); defaultpen(linewidth(.8pt)); draw(scale(4)*unitsquare); draw((0,3)--(4,3)); draw((1,3)--(1,4)); draw((2,3)--(2,4)); draw((3,3)--(3,4));[/asy]$ \textbf{(A)}\ \frac {5}{4} \qquad \textbf{(B)}\ \frac {4}{3} \qquad \textbf{(C)}\ \frac {3}{2} \qquad \textbf{(D)}\ 2 \qquad \textbf{(E)}\ 3$

If the base of a rectangle is increased by $ 10\%$ and the area is unchanged, then the altitude is decreased by: $ \textbf{(A)}\ 9\% \qquad\textbf{(B)}\ 10\% \qquad\textbf{(C)}\ 11\% \qquad\textbf{(D)}\ 11\frac {1}{9}\% \qquad\textbf{(E)}\ 9\frac {1}{11}\%$

If the perimeter of a rectangle is $ p$ and its diagonal is $ d$, the difference between the length and width of the rectangle is: $ \textbf{(A)}\ \frac {\sqrt {8d^2 \minus{} p^2}}{2} \qquad\textbf{(B)}\ \frac {\sqrt {8d^2 \plus{} p^2}}{2} \qquad\textbf{(C)}\ \frac {\sqrt {6d^2 \minus{} p^2}}{2}$ $ \textbf{(D)}\ \frac {\sqrt {6d^2 \plus{} p^2}}{2} \qquad\textbf{(E)}\ \frac {8d^2 \minus{} p^2}{4}$

We have a rectangle with it sides being a mirror.A light Ray enters from one of the corners of the rectangle and after being reflected several times enters to the opposite corner it started.Prove that at some time the light Ray passed the center of rectangle(Intersection of diagonals.)

$12.$ In a rectangle $ABCD$ $AB=8$ and $BC=20.$ Let $P$ be a point on $AD$ such that $\angle{BPC}=90^o.$ If $r_1,r_2,r_3.$ are the radii of the incircles of triangles $APB,$ $BPC,$ and $CPD.$ what is the value of $r_1+r_2+r_3 ?$

Show that for each integer $n > 0$, there is a polygon with vertices at lattice points and all sides parallel to the axes, which can be dissected into $1 \times 2$ (and / or $2 \times 1$) rectangles in exactly $n$ ways.

A polygon can be divided into 100 rectangles, but not into 99. Prove that it cannot be divided into 100 triangles. [i]A. Shapovalov[/i]

Let $ m,n\in \mathbb{N}^*$. Find the least $ n$ for which exists $ m$, such that rectangle $ (3m \plus{} 2)\times(4m \plus{} 3)$ can be covered with $ \dfrac{n(n \plus{} 1)}{2}$ squares, among which exist $ n$ squares of length $ 1$, $ n \minus{} 1$ of length $ 2$, $ ...$, $ 1$ square of length $ n$. For the found value of $ n$ give the example of covering.

A square-shaped pizza with side length $30$ cm is cut into pieces (not necessarily rectangular). All cuts are parallel to the sides, and the total length of the cuts is $240$ cm. Show that there is a piece whose area is at least $36$ cm$^2$

A rectangle \( m \times n \), where \( m \) and \( n \) are natural numbers strictly greater than 1, is partitioned into \( mn \) unit squares, each of which can be colored either black or white. An operation consists of changing the color of all the squares in a row or in a column to the opposite color. Is it possible that, although initially exactly one square is colored black and all the others are white, after a finite number of moves all squares have the same color?