In trapezoid $ABCD$, $AD$ is perpendicular to $DC$, $AD=AB=3$, and $DC=6$. In addition, E is on $DC$, and $BE$ is parallel to $AD$. Find the area of $\Delta BEC$. [asy] defaultpen(linewidth(0.7)); pair A=(0,3), B=(3,3), C=(6,0), D=origin, E=(3,0); draw(E--B--C--D--A--B); draw(rightanglemark(A, D, C)); label("$A$", A, NW); label("$B$", B, NW); label("$C$", C, SE); label("$D$", D, SW); label("$E$", E, NW); label("$3$", A--D, W); label("$3$", A--B, N); label("$6$", E, S);[/asy] $\textbf{(A)} \: 3\qquad \textbf{(B)} \: 4.5\qquad \textbf{(C)} \: 6\qquad \textbf{(D)} \: 9\qquad \textbf{(E)} \: 18\qquad $

In rectangular coordinate system, the equation $\frac{|x+y|}{2a}+\frac{|x-y|}{2b}=1$ ($a,b$ are different positive numbers) refers to $\text{(A)}$ a triangle $\text{(B)}$ a square $\text{(C)}$ rectangle, not square $\text{(D)}$ rhombus, not square

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]

[url=https://artofproblemsolving.com/community/c675693][b]Macedonia JBMO TST 2017[/b][/url] [url=http://artofproblemsolving.com/community/c6h1663908p10569198][b]Problem 1[/b][/url]. Let $p$ be a prime number such that $3p+10$ is a sum of squares of six consecutive positive integers. Prove that $p-7$ is divisible by $36$. [url=http://artofproblemsolving.com/community/c6h1663916p10569261][b]Problem 2[/b][/url]. In the triangle $ABC$, the medians $AA_1$, $BB_1$, and $CC_1$ are concurrent at a point $T$ such that $BA_1=TA_1$. The points $C_2$ and $B_2$ are chosen on the extensions of $CC_1$ and $BB_2$, respectively, such that $$C_1C_2 = \frac{CC_1}{3} \quad \text{and} \quad B_1B_2 = \frac{BB_1}{3}.$$ Show that $TB_2AC_2$ is a rectangle. [url=http://artofproblemsolving.com/community/c6h1663918p10569305][b]Problem 3[/b][/url]. Let $x,y,z$ be positive reals such that $xyz=1$. Show that $$\frac{x^2+y^2+z}{x^2+2} + \frac{y^2+z^2+x}{y^2+2} + \frac{z^2+x^2+y}{z^2+2} \geq 3.$$ When does equality happen? [url=http://artofproblemsolving.com/community/c6h1663920p10569326][b]Problem 4[/b][/url]. In triangle $ABC$, the points $X$ and $Y$ are chosen on the arc $BC$ of the circumscribed circle of $ABC$ that doesn't contain $A$ so that $\measuredangle BAX = \measuredangle CAY$. Let $M$ be the midpoint of the segment $AX$. Show that $$BM + CM > AY.$$ [url=http://artofproblemsolving.com/community/c6h1663922p10569370][b]Problem 5[/b][/url]. Find all the positive integers $n$ so that $n$ has the same number of digits as its number of different prime factors and the sum of these different prime factors is equal to the sum of exponents of all these primes in factorization of $n$.

A large collection of congruent right triangles is given, each with side length 3,4,5. Find the maximal number of such triangles you can place inside a 20x20 square, with no two triangles intersecting (in their interiors).

Consider a $m\times n$ rectangular board consisting of $mn$ unit squares. Two of its unit squares are called [i]adjacent[/i] if they have a common edge, and a [i]path[/i] is a sequence of unit squares in which any two consecutive squares are adjacent. Two parths are called [i]non-intersecting[/i] if they don't share any common squares. Each unit square of the rectangular board can be colored black or white. We speak of a [i]coloring[/i] of the board if all its $mn$ unit squares are colored. Let $N$ be the number of colorings of the board such that there exists at least one black path from the left edge of the board to its right edge. Let $M$ be the number of colorings of the board for which there exist at least two non-intersecting black paths from the left edge of the board to its right edge. Prove that $N^{2}\geq M\cdot 2^{mn}$.

In a Cartesian coordinate plane, call a rectangle $standard$ if all of its sides are parallel to the $x$- and $y$- axes, and call a set of points $nice$ if no two of them have the same $x$- or $y$- coordinate. First, Bert chooses a nice set $B$ of $2016$ points in the coordinate plane. To mess with Bert, Ernie then chooses a set $E$ of $n$ points in the coordinate plane such that $B\cup E$ is a nice set with $2016+n$ points. Bert returns and then miraculously notices that there does not exist a standard rectangle that contains at least two points in $B$ and no points in $E$ in its interior. For a given nice set $B$ that Bert chooses, define $f(B)$ as the smallest positive integer $n$ such that Ernie can find a nice set $E$ of size $n$ with the aforementioned properties. Help Bert determine the minimum and maximum possible values of $f(B)$. [i]Yannick Yao[/i]

Dirock has a very neat rectangular backyard that can be represented as a $32\times 32$ grid of unit squares. The rows and columns are each numbered $1,2,\ldots, 32$. Dirock is very fond of rocks, and places a rock in every grid square whose row and column number are both divisible by $3$. Dirock would like to build a rectangular fence with vertices at the centers of grid squares and sides parallel to the sides of the yard such that [list] [*] The fence does not pass through any grid squares containing rocks; [*] The interior of the fence contains exactly 5 rocks. [/list] In how many ways can this be done? [i]Ray Li[/i]

A rectangular piece of paper $ABCD$ has sides of lengths $AB = 1$, $BC = 2$. The rectangle is folded in half such that $AD$ coincides with $BC$ and $EF$ is the folding line. Then fold the paper along a line $BM$ such that the corner $A$ falls on line $EF$. How large, in degrees, is $\angle ABM$? [asy] size(180); pathpen = rgb(0,0,0.6)+linewidth(1); pointpen = black+linewidth(3); pointfontpen = fontsize(10); pen dd = rgb(0,0,0.6) + linewidth(0.7) + linetype("4 4"), dr = rgb(0.8,0,0), dg = rgb(0,0.6,0), db = rgb(0,0,0.6)+linewidth(1); pair A=(0,1), B=(0,0), C=(2,0), D=(2,1), E=A/2, F=(2,.5), M=(1/3^.5,1), N=reflect(B,M)*A; D(B--M--D("N",N,NE)--B--D("C",C,SE)--D("D",D,NE)--M); D(D("M",M,plain.N)--D("A",A,NW)--D("B",B,SW),dd); D(D("E",E,W)--D("F",F,plain.E),dd); [/asy]

Two congruent rectangles are located as shown in the figure. Find the area of the shaded part. [img]https://cdn.artofproblemsolving.com/attachments/2/e/10b164535ab5b3a3b98ce1a0b84892cd11d76f.png[/img]

For each positive integer $m$, we define $L_m$ as the figure that is obtained by overlapping two $1 \times m$ and $m \times 1$ rectangles in such a way that they coincide at the $1 \times 1$ square at their ends, as shown in the figure. [asy] pair h = (1, 0), v = (0, 1), o = (0, 0); for(int i = 1; i < 5; ++i) { o = (i*i/2 + i, 0); draw(o -- o + i*v -- o + i*v + h -- o + h + v -- o + i*h + v -- o + i*h -- cycle); string s = "$L_" + (string)(i) + "$"; label(s, o + ((i / 2), -1)); for(int j = 1; j < i; ++j) { draw(o + j*v -- o + j*v + h); draw(o + j*h -- o + j*h + v); } } label("...", (18, 0.5)); [/asy] Using some figures $L_{m_1}, L_{m_2}, \dots, L_{m_k}$, we cover an $n \times n$ board completely, in such a way that the edges of the figure coincide with lines in the board. Among all possible coverings of the board, find the minimal possible value of $m_1 + m_2 + \dots + m_k$. Note: In covering the board, the figures may be rotated or reflected, and they may overlap or not be completely contained within the board.

We divide up the plane into disjoint regions using a circle, a rectangle and a triangle. What is the greatest number of regions that we can get?

$12.$ 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 $r_1+r_2+r_3 ?$

Given three congruent rectangles in the space. Their centers coincide, but the planes they lie in are mutually perpendicular. For any two of the three rectangles, the line of intersection of the planes of these two rectangles contains one midparallel of one rectangle and one midparallel of the other rectangle, and these two midparallels have different lengths. Consider the convex polyhedron whose vertices are the vertices of the rectangles. [b]a.)[/b] What is the volume of this polyhedron ? [b]b.)[/b] Can this polyhedron turn out to be a regular polyhedron ? If yes, what is the condition for this polyhedron to be regular ?

You have a paper pentagon, $ABCDE$, such that $AB = BC = 3$ cm, $CD = DE= 5$ cm, $EA = 4$ cm, $\angle ABC = 100^o$ ,$ \angle CDE = 80^o$. You have to divide the pentagon into four triangles, by three straight cuts, so that with the four triangles assemble a rectangle, without gaps or overlays. (The triangles can be rotated and / or turned around.)

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}\%$

$ ABCD$ is a rectangle (see the accompanying diagram) with $ P$ any point on $ \overline{AB}$. $ \overline{PS} \perp \overline{BD}$ and $ \overline{PR} \perp \overline{AC}$. $ \overline{AF} \perp \overline{BD}$ and $ \overline{PQ} \perp \overline{AF}$. Then $ PR \plus{} PS$ is equal to: [asy]defaultpen(linewidth(.8pt)); unitsize(3cm); pair D = origin; pair C = (2,0); pair B = (2,1); pair A = (0,1); pair P = waypoint(B--A,0.2); pair S = foot(P,D,B); pair R = foot(P,A,C); pair F = foot(A,D,B); pair Q = foot(P,A,F); pair T = intersectionpoint(P--Q,A--C); pair X = intersectionpoint(A--C,B--D); draw(A--B--C--D--cycle); draw(A--C); draw(B--D); draw(P--S); draw(A--F); draw(P--R); draw(P--Q); label("$A$",A,NW); label("$B$",B,NE); label("$C$",C,SE); label("$D$",D,SW); label("$P$",P,N); label("$S$",S,SE); label("$T$",T,N); label("$E$",X,SW+SE); label("$R$",R,SW); label("$F$",F,SE); label("$Q$",Q,SW);[/asy] $ \textbf{(A)}\ PQ\qquad \textbf{(B)}\ AE\qquad \textbf{(C)}\ PT \plus{} AT\qquad \textbf{(D)}\ AF\qquad \textbf{(E)}\ EF$

A decorative window is made up of a rectangle with semicircles at either end. The ratio of $AD$ to $AB$ is $3:2$. And $AB$ is 30 inches. What is the ratio of the area of the rectangle to the combined area of the semicircle. [asy] import graph; size(5cm); real lsf=0; pen dps=linewidth(0.7)+fontsize(8); defaultpen(dps); pen ds=black; real xmin=-4.27,xmax=14.73,ymin=-3.22,ymax=6.8; draw((0,4)--(0,0)); draw((0,0)--(2.5,0)); draw((2.5,0)--(2.5,4)); draw((2.5,4)--(0,4)); draw(shift((1.25,4))*xscale(1.25)*yscale(1.25)*arc((0,0),1,0,180)); draw(shift((1.25,0))*xscale(1.25)*yscale(1.25)*arc((0,0),1,-180,0)); dot((0,0),ds); label("$A$",(-0.26,-0.23),NE*lsf); dot((2.5,0),ds); label("$B$",(2.61,-0.26),NE*lsf); dot((0,4),ds); label("$D$",(-0.26,4.02),NE*lsf); dot((2.5,4),ds); label("$C$",(2.64,3.98),NE*lsf); clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle);[/asy] $ \textbf{(A)}\ 2:3 \qquad\textbf{(B)}\ 3:2\qquad\textbf{(C)}\ 6:\pi \qquad\textbf{(D)}\ 9: \pi \qquad\textbf{(E)}\ 30 : \pi$

Rectangle $HOMF$ has $HO=11$ and $OM=5$. Triangle $ABC$ has orthocenter $H$ and circumcenter $O$. $M$ is the midpoint of $BC$ and altitude $AF$ meets $BC$ at $F$. Find the length of $BC$.

Rectangle $ABCD$ has sides $\overline {AB}$ of length 4 and $\overline {CB}$ of length 3. Divide $\overline {AB}$ into 168 congruent segments with points $A=P_0, P_1, \ldots, P_{168}=B$, and divide $\overline {CB}$ into 168 congruent segments with points $C=Q_0, Q_1, \ldots, Q_{168}=B$. For $1 \le k \le 167$, draw the segments $\overline {P_kQ_k}$. Repeat this construction on the sides $\overline {AD}$ and $\overline {CD}$, and then draw the diagonal $\overline {AC}$. Find the sum of the lengths of the 335 parallel segments drawn.

A triangle has side lengths 10, 10, and 12. A rectangle has width 4 and area equal to the area of the triangle. What is the perimeter of this rectangle? $ \textbf{(A)}\ 16\qquad\textbf{(B)}\ 24\qquad\textbf{(C)}\ 28\qquad\textbf{(D)}\ 32\qquad\textbf{(E)}\ 36$

[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 $ABCD$ be a rectangle with $AB \ne BC$ and the center the point $O$. Perpendicular from $O$ on $BD$ intersects lines $AB$ and $BC$ in points $E$ and $F$ respectively. Points $M$ and $N$ are midpoints of segments $[CD]$ and $[AD]$ respectively. Prove that $FM \perp EN$ .

A rectangle $ABCD$ is given. Segment $DK$ is equal to $BD$ and lies on the half-line $DC$. $M$ is the midpoint of $BK$. Prove that $AM$ is the angle bisector of $\angle BAC$. [i]Proposed by S. Berlov[/i]

We put pebbles on some unit squares of a $2013 \times 2013$ chessboard such that every unit square contains at most one pebble. Determine the minimum number of pebbles on the chessboard, if each $19\times 19$ square formed by unit squares contains at least $21$ pebbles.