Let $A=(a_{jk})$ be a $10\times 10$ array of positive real numbers such that the sum of numbers in row as well as in each column is $1$. Show that there exists $j<k$ and $l<m$ such that \[a_{jl}a_{km}+a_{jm}a_{kl}\ge \frac{1}{50}\]

From a $n\times (n-1)$ rectangle divided into unit squares, we cut the [i]corner[/i], which consists of the first row and the first column. (that is, the corner has $2n-2$ unit squares). For the following, when we say [i]corner[/i] we reffer to the above definition, along with rotations and symmetry. Consider an infinite lattice of unit squares. We will color the squares with $k$ colors, such that for any corner, the squares in that corner are coloured differently (that means that there are no squares coloured with the same colour). Find out the minimum of $k$. [i]Proposed by S. Berlov[/i]

Consider a rhombus $ABCD$ with center $O$. A point $P$ is given inside the rhombus, but not situated on the diagonals. Let $M,N,Q,R$ be the projections of $P$ on the sides $(AB), (BC), (CD), (DA)$, respectively. The perpendicular bisectors of the segments $MN$ and $RQ$ meet at $S$ and the perpendicular bisectors of the segments $NQ$ and $MR$ meet at $T$. Prove that $P, S, T$ and $O$ are the vertices of a rectangle.

We call a tiling of an $m\times$ n rectangle with arabos (see figure below) [i]regular[/i] if there is no sub-rectangle which is tiled with arabos. Prove that if for some $m$ and $n$ there exists a [i]regular[/i] tiling of the $m\times n$ rectangle then there exists a [i]regular[/i] tiling also for the $2m \times 2n$ rectangle. [img]https://cdn.artofproblemsolving.com/attachments/1/1/2ab41cc5107a21760392253ed52d9e4ecb22d1.png[/img]

5. Given the point T in rectangle ABCD, the distances from T to A,B,C is 15,20,25. Find the distance from T to D.

Let $S$ be a square of side length $1$. Two points are chosen independently at random on the sides of $S$. The probability that the straight-line distance between the points is at least $\tfrac12$ is $\tfrac{a-b\pi}c$, where $a$, $b$, and $c$ are positive integers and $\gcd(a,b,c)=1$. What is $a+b+c$? $\textbf{(A) }59\qquad\textbf{(B) }60\qquad\textbf{(C) }61\qquad\textbf{(D) }62\qquad\textbf{(E) }63$

The rectangle is cut into $10$ squares as shown in the figure on the right. Find its sides if the side of the smallest square is $3$.[img]https://cdn.artofproblemsolving.com/attachments/e/5/1fe3a0e41b2d3182338a557d3d44ff5ef9385d.png[/img]

Cells of a $2000\times2000$ board are colored according to the following rules: 1)At any moment a cell can be colored, if none of its neighbors are colored 2)At any moment a $1\times2$ rectangle can be colored, if exactly two of its neighbors are colored. 3)At any moment a $2\times2$ squared can be colored, if 8 of its neighbors are colored (Two cells are considered to be neighboring, if they share a common side). Can the entire $2000\times2000$ board be colored? [I]Proposed by K. Kohas[/i]

Let $ABCD$ be a rectangle with diagonals of length $10.$ Let $P$ be the midpoint of $\overline{AD},$ $S$ be the midpoint of $\overline{BC},$ and $T$ be the midpoint of $\overline{CD}.$ Points $Q$ and $R$ are chosen on $\overline{AB}$ such that $AP=AQ$ and $BR=BS,$ and minor arcs $\widehat{PQ}$ and $\widehat{RS}$ centered at $A$ and $B,$ respectively, are drawn. Circle $\omega$ is tangent to $\overline{CD}$ at $T$ and externally tangent to $\widehat{PQ}$ and $\widehat{RS}.$ Suppose that the radius of $\omega$ is $\tfrac{43}{18}.$ Then the sum of all possible values of the area of $ABCD$ can be written in the form $\tfrac{a+b\sqrt{c}}{d},$ where $a,\ b,\ c,$ and $d$ are positive integers, $b$ and $d$ are relatively prime, and $c$ is prime. Find $a+b+c+d.$

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.

Suppose from $(m+2)\times(n+2)$ rectangle we cut $4$, $1\times1$ corners. Now on first and last row first and last columns we write $2(m+n)$ real numbers. Prove we can fill the interior $m\times n$ rectangle with real numbers that every number is average of it's $4$ neighbors.

A rectangle has sides of length $\sin x$ and $\cos x$ for some $x$. What is the largest possible area of such a rectangle?

Every point in the plane is coloured one of seven distinct colours. Is there an inscribed trapezoid whose vertices are all of the same colour?

Let $ABCD$ be a rectangle with sides $AB,BC,CD$ and $DA$. Let $K,L$ be the midpoints of the sides $BC,DA$ respectivily. The perpendicular from $B$ to $AK$ hits $CL$ at $M$. Find $$\frac{[ABKM]}{[ABCL]}$$

Let $ n $ and $ r $ be positive integers and $ A $ be a set of lattice points in the plane such that any open disc of radius $ r $ contains a point of $ A $. Show that for any coloring of the points of $ A $ in $ n $ colors there exists four points of the same color which are the vertices of a rectangle.

A rectangle that is inscribed in a larger rectangle (with one vertex on each side) is called [i]unstuck[/i] if it is possible to rotate (however slightly) the smaller rectangle about its center within the confines of the larger. Of all the rectangles that can be inscribed unstuck in a 6 by 8 rectangle, the smallest perimeter has the form $\sqrt{N}$, for a positive integer $N$. Find $N$.

[4] Let $ABCDEF$ be a regular hexagon. Let $P$ be the circle inscribed in $\triangle{BDF}$. Find the ratio of the area of circle $P$ to the area of rectangle $ABDE$.

$ABCD$ 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 $P$.

From each vertex of triangle $ABC$ we draw 3 arbitary parrallell lines, and from each vertex we draw a perpendicular to these lines. There are 3 rectangles that one of their diagnals is triangle's side. We draw their other diagnals and call them $\ell_1$, $\ell_2$ and $\ell_3$. a) Prove that $\ell_1$, $\ell_2$ and $\ell_3$ are concurrent at a point $P$. b) Find the locus of $P$ as we move the 3 arbitary lines.

A square is cut into several rectangles, none of which is a square, so that the sides of each rectangle are parallel to the sides of the square. For each rectangle with sides $a, b,a<b$, compute the ratio $a/b$. Prove that sum of these ratios is at least $1$.

Two tangents $AT$ and $BT$ touch a circle at $A$ and $B$, respectively, and meet perpendicularly at $T$. $Q$ is on $AT$, $S$ is on $BT$, and $R$ is on the circle, so that $QRST$ is a rectangle with $QT = 8$ and $ST = 9$. Determine the radius of the circle.

[asy] //rectangles above problem statement size(15cm); for(int i=0;i<5;++i){ draw((6*i-14,-1.2)--(6*i-14,1.2)--(6*i-10,1.2)--(6*i-10,-1.2)--cycle); } label("$A$", (-12,2.25)); label("$B$", (-6,2.25)); label("$C$", (0,2.25)); label("$D$", (6,2.25)); label("$E$", (12,2.25)); //top numbers label("$1$", (-12,1.25),dir(-90)); label("$0$", (-6,1.25),dir(-90)); label("$8$", (0,1.25),dir(-90)); label("$5$", (6,1.25),dir(-90)); label("$2$", (12,1.25),dir(-90)); //bottom numbers label("$9$", (-12,-1.25),dir(90)); label("$6$", (-6,-1.25),dir(90)); label("$2$", (0,-1.25),dir(90)); label("$8$", (6,-1.25),dir(90)); label("$0$", (12,-1.25),dir(90)); //left numbers label("$4$", (-14,0),dir(0)); label("$1$", (-8,0),dir(0)); label("$3$", (-2,0),dir(0)); label("$7$", (4,0),dir(0)); label("$9$", (10,0),dir(0)); //right numbers label("$6$", (-10,0),dir(180)); label("$3$", (-4,0),dir(180)); label("$5$", (2,0),dir(180)); label("$4$", (8,0),dir(180)); label("$7$", (14,0),dir(180)); [/asy] Each of the sides of the five congruent rectangles is labeled with an integer, as shown above. These five rectangles are placed, without rotating or reflecting, in positions $I$ through $V$ so that the labels on coincident sides are equal. [asy] //diagram below problem statement size(7cm); for(int i=-3;i<=1;i+=2){ for(int j=-1;j<=0;++j){ if(i==1 && j==-1) continue; draw((i,j)--(i+2,j)--(i+2,j-1)--(i,j-1)--cycle); }} label("$I$",(-2,-0.5)); label("$II$",(0,-0.5)); label("$III$",(2,-0.5)); label("$IV$",(-2,-1.5)); label("$V$",(0,-1.5)); [/asy] Which of the rectangles is in position $I$? $\textbf{(A)} \ A \qquad \textbf{(B)} \ B \qquad \textbf{(C)} \ C \qquad \textbf{(D)} \ D \qquad \textbf{(E)} \ E$

Suppose from $(m+2)\times(n+2)$ rectangle we cut $4$, $1\times1$ corners. Now on first and last row first and last columns we write $2(m+n)$ real numbers. Prove we can fill the interior $m\times n$ rectangle with real numbers that every number is average of it's $4$ neighbors.

Andy has 2010 square tiles, each of which has a side length of one unit. He plans to arrange the tiles in an m x n rectangle, where mn = 2010. Compute the sum of the perimeters of all of the different possible rectangles he can make. Two rectangles are considered to be the same if one can be rotated to become the other, so, for instance, a 1 x 2010 rectangle is considered to be the same as a 2010 x 1 rectangle.

Find the locus of the center of a rectangle, whose four vertices lies on the sides of a given triangle.