Suppose $l(M, XYZ)$ is a Simson line of the triangle $XYZ$ that passes through $M$. Suppose $ABCDEF$ is a cyclic hexagon such that $l(A, BDF), l(B, ACE), l(D, ABF), l(E, ABC)$ intersect at a single point. Prove that $CDEF$ is a rectangle. [color=blue]Should the first sentence read: Suppose $l(M, XYZ)$ is a Simson line of the triangle $XYZ$ [u]with respect to[/u] $M$. ? Since it appears weird that a Simson line that passes a point is to be constructed. However, this is Unsolved after all, so I'm not sure.[/color]

Let $ S$ be any point on the circumscribed circle of $ PQR.$ Then the feet of the perpendiculars from S to the three sides of the triangle lie on the same straight line. Denote this line by $ l(S, PQR).$ Suppose that the hexagon $ ABCDEF$ is inscribed in a circle. Show that the four lines $ l(A,BDF),$ $ l(B,ACE),$ $ l(D,ABF),$ and $ l(E,ABC)$ intersect at one point if and only if $ CDEF$ is a rectangle.

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.

A rectangle with the side lengths $a$ and $b$ with $a <b$ should be placed in a right-angled coordinate system so that there is no point with integer coordinates in its interior or on its edge. Under what necessary and at the same time sufficient conditions for $a$ and $b$ is this possible?

What is the smallest possible area of a rectangle that can completely contain the shape formed by joining six squares of side length $8$ cm as shown below? [asy] size(5cm,0); draw((0,2)--(0,3)); draw((1,1)--(1,3)); draw((2,0)--(2,3)); draw((3,0)--(3,2)); draw((4,0)--(4,1)); draw((2,0)--(4,0)); draw((1,1)--(4,1)); draw((0,2)--(3,2)); draw((0,3)--(2,3)); [/asy] $\text{(A) }384\text{ cm}^2\qquad\text{(B) }576\text{ cm}^2\qquad\text{(C) }672\text{ cm}^2\qquad\text{(D) }768\text{ cm}^2\qquad\text{(E) }832\text{ cm}^2$

Andriyko has rectangle desk and a lot of stripes that lie parallel to sides of the desk. For every pair of stripes we can say that first of them is under second one. In desired configuration for every four stripes such that two of them are parallel to one side of the desk and two others are parallel to other side, one of them is under two other stripes that lie perpendicular to it. Prove that Andriyko can put stripes one by one such way that every next stripe lie upper than previous and get desired configuration. [i]Proposed by Denys Smirnov[/i]

Given a $ 1 \times 25$ rectangle divided into $ 25$ "boxes" ($ 1 \times 1$), is it possible to write integers $ 1$ to $ 25$ so that the sum of any two adjacent "boxes" is a perfect square?

A ball was launched on a rectangular billiard table at an angle of $45^o$ to one of the sides. Reflected from all sides (the angle of incidence is equal to the angle of reflection), he returned to his original position . It is known that one of the sides of the table has a length of one meter. Find the length of the second side. [img]https://cdn.artofproblemsolving.com/attachments/3/d/e0310ea910c7e3272396cd034421d1f3e88228.png[/img]

All of the rectangles in the figure below, which is drawn to scale, are similar to the enclosing rectangle. Each number represents the area of the rectangle. What is length $AB$? [img]https://cdn.artofproblemsolving.com/attachments/3/b/298cf96ec8fc90c438e4936a05c260170eda01.png[/img] $\textbf{(A) }4+4\sqrt5\qquad\textbf{(B) }10\sqrt2\qquad\textbf{(C) }5+5\sqrt5\qquad\textbf{(D) }10\sqrt[4]{8}\qquad\textbf{(E) }20$

Amy has divided a square into finitely many white and red rectangles, each with sides parallel to the sides of the square. Within each white rectangle, she writes down its width divided by its height. Within each red rectangle, she writes down its height divided by its width. Finally, she calculates $x$, the sum of these numbers. If the total area of white equals the total area of red, determine the minimum of $x$.

A $9\times 9$ board has a number written on each square: all squares in the first row have $1$, all squares in the second row have $2$, $\ldots$, all squares in the ninth row have $9$. We will call [i]special [/i] rectangle any rectangle of $2\times 3$ or $3\times 2$ or $4\times 5$ or $5\times 4$ on the board. The permitted operations are: $\bullet$ Simultaneously add $1$ to all the numbers located in a special rectangle. $\bullet$ Simultaneously subtract $1$ from all numbers located in a special rectangle. Demonstrate that it is possible to achieve, through a succession of permitted operations, that $80$ squares to have $0$ (zero). What number is left in the remaining box?

Let $ABCD$ be a rectangle, and let $\omega$ be a circular arc passing through the points $A$ and $C$. Let $\omega_{1}$ be the circle tangent to the lines $CD$ and $DA$ and to the circle $\omega$, and lying completely inside the rectangle $ABCD$. Similiarly let $\omega_{2}$ be the circle tangent to the lines $AB$ and $BC$ and to the circle $\omega$, and lying completely inside the rectangle $ABCD$. Denote by $r_{1}$ and $r_{2}$ the radii of the circles $\omega_{1}$ and $\omega_{2}$, respectively, and by $r$ the inradius of triangle $ABC$. [b](a)[/b] Prove that $r_{1}+r_{2}=2r$. [b](b)[/b] Prove that one of the two common internal tangents of the two circles $\omega_{1}$ and $\omega_{2}$ is parallel to the line $AC$ and has the length $\left|AB-AC\right|$.

A picture with an area of $160$ square inches is surrounded by a $2$ inch border. The picture with its border is a rectangle twice as long as it is wide. How many inches long is that rectangle?

Ali Barber, the carpet merchant, has a rectangular piece of carpet whose dimensions are unknown. Unfortunately, his tape measure is broken and he has no other measuring instruments. However, he finds that if he lays it flat on the floor of either of his storerooms, then each corner of the carpet touches a different wall of that room. He knows that the sides of the carpet are integral numbers of feet and that his two storerooms have the same (unknown) length, but widths of 38 feet and 50 feet respectively. What are the carpet dimensions?

The points $K$ and $M$ are the centres of the $AB$ and $CD$ sides of the convex quadrangle $ABCD$. The points $L$ and $M$ belong to two other sides and $KLMN$ is a rectangle. Prove that $KLMN$ area is a half of $ABCD$ area.

In the figure, $ ABCD$ is a $ 2\times 2$ square, $ E$ is the midpoint of $ \overline{AD}$, and $ F$ is on $ \overline{BE}$. If $ \overline{CF}$ is perpendicular to $ \overline{BE}$, then the area of quadrilateral $ CDEF$ is [asy]defaultpen(linewidth(.8pt)); dotfactor=4; pair A = (0,2); pair B = origin; pair C = (2,0); pair D = (2,2); pair E = midpoint(A--D); pair F = foot(C,B,E); dot(A);dot(B);dot(C);dot(D);dot(E);dot(F); label("$A$",A,N);label("$B$",B,S);label("$C$",C,S);label("$D$",D,N);label("$E$",E,N);label("$F$",F,NW); draw(A--B--C--D--cycle); draw(B--E); draw(C--F); draw(rightanglemark(B,F,C,4));[/asy]$ \textbf{(A)}\ 2\qquad \textbf{(B)}\ 3 \minus{} \frac {\sqrt {3}}{2}\qquad \textbf{(C)}\ \frac {11}{5}\qquad \textbf{(D)}\ \sqrt {5}\qquad \textbf{(E)}\ \frac {9}{4}$