A $(3n + 1) \times (3n + 1)$ table $(n \in \mathbb{N})$ is given. Prove that deleting any one of its squares yields a shape cuttable into pieces of the following form and its rotations: ''L" shape formed by cutting one square from a $2 \times 2$ squares.

The diagonals of a rectangle $ABCD$ meet at point $E$. A circle centered at $E$ lies inside the rectangle. Let $CF$, $DG$, $AH$ be the tangents to this circle from $C$, $D$, $A$; let $CF$ meet $DG$ at point $I$, $EI$ meet $AD$ at point $J$, and $AH$ meet $CF$ at point $L$. Prove that $LJ$ is perpendicular to $AD$.

The [i]cross[/i] of a convex $n$-gon is the quadratic mean of the lengths between the possible pairs of vertices. For example, the cross of a $3 \times 4$ rectangle is $\sqrt{ \dfrac{3^2 + 3^2 + 4^2 + 4^2 + 5^2 + 5^2}{6} } = \dfrac{5}{3} \sqrt{6}$. Suppose $S$ is a dodecagon ($12$-gon) inscribed in a unit circle. Find the greatest possible cross of $S$.

The fence consists of $25$ vertical bars. The heights of the bars are pairwise distinct positive integers from $1$ to $25$. The width of every bar is $1$. Find the maximum $S$ for which regardless of the order of the bars one can find a rectangle of area $S$ formed by the fence.

(a) A regular $4k$-gon is cut into parallelograms. Prove that among these there are at least $k$ rectangles. (b) Find the total area of the rectangles in (a) if the lengths of the sides of the $4k$-gon equal $a$. (VV Proizvolov, Moscow)

$C$, $C'$ are concentric circles with radii $R$, $R'$. A rectangle has two adjacent vertices on $C$ and the other two vertices on $C'$. Find its sides if its area is as large as possible.

Consider the set of points that are inside or within one unit of a rectangular parallelepiped (box) that measures 3 by 4 by 5 units. Given that the volume of this set is $(m + n \pi)/p$, where $m$, $n$, and $p$ are positive integers, and $n$ and $p$ are relatively prime, find $m + n + p$.

We say that a covering of a $m\times n$ rectangle with dominos has a wall if there exists a horizontal or vertical line that splits the rectangle into two smaller rectangles and doesn't cut any of the dominos. prove that if these three conditions are satisfied: [b]a)[/b] $mn$ is an even number [b]b)[/b] $m\ge 5$ and $n\ge 5$ [b]c)[/b] $(m,n)\neq(6,6)$ then we can cover the rectangle with dominos in such a way that we have no walls. (20 points)

Chim Tu has a large rectangular table. On it, there are finitely many pieces of paper with nonoverlapping interiors, each one in the shape of a convex polygon. At each step, Chim Tu is allowed to slide one piece of paper in a straight line such that its interior does not touch any other piece of paper during the slide. Can Chim Tu always slide all the pieces of paper off the table in finitely many steps?

Consider the rectangle in the coordinate plane with corners $(0, 0)$, $(16, 0)$, $(16, 4)$, and $(0, 4)$. For a constant $x_0 \in [0, 16]$, the curves \[ \{(x, y) : y = \sqrt{x} \,\text{ and }\, 0 \leq x \leq 16\} \quad \text{and} \quad \{(x_0, y) : 0 \leq y \leq 4\} \] partition this rectangle into four 2D regions. Over all choices of $x_0$, determine the smallest possible sum of the areas of the bottom-left and top-right 2D regions in this partition. (The bottom-left region is $\{(x, y) : 0 \leq x < x_0 \,\text{ and }\, 0 \leq y < \sqrt{x}\}$, and the top-right region is $\{(x, y) : x_0 < x \leq 16 \,\text{ and }\, \sqrt{x} < y \leq 4\}$.)

Six rectangles each with a common base width of $2$ have lengths of $1, 4, 9, 16, 25,$ and $36$. What is the sum of the areas of the six rectangles? $\textbf{(A) }91\qquad\textbf{(B) }93\qquad\textbf{(C) }162\qquad\textbf{(D) }182\qquad \textbf{(E) }202$

Let $ABC$ be a triangle with incenter $I$, and suppose the incircle is tangent to $CA$ and $AB$ at $E$ and $F$. Denote by $G$ and $H$ the reflections of $E$ and $F$ over $I$. Let $Q$ be the intersection of $BC$ with $GH$, and let $M$ be the midpoint of $BC$. Prove that $IQ$ and $IM$ are perpendicular.

A figure is an equiangular parallelogram if and only if it is a $ \textbf{(A)}\ \text{rectangle}\qquad\textbf{(B)}\ \text{regular polygon}\qquad\textbf{(C)}\ \text{rhombus}\qquad\textbf{(D)}\ \text{square}\qquad\textbf{(E)}\ \text{trapezoid} $

A non-square rectangle is cut into $N$ rectangles of various shapes and sizes. Prove that one can always cut each of these rectangles into two rectangles so that one can construct a square and rectangle, each figure consisting of $N$ pieces. [i](6 points)[/i]

The bases of a trapezoid have lengths 10 and 21, and the legs have lengths $\sqrt{34}$ and $3 \sqrt{5}$. What is the area of the trapezoid?

Triangle $ABC$ is given. Consider ellipse $ \Omega _1$, passes through $C$ with focuses in $A$ and $B$. Similarly define ellipses $ \Omega _2 , \Omega _3$ with focuses $B,C$ and $C,A$ respectively. Prove, that if all ellipses have common point $D$ then $A,B,C,D$ lies on the circle. Ellipse with focuses $X,Y$, passes through $Z$- locus of point $T$, such that $XT+YT=XZ+YZ$

A rectangle in xy Cartesian System is called latticed if all it's vertices have integer coordinates. a) Find a latticed rectangle of area $2013$, whose sides are not parallel to the axes. b) Show that if a latticed rectangle has area $2011$, then their sides are parallel to the axes.

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 $ABCD$ be a rectangle with $\angle ABD = 15^o, BD = 6$ cm. Compute the area of the rectangle. A. $9$ cm$^2$ B. $9 \sqrt3$ cm$^2$ C. $18$ cm$^2$ D. $18 \sqrt3$ cm$^2$ E. $24 \sqrt3$ cm$^2$

Find all rectangles that can be cut into $13$ equal squares.

Let $ABCD$ be a rectangle with: $AB=a$, $BC=b$. Inside the rectangle we have to exteriorly tangents circles such that one is tangent to the sides $AB$ and $AD$,the other is tangent to the sides $CB$ and $CD$. 1. Find the distance between the centers of the circles(using $a$ and $b$). 2. When the radiums of both circles change the tangency point between both of them changes, and describes a locus. Find that locus.

Let $S > 1$ be a real number. The Cartesian plane is partitioned into rectangles whose sides are parallel to the axes of the coordinate system. and whose vertices have integer coordinates. Prove that if the area of each triangle if at most $S$, then for any positive integer $k$ there exist $k$ vertices of these rectangles which lie on a line.

A circle of diameter $1$ is removed from a $2\times 3$ rectangle, as shown. Which whole number is closest to the area of the shaded region? [asy] fill((0,0)--(0,2)--(3,2)--(3,0)--cycle,gray); draw((0,0)--(0,2)--(3,2)--(3,0)--cycle,linewidth(1)); fill(circle((1,5/4),1/2),white); draw(circle((1,5/4),1/2),linewidth(1)); [/asy] $\text{(A)}\ 1 \qquad \text{(B)}\ 2 \qquad \text{(C)}\ 3 \qquad \text{(D)}\ 4 \qquad \text{(E)}\ 5$

In rectangle $ ABCD$, $ AB\equal{}5$ and $ BC\equal{}3$. Points $ F$ and $ G$ are on $ \overline{CD}$ so that $ DF\equal{}1$ and $ GC\equal{}2$. Lines $ AF$ and $ BG$ intersect at $ E$. Find the area of $ \triangle{AEB}$. [asy]unitsize(6mm); defaultpen(linewidth(.8pt)+fontsize(8pt)); pair A=(0,0), B=(5,0), C=(5,3), D=(0,3), F=(1,3), G=(3,3); pair E=extension(A,F,B,G); draw(A--B--C--D--A--E--B); label("$A$",A,SW); label("$B$",B,SE); label("$C$",C,NE); label("$D$",D,NW); label("$E$",E,N); label("$F$",F,SE); label("$G$",G,SW); label("$B$",B,SE); label("1",midpoint(D--F),N); label("2",midpoint(G--C),N); label("3",midpoint(B--C),E); label("3",midpoint(A--D),W); label("5",midpoint(A--B),S);[/asy]$ \textbf{(A)}\ 10 \qquad \textbf{(B)}\ \frac{21}{2} \qquad \textbf{(C)}\ 12 \qquad \textbf{(D)}\ \frac{25}{2} \qquad \textbf{(E)}\ 15$

Let $P_1, P_2, P_3, P_4$ be four points on a circle, let $I_1$ be incenter of the triangle of vertices $P_2P_3P_4$, $I_2$ the incenter of the triangle $P_1P_3P_4$, $I_3$ the incenter of the triangle $P_1P_2P_4$, $I_4$ the incenter of the triangle $P_2P_3P_1$. Prove that $I_1I_2I_3I_4$ is a rectangle.