Several identical square sheets of paper are laid out on a rectangular table so that their sides are parallel to the edges of the table (sheets may overlap). Prove that you can stick a few pins in such a way that each sheet will be attached to the table exactly by one pin.

In a concyclic quadrilateral $PQRS$,$\angle PSR=\frac{\pi}{2}$ , $H,K$ are perpendicular foot from $Q$ to sides $PR,RS$ , prove that $HK$ bisect segment$SQ$.

In a rectangle with sides of length 20 and 25 there are 120 squares of side length 1. Prove that there is a circle with a diameter of 1 contained in this rectangle and having no points in common with any of these squares.

Inside a rectangular piece of paper $n$ rectangular holes with sides parallel to the sides of the paper have been cut out. Into what minimal number of rectangular pieces (without holes) is it always possible to cut this piece of paper? (A Shapovalov)

A certain number of cubes are painted in six colours, each cube having six faces of different colours (the colours in different cubes may be arranged differently) . The cubes are placed on a table so as to form a rectangle. We are allowed to take out any column of cubes, rotate it (as a whole) along its long axis and replace it in the rectangle. A similar operation with rows is also allowed. Can we always make the rectangle monochromatic (i.e. such that the top faces of all the cubes are the same colour) by means of such operations? ( D. Fomin , Leningrad)

Consider $120$ unit squares arbitrarily situated in a $20\times 25$ rectangle. Prove that one can place a circle with unit diameter in the rectangle without intersecting any of the squares.

Two circles with different radii intersect in two points $A$ and $B$. Let the common tangents of the two circles be $MN$ and $ST$ such that $M,S$ lie on the first circle, and $N,T$ on the second. Prove that the orthocenters of the triangles $AMN$, $AST$, $BMN$ and $BST$ are the four vertices of a rectangle.

Given the true statement: If a quadrilateral is a square, then it is a rectangle. It follows that, of the converse and the inverse of this true statement is: $ \textbf{(A)}\ \text{only the converse is true} \qquad\textbf{(B)}\ \text{only the inverse is true }\qquad \textbf{(C)}\ \text{both are true} \qquad$ $\textbf{(D)}\ \text{neither is true} \qquad\textbf{(E)}\ \text{the inverse is true, but the converse is sometimes true} $

On a chessboard ($8\times 8$ squares with sides of length $1$) two diagonally opposite corner squares are taken away. Can the board now be covered with nonoverlapping rectangles with sides of lengths $1$ and $2$?

Each unit square of a 3-by-3 unit-square grid is to be colored either blue or red. For each square, either color is equally likely to be used. The probability of obtaining a grid that does not have a 2-by-2 red square is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

The quadrilateral $ABCD$ is circumscribed about the circle, touches the sides $AB, BC, CD, DA$ in the points $K, L, M, N,$ respectively. Let $P, Q, R, S$ midpoints of the sides $KL, LM, MN, NK$. Prove that $PR = QS$ if and only if $ABCD$ is inscribed.

Let $ ABCD$ be a convex cuadrilateral inscribed in a circumference centered at $ O$ such that $ AC$ is a diameter. Pararellograms $ DAOE$ and $ BCOF$ are constructed. Show that if $ E$ and $ F$ lie on the circumference then $ ABCD$ is a rectangle.

All the vertices of a convex pentagon are on lattice points. Prove that the area of the pentagon is at least $\frac{5}{2}$. [i]Bogdan Enescu[/i]

A rectangle $\mathcal{R}$ with odd integer side lengths is divided into small rectangles with integer side lengths. Prove that there is at least one among the small rectangles whose distances from the four sides of $\mathcal{R}$ are either all odd or all even. [i]Proposed by Jeck Lim, Singapore[/i]

A strip of width $1$ is to be divided by rectangular panels of common width $1$ and denominations long $a_1$, $a_2$, $a_3$, $. . .$ be paved without gaps ($a_1 \ne 1$). From the second panel on, each panel is similar but not congruent to the already paved part of the strip. When the first $n$ slabs are laid, the length of the paved part of the strip is $sn$. Given $a_1$, is there a number that is not surpassed by any $s_n$? The accuracy answer has to be proven.

Let $21$ pieces, some white and some black, be placed on the squares of a $3\times 7$ rectangle. Prove that there always exist four pieces of the same color standing at the vertices of a rectangle.

It is given a triangle $\triangle ABC$. Determine the locus of center of rectangle inscribed in triangle $ABC$ such that one side of rectangle lies on side $AB$.

In this diagram the center of the circle is $O$, the radius is $a$ inches, chord $EF$ is parallel to chord $CD, O, G, H, J$ are collinear, and $G$ is the midpoint of $CD$. Let $K$ (sq. in.) represent the area of trapezoid $CDFE$ and let $R$ (sq. in.) represent the area of rectangle $ELMF$. Then, as $CD$ and $EF$ are translated upward so that $OG$ increases toward the value $a$, while $JH$ always equals $HG$, the ratio $K:R$ become arbitrarily close to: [asy] size((270)); draw((0,0)--(10,0)..(5,5)..(0,0)); draw((5,0)--(5,5)); draw((9,3)--(1,3)--(1,1)--(9,1)--cycle); draw((9.9,1)--(.1,1)); label("O", (5,0), S); label("a", (7.5,0), S); label("G", (5,1), SE); label("J", (5,5), N); label("H", (5,3), NE); label("E", (1,3), NW); label("L", (1,1), S); label("C", (.1,1), W); label("F", (9,3), NE); label("M", (9,1), S); label("D", (9.9,1), E); [/asy] $\textbf{(A)}\ 0 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ \sqrt{2} \qquad\textbf{(D)}\ \frac{1}{\sqrt{2}}+\frac{1}{2} \qquad\textbf{(E)}\ \frac{1}{\sqrt{2}}+1$

In a square $10^{2019} \times 10^{2019}, 10^{4038}$ points are marked. Prove that there is such a rectangle with sides parallel to the sides of a square whose area differs from the number of points located in it by at least $6$.

How many rectangles can be formed by the vertices of a cube? (Note: square is also a special rectangle). A. $6$ B. $8$ C. $12$ D. $18$ E. $16$

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]

Let $O, H$ be circumcenter and orthogonal center of triangle $ABC$. $M,N$ are midpoints of $BH$ and $CH$. $BB'$ is diagonal of circumcircle. If $HONM$ is a cyclic quadrilateral, prove that $B'N=\frac12AC$.

Suppose $F$ is a polygon with lattice vertices and sides parralell to x-axis and y-axis.Suppose $S(F),P(F)$ are area and perimeter of $F$. Find the smallest k that: $S(F) \leq k.P(F)^2$

Let $ABCD$ be a quadrilateral inscribed in circle $\omega$. Define $E = AA \cap CD$, $F = AA \cap BC$, $G = BE \cap \omega$, $H = BE \cap AD$, $I = DF \cap \omega$, and $J = DF \cap AB$. Prove that $GI$, $HJ$, and the $B$-symmedian are concurrent. [i]Proposed by Robin Park[/i]

Given a rectangle $ABCD$, which is located on the line $\ell$ They want it "turn over" by first turning around the vertex $D$, and then as point $C$ appears on the line $\ell$ - by making a turn around the vertex $C$ (see figure). What is the length of the curve along which the vertex $A$ is moving , at such movement, if $AB = 30$ cm, $BC = 40$ cm? (Alexey Panasenko) [img]https://cdn.artofproblemsolving.com/attachments/d/9/3cca36b08771b1897e385d43399022049bbcde.png[/img]