Show that amongst any $ 8$ points in the interior of a $7 \times 12$ rectangle, there exists a pair whose distance is less than $5$. Note: The interior of a rectangle excludes points lying on the sides of the rectangle.

In the cartesian plane consider rectangles with sides parallel to the coordinate axes. We say that one rectangle is [i]below[/i] another rectangle if there is a line $g$ parallel to the $x$-axis such that the first rectangle is below $g$, the second one above $g$ and both rectangles do not touch $g$. Similarly, we say that one rectangle is [i]to the right of[/i] another rectangle if there is a line $h$ parallel to the $y$-axis such that the first rectangle is to the right of $h$, the second one to the left of $h$ and both rectangles do not touch $h$. Show that any finite set of $n$ pairwise disjoint rectangles with sides parallel to the coordinate axes can be enumerated as a sequence $(R_1,\dots,R_n)$ so that for all indices $i,j$ with $1 \le i<j \le n$ the rectangle $R_i$ is to the right of or below the rectangle $R_j$

In the diagram below, square $ABCD$ with side length 23 is cut into nine rectangles by two lines parallel to $\overline{AB}$ and two lines parallel to $\overline{BC}$. The areas of four of these rectangles are indicated in the diagram. Compute the largest possible value for the area of the central rectangle. [asy] size(250); defaultpen (linewidth (0.7) + fontsize (10)); draw ((0,0)--(23,0)--(23,23)--(0,23)--cycle); label("$A$", (0,23), NW); label("$B$", (23, 23), NE); label("$C$", (23,0), SE); label("$D$", (0,0), SW); draw((0,6)--(23,6)); draw((0,19)--(23,19)); draw((5,0)--(5,23)); draw((12,0)--(12,23)); label("13", (17/2, 21)); label("111",(35/2,25/2)); label("37",(17/2,3)); label("123",(2.5,12.5));[/asy] [i]Proposed by Lewis Chen[/i]

In a table with $n$ rows and $2n$ columns where $n$ is a fixed positive integer, we write either zero or one into each cell so that each row has $n$ zeros and $n$ ones. For $1 \le k \le n$ and $1 \le i \le n$, we define $a_{k,i}$ so that the $i^{\text{th}}$ zero in the $k^{\text{th}}$ row is the $a_{k,i}^{\text{th}}$ column. Let $\mathcal F$ be the set of such tables with $a_{1,i} \ge a_{2,i} \ge \dots \ge a_{n,i}$ for every $i$ with $1 \le i \le n$. We associate another $n \times 2n$ table $f(C)$ from $C \in \mathcal F$ as follows: for the $k^{\text{th}}$ row of $f(C)$, we write $n$ ones in the columns $a_{n,k}-k+1, a_{n-1,k}-k+2, \dots, a_{1,k}-k+n$ (and we write zeros in the other cells in the row). (a) Show that $f(C) \in \mathcal F$. (b) Show that $f(f(f(f(f(f(C)))))) = C$ for any $C \in \mathcal F$.

Given two points $p_1=(x_1, y_1)$ and $p_2=(x_2, y_2)$ on the plane, denote by $\mathcal{R}(p_1,p_2)$ the rectangle with sides parallel to the coordinate axes and with $p_1$ and $p_2$ as opposite corners, that is, \[\{(x,y)\in \mathbb{R}^2:\min\{x_1, x_2\}\leq x\leq \max\{x_1, x_2\},\min\{y_1, y_2\}\leq y\leq \max\{y_1, y_2\}\}.\] Find the largest value of $k$ for which the following statement is true: for all sets $\mathcal{S}\subset\mathbb{R}^2$ with $|\mathcal{S}|=2024$, there exist two points $p_1, p_2\in\mathcal{S}$ such that $|\mathcal{S}\cap\mathcal{R}(p_1, p_2)|\geq k$.

How many ways are there to divide a $2\times n$ rectangle into rectangles having integral sides, where $n$ is a positive integer?

A rectangle can be divided by parallel lines to its sides into 200 congruent squares, and also in 288 congruent squares. Prove that the rectangle can also be divided into 392 congruent squares.

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?

Consider a $9 \times 9$ array of white squares. Find the largest $n \in\mathbb N$ with the property: No matter how one chooses $n$ out of 81 white squares and color in black, there always remains a $1 \times 4$ array of white squares (either vertical or horizontal).

In the plane we consider rectangles whose sides are parallel to the coordinate axes and have positive length. Such a rectangle will be called a [i]box[/i]. Two boxes [i]intersect[/i] if they have a common point in their interior or on their boundary. Find the largest $ n$ for which there exist $ n$ boxes $ B_1$, $ \ldots$, $ B_n$ such that $ B_i$ and $ B_j$ intersect if and only if $ i\not\equiv j\pm 1\pmod n$. [i]Proposed by Gerhard Woeginger, Netherlands[/i]

A cube with side length $2$ is inscribed in a sphere. A second cube, with faces parallel to the first, is inscribed between the sphere and one face of the first cube. What is the length of a side of the smaller cube?

Triangle $ABC$ has side lengths $AB=12$, $BC=25$, and $CA=17$. Rectangle $PQRS$ has vertex $P$ on $\overline{AB}$, vertex $Q$ on $\overline{AC}$, and vertices $R$ and $S$ on $\overline{BC}$. In terms of the side length $PQ=w$, the area of $PQRS$ can be expressed as the quadratic polynomial \[\text{Area}(PQRS)=\alpha w-\beta\cdot w^2\] Then the coefficient $\beta=\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

Is it possible to write a natural number in every cell of an infinite chessboard in such a manner that for all integers $m, n > 100$, the sum of numbers in every $m\times n$ rectangle is divisible by $m + n \ ?$

The diagram below shows a rectangle with one side divided into seven equal segments and the opposite side divided in half. The rectangle has area 350. Find the area of the shaded region. For Diagram go to purplecomet.org/welcome/practice, the $2015$ middle school contest, and #5.

A square-shaped pizza with side length $30$ cm is cut into pieces (not necessarily rectangular). All cuts are parallel to the sides, and the total length of the cuts is $240$ cm. Show that there is a piece whose area is at least $36$ cm$^2$

Let $S$ be the set of ordered pairs $(x, y)$ such that $0<x\le 1$, $0<y\le 1$, and $\left[\log_2{\left(\frac 1x\right)}\right]$ and $\left[\log_5{\left(\frac 1y\right)}\right]$ are both even. Given that the area of the graph of $S$ is $m/n$, where $m$ and $n$ are relatively prime positive integers, find $m+n$. The notation $[z]$ denotes the greatest integer that is less than or equal to $z$.

In rectangle $ABCD$, point $E$ is chosen on $AB$ and $F$ is the foot of $E$ onto side $CD$ such that the circumcircle of $\vartriangle ABF$ intersects line segments $AD$ and $BC$ at points $G$ and $H$ respectively. Let $S$ be the intersection of $EF$ and $GH$, and $T$ the intersection of lines $EC$ and $DS$. If $\angle SF T = 15^o$ , compute the measure of $\angle CSD$.

$ (a)$ The rectangle $ PQRS$ with $ PQ\equal{}l$ and $ QR\equal{}m$ $ (l,m \in \mathbb{N})$ is divided into $ lm$ unit squares. Prove that the diagonal $ PR$ intersects exactly $ l\plus{}m\minus{}d$ of these squares, where $ d\equal{}(l,m)$. $ (b)$ A box with edge lengths $ l,m,n \in \mathbb{N}$ is divided into $ lmn$ unit cubes. How many of the cubes does a main diagonal of the box intersect?

Let $k$ and $n$ be positive integers. Consider an array of $2\left(2^n-1\right)$ rows by $k$ columns. A $2$-coloring of the elements of the array is said to be [i]acceptable[/i] if any two columns agree on less than $2^n-1$ entries on the same row. Given $n$, determine the maximum value of $k$ for an acceptable $2$-coloring to exist.

Let ABCD be a rectangle with $45^o < \angle ADB < 60^o$. The diagonals $AC$ and$ BD$ intersect at $O$. A line passing through $O$ and perpendicular to $BD$ meets $AD$ and $CD$ at $M$ and $N$ respectively. Let $K$ be a point on side $BC$ such that $MK \parallel AC$. Show that $\angle MKN = 90^o$. [img]https://cdn.artofproblemsolving.com/attachments/4/1/1d37b96cebaea3409ade7ce6711ac2d3fc2ef9.png[/img]

Suppose $ABCD$ and $AEFG$ are rectangles such that the points $B,E,D,G$ are collinear (in this order). Let the lines $BC$ and $GF$ intersect at point $T$ and let the lines $DC$ and $EF$ intersect at point $H$. Prove that points $A, H$ and $T$ are collinear.

Jerry cuts a wedge from a $6$-cm cylinder of bologna as shown by the dashed curve. Which answer choice is closest to the volume of his wedge in cubic centimeters? [asy] defaultpen(linewidth(0.65)); real d=90-63.43494882; draw(ellipse((origin), 2, 4)); fill((0,4)--(0,-4)--(-8,-4)--(-8,4)--cycle, white); draw(ellipse((-4,0), 2, 4)); draw((0,4)--(-4,4)); draw((0,-4)--(-4,-4)); draw(shift(-2,0)*rotate(-d-5)*ellipse(origin, 1.82, 4.56), linetype("10 10")); draw((-4,4)--(-8,4), dashed); draw((-4,-4)--(-8,-4), dashed); draw((-4,4.3)--(-4,5)); draw((0,4.3)--(0,5)); draw((-7,4)--(-7,-4), Arrows(5)); draw((-4,4.7)--(0,4.7), Arrows(5)); label("$8$ cm", (-7,0), W); label("$6$ cm", (-2,4.7), N);[/asy] $\textbf{(A)} 48 \qquad \textbf{(B)} 75 \qquad \textbf{(C)}151\qquad \textbf{(D)}192 \qquad \textbf{(E)}603$

Two players, Red and Blue, play in alternating turns on a 10x10 board. Blue goes first. In his turn, a player picks a row or column (not chosen by any player yet) and color all its squares with his own color. If any of these squares was already colored, the new color substitutes the old one. The game ends after 20 turns, when all rows and column were chosen. Red wins if the number of red squares in the board exceeds at least by 10 the number of blue squares; otherwise Blue wins. Determine which player has a winning strategy and describe this strategy.

A solitaire game is played on an $m\times n$ rectangular board, using $mn$ markers which are white on one side and black on the other. Initially, each square of the board contains a marker with its white side up, except for one corner square, which contains a marker with its black side up. In each move, one may take away one marker with its black side up, but must then turn over all markers which are in squares having an edge in common with the square of the removed marker. Determine all pairs $(m,n)$ of positive integers such that all markers can be removed from the board.

In the plane we have $n$ rectangles with parallel sides. The sides of distinct rectangles lie on distinct lines. The boundaries of the rectangles cut the plane into connected regions. A region is [i]nice[/i] if it has at least one of the vertices of the $n$ rectangles on the boundary. Prove that the sum of the numbers of the vertices of all nice regions is less than $40n$. (There can be nonconvex regions as well as regions with more than one boundary curve.)