There are given points with integer coordinate $(m,n)$ such that $1\leq m,n\leq 4$. Two players, Ana and Ben, are playing a game: First Ana color one of the coordinates with red one, then she pass the turn to Ben who color one of the remaining coordinates with yellow one, then this process they repeate again one after other. The game win the first player who can create a rectangle with same color of vertices and the length of sides are positive integer numbers, otherwise the game is a tie. Does there exist a strategy for any of the player to win the game?

Point $K$ lies on the circumcircle of a rectangle $ABCD$. Line $CK$ intersects line segment $AD$ at point $M$ so that $AM:MD=2$. $O$ is the center the rectangle. Prove that the centroid of triangle $OKD$ belongs to the circumcircle of triangle $COD$. (Author: V. Shmarov)

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]

Given an $ n\times n$ chessboard. a) Find the number of rectangles on the chessboard. b) Assume there exists an $ r\times r$ square (label $ B$) with $ r<n$ which is located on the upper left corner of the board. Define "inner border" of $ A$ as the border of $ A$ which is not the border of the chessboard. How many rectangles in $ B$ that touch exactly one inner border of $ B$?

The columns and the row of a $3n \times 3n$ square board are numbered $1,2,\ldots ,3n$. Every square $(x,y)$ with $1 \leq x,y \leq 3n$ is colored asparagus, byzantium or citrine according as the modulo $3$ remainder of $x+y$ is $0,1$ or $2$ respectively. One token colored asparagus, byzantium or citrine is placed on each square, so that there are $3n^2$ tokens of each color. Suppose that one can permute the tokens so that each token is moved to a distance of at most $d$ from its original position, each asparagus token replaces a byzantium token, each byzantium token replaces a citrine token, and each citrine token replaces an asparagus token. Prove that it is possible to permute the tokens so that each token is moved to a distance of at most $d+2$ from its original position, and each square contains a token with the same color as the square.

A rectangle $ABCD$ and a point $P$ are given. Lines passing through $A$ and $B$ and perpendicular to $PC$ and $PD$ respectively, meet at a point $Q$. Prove that $PQ \perp AB$.

Show that for every integer $n \geq 6$, there exists a convex hexagon which can be dissected into exactly $n$ congruent triangles.

Given that $A,B,$ and $C$ are noncollinear points in the plane with integer coordinates such that the distances $AB,AC,$ and $BC$ are integers, what is the smallest possible value of $AB?$

Rectangle $ABCD$ is given with $AB=63$ and $BC=448.$ Points $E$ and $F$ lie on $AD$ and $BC$ respectively, such that $AE=CF=84.$ The inscribed circle of triangle $BEF$ is tangent to $EF$ at point $P,$ and the inscribed circle of triangle $DEF$ is tangent to $EF$ at point $Q.$ Find $PQ.$

Cut the $19 \times 20$ grid rectangle along the grid lines into several squares so that there are exactly four of them with odd sidelengths.

$\textbf{Bake Sale}$ Four friends, Art, Roger, Paul and Trisha, bake cookies, and all cookies have the same thickness. The shapes of the cookies differ, as shown. $\circ$ Art's cookies are trapezoids: [asy]size(80);defaultpen(linewidth(0.8));defaultpen(fontsize(8)); draw(origin--(5,0)--(5,3)--(2,3)--cycle); draw(rightanglemark((5,3), (5,0), origin)); label("5 in", (2.5,0), S); label("3 in", (5,1.5), E); label("3 in", (3.5,3), N);[/asy] $\circ$ Roger's cookies are rectangles: [asy]size(80);defaultpen(linewidth(0.8));defaultpen(fontsize(8)); draw(origin--(4,0)--(4,2)--(0,2)--cycle); draw(rightanglemark((4,2), (4,0), origin)); draw(rightanglemark((0,2), origin, (4,0))); label("4 in", (2,0), S); label("2 in", (4,1), E);[/asy] $\circ$ Paul's cookies are parallelograms: [asy]size(80);defaultpen(linewidth(0.8));defaultpen(fontsize(8)); draw(origin--(3,0)--(2.5,2)--(-0.5,2)--cycle); draw((2.5,2)--(2.5,0), dashed); draw(rightanglemark((2.5,2),(2.5,0), origin)); label("3 in", (1.5,0), S); label("2 in", (2.5,1), W);[/asy] $\circ$ Trisha's cookies are triangles: [asy]size(80);defaultpen(linewidth(0.8));defaultpen(fontsize(8)); draw(origin--(3,0)--(3,4)--cycle); draw(rightanglemark((3,4),(3,0), origin)); label("3 in", (1.5,0), S); label("4 in", (3,2), E);[/asy] Each friend uses the same amount of dough, and Art makes exactly 12 cookies. How many cookies will be in one batch of Trisha's cookies? $ \textbf{(A)}\ 10\qquad\textbf{(B)}\ 12\qquad\textbf{(C)}\ 16\qquad\textbf{(D)}\ 18\qquad\textbf{(E)}\ 24$

Let $a_1,a_2,a_3,\cdots$ be a non-decreasing sequence of positive integers. For $m\ge1$, define $b_m=\min\{n: a_n \ge m\}$, that is, $b_m$ is the minimum value of $n$ such that $a_n\ge m$. If $a_{19}=85$, determine the maximum value of \[a_1+a_2+\cdots+a_{19}+b_1+b_2+\cdots+b_{85}.\]

A domino is a $2$ by $1$ rectangle . For what integers $m$ and $n$ can we cover an $m*n$ rectangle with non-overlapping dominoes???

Determine all values of $n$ for which there is a set $S$ with $n$ points, with no 3 collinear, with the following property: it is possible to paint all points of $S$ in such a way that all angles determined by three points in $S$, all of the same color or of three different colors, aren't obtuse. The number of colors available is unlimited.

We are given a positive integer $ r$ and a rectangular board $ ABCD$ with dimensions $ AB \equal{} 20, BC \equal{} 12$. The rectangle is divided into a grid of $ 20 \times 12$ unit squares. The following moves are permitted on the board: one can move from one square to another only if the distance between the centers of the two squares is $ \sqrt {r}$. The task is to find a sequence of moves leading from the square with $ A$ as a vertex to the square with $ B$ as a vertex. (a) Show that the task cannot be done if $ r$ is divisible by 2 or 3. (b) Prove that the task is possible when $ r \equal{} 73$. (c) Can the task be done when $ r \equal{} 97$?