Ana and Beatriz take turns in a game that starts with a square of side $1$ drawn on an infinite grid. Each turn consists of drawing a square that does not overlap with the rectangle already drawn, in such a way that one of its sides is a (complete) side of the figure already drawn. A player wins if she completes a rectangle whose area is a multiple of $5$. If Ana goes first, does either player have a winning strategy?

Square $ABCD$ has side length $13$, and points $E$ and $F$ are exterior to the square such that $BE=DF=5$ and $AE=CF=12$. Find $EF^{2}$. [asy] size(200); defaultpen(fontsize(10)); real x=22.61986495; pair A=(0,26), B=(26,26), C=(26,0), D=origin, E=A+24*dir(x), F=C+24*dir(180+x); draw(B--C--F--D--C^^D--A--E--B--A, linewidth(0.7)); dot(A^^B^^C^^D^^E^^F); pair point=(13,13); label("$A$", A, dir(point--A)); label("$B$", B, dir(point--B)); label("$C$", C, dir(point--C)); label("$D$", D, dir(point--D)); label("$E$", E, dir(point--E)); label("$F$", F, dir(point--F));[/asy]

We have a rectangle with it sides being a mirror.A light Ray enters from one of the corners of the rectangle and after being reflected several times enters to the opposite corner it started.Prove that at some time the light Ray passed the center of rectangle(Intersection of diagonals.)

The octagon $P_1P_2P_3P_4P_5P_6P_7P_8$ is inscribed in a circle with the vertices around the circumference in the given order. Given that the polygon $P_1P_3P_5P_7$ is a square of area $5$, and the polygon $P_2P_4P_6P_8$ is a rectangle of area $4$, find the maximum possible area of the octagon.

Let $ABC$ be a triangle . Let point $X$ be in the triangle and $AX$ intersects $BC$ in $Y$ . Draw the perpendiculars $YP,YQ,YR,YS$ to lines $CA,CX,BX,BA$ respectively. Find the necessary and sufficient condition for $X$ such that $PQRS$ be cyclic .

Consider on the coordinate plane all rectangles whose (i) vertices have integer coordinates; (ii) edges are parallel to coordinate axes; (iii) area is $2^k$, where $k = 0,1,2....$ Is it possible to color all points with integer coordinates in two colors so that no such rectangle has all its vertices of the same color?

Let $n\geq 3$ be a positive integer. Inside a $n\times n$ array there are placed $n^2$ positive numbers with sum $n^3$. Prove that we can find a square $2\times 2$ of 4 elements of the array, having the sides parallel with the sides of the array, and for which the sum of the elements in the square is greater than $3n$. [i]Radu Gologan[/i]

Let $K$ and $L$ be positive integers. On a board consisting of $2K \times 2L$ unit squares an ant starts in the lower left corner square and walks to the upper right corner square. In each step it goes horizontally or vertically to a neighbouring square. It never visits a square twice. At the end some squares may remain unvisited. In some cases the collection of all unvisited squares forms a single rectangle. In such cases, we call this rectangle [i]MEMOrable[/i]. Determine the number of different MEMOrable rectangles. [i]Remark: Rectangles are different unless they consist of exactly the same squares.[/i]

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)

In rectangle $ ABCD$, points $ F$ and $ G$ lie on $ \overline{AB}$ so that $ AF \equal{} FG \equal{} GB$ and $ E$ is the midpoint of $ \overline{DC}$. Also, $ \overline{AC}$ intersects $ \overline{EF}$ at $ H$ and $ \overline{EG}$ at $ J$. The area of the rectangle $ ABCD$ is $ 70$. Find the area of triangle $ EHJ$. [asy] size(180); pair A, B, C, D, E, F, G, H, J; A = origin; real length = 6; real width = 3.5; B = length*dir(0); C = (length, width); D = width*dir(90); F = length/3*dir(0); G = 2*length/3*dir(0); E = (length/2, width); H = extension(A, C, E, F); J = extension(A, C, E, G); draw(A--B--C--D--cycle); draw(G--E--F); draw(A--C); label("$A$", A, dir(180)); label("$D$", D, dir(180)); label("$B$", B, dir(0)); label("$C$", C, dir(0)); label("$F$", F, dir(270)); label("$E$", E, dir(90)); label("$G$", G, dir(270)); label("$H$", H, dir(140)); label("$J$", J, dir(340)); [/asy] $ \displaystyle \textbf{(A)} \ \frac {5}{2} \qquad \textbf{(B)} \ \frac {35}{12} \qquad \textbf{(C)} \ 3 \qquad \textbf{(D)} \ \frac {7}{2} \qquad \textbf{(E)} \ \frac {35}{8}$

A finite set $S$ of unit squares is chosen out of a large grid of unit squares. The squares of $S$ are tiled with isosceles right triangles of hypotenuse $2$ so that the triangles do not overlap each other, do not extend past $S$, and all of $S$ is fully covered by the triangles. Additionally, the hypotenuse of each triangle lies along a grid line, and the vertices of the triangles lie at the corners of the squares. Show that the number of triangles must be a multiple of $4$.

A $9\times 7$ rectangle is tiled with tiles of the two types: L-shaped tiles composed by three unit squares (can be rotated repeatedly with $90^\circ$) and square tiles composed by four unit squares. Let $n\ge 0$ be the number of the $2 \times 2 $ tiles which can be used in such a tiling. Find all the values of $n$.

An interior point in a rectangle is connected by line segments to the midpoints of its four sides. Thus four domains (polygons) with the areas $a, b, c$ and $d$ appear (see the figure). Prove that $a + c = b + d$. [img]https://1.bp.blogspot.com/-BipDNHELjJI/XzcCa68P3HI/AAAAAAAAMXY/H2Iqya9VItMLXrRqsdyxHLTXCAZ02nEtgCLcBGAsYHQ/s0/2002%2BMohr%2Bp1.png[/img]

Determine the values of $n \in N$ such that a square of side $n$ can be split into a square of side $1$ and five rectangles whose side measures are $10$ distinct natural numbers and all greater than $1$.

In the plane are given two mutually perpendicular lines and $n$ rectangles with sides parallel to the two lines. Show that if every two rectangles have a common point, then all the rectangles have a common point.

Let $ABC$ be an acute triangle. Find the locus of the centers of the rectangles which have their vertices on the sides of $ABC$.

The length of a rectangle is increased by $ 10\%$ and the width is decreased by $ 10\%$. What percent of the old area is the new area? $ \textbf{(A)}\ 90 \qquad \textbf{(B)}\ 99 \qquad \textbf{(C)}\ 100 \qquad \textbf{(D)}\ 101 \qquad \textbf{(E)}\ 110$

In the rectangle $ABCD, M, N, P$ and $Q$ are the midpoints of the sides. If the area of the shaded triangle is $1$, calculate the area of the rectangle $ABCD$. [img]https://2.bp.blogspot.com/-9iyKT7WP5fc/XNYuXirLXSI/AAAAAAAAKK4/10nQuSAYypoFBWGS0cZ5j4vn_hkYr8rcwCK4BGAYYCw/s400/may3.gif[/img]

Let $ABCD$ be a rectangle and $P$ a point outside of it such that $\angle{BPC} = 90^{\circ}$ and the area of the pentagon $ABPCD$ is equal to $AB^{2}$. Show that $ABPCD$ can be divided in 3 pieces with straight cuts in such a way that a square can be built using those 3 pieces, without leaving any holes or placing pieces on top of each other. Note: the pieces can be rotated and flipped over.

Three points inside a rectangle determine a triangle. A fourth point is taken inside the triangle. i) Prove at least one of the three concave quadrilaterals formed by these four points has perimeter lesser than that of the rectangle. ii) Assuming the three points inside the rectangle are three corners of it, prove at least two of the three concave quadrilaterals formed by these four points have perimeters lesser than that of the rectangle. [i](Dan Schwarz)[/i]

Rectangle $DEFA$ below is a $3 \times 4$ rectangle with $DC=CB=BA$. The area of the "bat wings" is [asy] size(180); defaultpen(fontsize(11pt)); draw((0,0)--(3,0)--(3,4)--(0,4)--(0,0)--(2,4)--(3,0)); draw((3,0)--(1,4)--(0,0)); fill((0,0)--(1,4)--(1.5,3)--cycle, black); fill((3,0)--(2,4)--(1.5,3)--cycle, black); label("$D$",(0,4),NW); label("$C$",(1,4),N); label("$B$",(2,4),N); label("$A$",(3,4),NE); label("$E$",(0,0),SW); label("$F$",(3,0),SE);[/asy] $\textbf{(A) }2\qquad\textbf{(B) }2 \frac{1}{2}\qquad\textbf{(C) }3\qquad\textbf{(D) }3 \frac{1}{2}\qquad \textbf{(E) }5$

A square is cut into several rectangles, none of which is a square, so that the sides of each rectangle are parallel to the sides of the square. For each rectangle with sides $a, b,a<b$, compute the ratio $a/b$. Prove that sum of these ratios is at least $1$.

A cubic block with dimensions $n$ by $n$ by $n$ is made up of a collection of $1$ by $1$ by $1$ unit cubes. What is the smallest value of $n$ so that if the outer layer of unit cubes are removed from the block, more than half the original unit cubes will still remain?

Let $ABCD$ be a tetrahedron such that $AD \perp BD$, $BD \perp CD$, $CD \perp AD$ and $AD=10$, $BD=15$, $CD=20$. Let $E$ and $F$ be points such that $E$ lies on $BC$, $DE \perp BC$, and $ADEF$ is a rectangle. If $S$ is the solid consisting of all points inside $ABCD$ but outside $FBCD$, compute the volume of $S$. [i]2015 CCA Math Bonanza Individual Round #13[/i]

In a billiard with shape of a rectangle $ABCD$ with $AB=2013$ and $AD=1000$, a ball is launched along the line of the bisector of $\angle BAD$. Supposing that the ball is reflected on the sides with the same angle at the impact point as the angle shot , examine if it shall ever reach at vertex B.