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 $AB?$

Prove that every rectangle circumscribed by a square is itself a square. (A rectangle is circumscribed by a square if there is exactly one corner point of the square on each side of the rectangle.)

A rectangle that is inscribed in a larger rectangle (with one vertex on each side) is called [i]unstuck[/i] if it is possible to rotate (however slightly) the smaller rectangle about its center within the confines of the larger. Of all the rectangles that can be inscribed unstuck in a 6 by 8 rectangle, the smallest perimeter has the form $\sqrt{N}$, for a positive integer $N$. Find $N$.

Dürer's $n \times m$ garden is surgically divided into $n \times m$ unit squares, and in the middle of one of these squares, he planted his favourite petunia. Dürer's gardener struggles with a mole, trying to drive him out of the magnificent garden, so he builds an underground wall on the edge of the garden. The only problem is that the mole managed to stay inside the walls.. When the mole meets a wall, it changes it's direction as if it was "reflected", that is, proceeding his route in the direction that includes the same angle with the wall as his direction before. The mole starts beneath the petunia, in a direction that includes a $45^o$ angle with the walls. Is it possible for the mole to cross the petunia in a direction perpendicular to it's original direction? (Think in terms of $n,m$.)

Two tangents $AT$ and $BT$ touch a circle at $A$ and $B$, respectively, and meet perpendicularly at $T$. $Q$ is on $AT$, $S$ is on $BT$, and $R$ is on the circle, so that $QRST$ is a rectangle with $QT = 8$ and $ST = 9$. Determine the radius of the circle.

The pentagon below has three right angles. Find its area. [asy] size(150); defaultpen(linewidth(1)); draw((4,10)--(0,10)--origin--(10,0)--(10,2)--cycle); label("4",(2,10),N); label("10",(0,5),W); label("10",(5,0),S); label("2",(10,1),E); label("10",(7,6),NE); [/asy]

The graph of the equation $9x+223y=2007$ is drawn on graph paper with each square representing one unit in each direction. How many of the $1$ by $1$ graph paper squares have interiors lying entirely below the graph and entirely in the first quadrant?

A square sheet of paper $ABCD$ is so folded that $B$ falls on the mid point of $M$ of $CD$. Prove that the crease will divide $BC$ in the ration $5 : 3$.

a) * In a rectangle of area $5$ sq. units, $9$ rectangles of area $1$ are arranged. Prove that the area of the overlap of some two of these rectangles is $\ge 1/9$ b) In a rectangle of area $5$ sq. units, lie $9$ arbitrary polygons each of area $1$. Prove that the area of the overlap of some two of these rectangles is $\ge 1/9$

Ali Barber, the carpet merchant, has a rectangular piece of carpet whose dimensions are unknown. Unfortunately, his tape measure is broken and he has no other measuring instruments. However, he finds that if he lays it flat on the floor of either of his storerooms, then each corner of the carpet touches a different wall of that room. If the two rooms have dimensions of 38 feet by 55 feet and 50 feet by 55 feet, what are the carpet dimensions?

Given a natural number $n$, calculate the number of rectangles in the plane, the coordinates of whose vertices are integers in the range $0$ to $n$, and whose sides are parallel to the axes.

Consider a $2n \times 2n$ board. From the $i$th line we remove the central $2(i-1)$ unit squares. What is the maximal number of rectangles $2 \times 1$ and $1 \times 2$ that can be placed on the obtained figure without overlapping or getting outside the board?

A rectangle has been divided into 8 smaller rectangles as shown below. Given the area of seven of these rectangles, find the area of the shaded rectangle. [i]Lightning 2.1[/i]

In the figure, $ABCD$ is a square of side length 1. The rectangles $JKHG$ and $EBCF$ are congruent. What is $BE$? [asy] unitsize(150); pair A,B,C,D,E,F,G,H,J,K; A=(1,0); B=(0,0); C=(0,1); D=(1,1); draw(A--B--C--D--A); E=(2-sqrt(3),0); F=(2-sqrt(3),1); draw(E--F); G=(1,sqrt(3)/2); H=(2.5-sqrt(3),1); K=(2-sqrt(3),1-sqrt(3)/2); J=(0.5,0); draw(G--H--K--J--G); label("$A$",A,SE); label("$B$",B,SW); label("$C$",C,NW); label("$D$",D,NE); label("$E$",E,S); label("$F$",F,N); label("$G$",G,E); label("$H$",H,N); label("$K$",K,W); label("$J$",J,S); [/asy] $ \textbf{(A) }\dfrac{1}{2}(\sqrt{6}-2)\qquad\textbf{(B) }\dfrac{1}{4}\qquad\textbf{(C) }2-\sqrt{3}\qquad\textbf{(D) }\dfrac{\sqrt{3}}{6}\qquad\textbf{(E) }1-\dfrac{\sqrt{2}}{2} $

Let $ABCD$ be a square with the side of $20$ units. Amir divides this square into $400$ unit squares. Reza then picks $4$ of the vertices of these unit squares. These vertices lie inside the square $ABCD$ and define a rectangle with the sides parallel to the sides of the square $ABCD.$ There are exactly $24$ unit squares which have at least one point in common with the sides of this rectangle. Find all possible values for the area of a rectangle with these properties. [hide="Note"][i]Note:[/i] Vid changed to Amir, and Eva change to Reza![/hide]

In the isosceles triangle $ABC$ with the vertex at the point $B$, the altitudes $BH$ and $CL$ are drawn. The point $D$ is such that $BDCH$ is a rectangle. Find the value of the angle $DLH$. (Bogdan Rublev)

Two equal rectangles are intersecting in $8$ points. Prove that the common part area is greater than the half of the rectangle's area.

Let $ABC$ an acute triangle. (a) Find the locus of points that are centers of rectangles whose vertices lie on the sides of $ABC$; (b) Determine if exist some points that are centers of $3$ distinct rectangles whose vertices lie on the sides of $ABC$.

A table $2 \times 2010$ is divided to unit cells. Ivan and Peter are playing the following game. Ivan starts, and puts horizontal $2 \times 1$ domino that covers exactly two unit table cells. Then Peter puts vertical $1 \times 2$ domino that covers exactly two unit table cells. Then Ivan puts horizontal domino. Then Peter puts vertical domino, etc. The person who cannot put his domino will lose the game. Find who have winning strategy.

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]

A rectangle has side lengths $6$ and $8$. There are relatively prime positive integers $m$ and $n$ so that $\tfrac{m}{n}$ is the probability that a point randomly selected from the inside of the rectangle is closer to a side of the rectangle than to either diagonal of the rectangle. Find $m + n$.

The $8$ eyelets for the lace of a sneaker all lie on a rectangle, four equally spaced on each of the longer sides. The rectangle has a width of $50$ mm and a length of $80$ mm. There is one eyelet at each vertex of the rectangle. The lace itself must pass between the vertex eyelets along a width side of the rectangle and then crisscross between successive eyelets until it reaches the two eyelets at the other width side of the rectrangle as shown. After passing through these final eyelets, each of the ends of the lace must extend at least $200$ mm farther to allow a knot to be tied. Find the minimum length of the lace in millimeters. [asy] size(200); defaultpen(linewidth(0.7)); path laceL=(-20,-30)..tension 0.75 ..(-90,-135)..(-102,-147)..(-152,-150)..tension 2 ..(-155,-140)..(-135,-40)..(-50,-4)..tension 0.8 ..origin; path laceR=reflect((75,0),(75,-240))*laceL; draw(origin--(0,-240)--(150,-240)--(150,0)--cycle,gray); for(int i=0;i<=3;i=i+1) { path circ1=circle((0,-80*i),5),circ2=circle((150,-80*i),5); unfill(circ1); draw(circ1); unfill(circ2); draw(circ2); } draw(laceL--(150,-80)--(0,-160)--(150,-240)--(0,-240)--(150,-160)--(0,-80)--(150,0)^^laceR,linewidth(1));[/asy]

Determine all integer $n$ such that a surface of an $n \times n \times n$ grid cube can be pasted in one layer by paper $1 \times 2$ rectangles so that each rectangle has exactly five neighbors (by a line segment). (A.Shapovalov)

In rectangle $ABCD$, angle $C$ is trisected by $\overline{CF}$ and $\overline{CE}$, where $E$ is on $\overline{AB}$, $F$ is on $\overline{AD}$, $BE = 6,$ and $AF = 2$. Which of the following is closest to the area of the rectangle $ABCD$? [asy] size(140); pair A, B, C, D, E, F, X, Y; real length = 12.5; real width = 10; A = origin; B = (length, 0); C = (length, width); D = (0, width); X = rotate(330, C)*B; E = extension(C, X, A, B); Y = rotate(30, C)*D; F = extension(C, Y, A, D); draw(E--C--F); label("$2$", A--F, dir(180)); label("$6$", E--B, dir(270)); draw(A--B--C--D--cycle); dot(A);dot(B);dot(C);dot(D);dot(E);dot(F); label("$A$", A, dir(225)); label("$B$", B, dir(315)); label("$C$", C, dir(45)); label("$D$", D, dir(135)); label("$E$", E, dir(270)); label("$F$", F, dir(180)); [/asy] $\textbf{(A)} \ 110 \qquad \textbf{(B)} \ 120 \qquad \textbf{(C)} \ 130 \qquad \textbf{(D)} \ 140 \qquad \textbf{(E)} \ 150$

Rectangles $1\times20$, $1\times 19$, ..., $1\times 1$ were cut out of $20\times20$ table. Prove that at least 85 dominoes(1×2 rectangle) can be removed from the remainder. Proposed by S. Berlov