A hexagonal table is given, as the one on the drawing, which has $~$ $2012$ $~$ columns. There are $~$ $2012$ $~$ hexagons in each of the odd columns, and there are $~$ $2013$ $~$ hexagons in each of the even columns. The number $~$ $i$ $~$ is written in each hexagon from the $~$ $i$-th column. Changing the numbers in the table is allowed in the following way: We arbitrarily select three adjacent hexagons, we rotate the numbers, and if the rotation is clockwise then the three numbers decrease by one, and if we rotate them counterclockwise the three numbers increase by one (see the drawing below). What's the maximum number of zeros that can be obtained in the table by using the above-defined steps.

A square is divided in four parts by two perpendicular lines, in such a way that three of these parts have areas equal to 1. Show that the square has area equal to 4.

$ABCD$ has $AB = CD$, but $AB$ not parallel to $CD$, and $AD$ parallel to $BC$. The triangle is $ABC$ is rotated about $C$ to $A'B'C$. Show that the midpoints of $BC, B'C$ and $A'D$ are collinear.

[asy]size(8cm); real w = 2.718; // width of block real W = 13.37; // width of the floor real h = 1.414; // height of block real H = 7; // height of block + string real t = 60; // measure of theta pair apex = (w/2, H); // point where the strings meet path block = (0,0)--(w,0)--(w,h)--(0,h)--cycle; // construct the block draw(shift(-W/2,0)*block); // draws white block path arrow = (w,h/2)--(w+W/8,h/2); // path of the arrow draw(shift(-W/2,0)*arrow, EndArrow); // draw the arrow picture pendulum; // making a pendulum... draw(pendulum, block); // block fill(pendulum, block, grey); // shades block draw(pendulum, (w/2,h)--apex); // adds in string add(pendulum); // adds in block + string add(rotate(t, apex) * pendulum); // adds in rotated block + string dot("$\theta$", apex, dir(-90+t/2)*3.14); // marks the apex and labels it with theta draw((apex-(w,0))--(apex+(w,0))); // ceiling draw((-W/2-w/2,0)--(w+W/2,0)); // floor[/asy] A block of mass $m=\text{4.2 kg}$ slides through a frictionless table with speed $v$ and collides with a block of identical mass $m$, initially at rest, that hangs on a pendulum as shown above. The collision is perfectly elastic and the pendulum block swings up to an angle $\theta=12^\circ$, as labeled in the diagram. It takes a time $ t = \text {1.0 s} $ for the block to swing up to this peak. Find $10v$, in $\text{m/s}$ and round to the nearest integer. Do not approximate $ \theta \approx 0 $; however, assume $\theta$ is small enough as to use the small-angle approximation for the period of the pendulum. [i](Ahaan Rungta, 6 points)[/i]

Let $P$ be a point on a circle $C$ and let $\phi$ be a given angle incommensurable with $2\pi$. For each $n \in N, P_n$ denotes the image of $P$ under the rotation about the center $O$ of $C$ by the angle $\alpha_n = n \phi$. Prove that the set $M = \{P_n | n \ge 0\}$ is dense in $C$.

Let $n$ be a positive with $n\geq 3$. Consider a board of $n \times n$ boxes. In each step taken the colors of the $5$ boxes that make up the figure bellow change color (black boxes change to white and white boxes change to black) The figure can be rotated $90°, 180°$ or $270°$. Firstly, all the boxes are white.Determine for what values of $n$ it can be achieved, through a series of steps, that all the squares on the board are black.

On a circle there are 2009 nonnegative integers not greater than 100. If two numbers sit next to each other, we can increase both of them by 1. We can do this at most $ k$ times. What is the minimum $ k$ so that we can make all the numbers on the circle equal?

The $ 8\times 18$ rectangle $ ABCD$ is cut into two congruent hexagons, as shown, in such a way that the two hexagons can be repositioned without overlap to form a square. What is $ y$? [asy] unitsize(2mm); defaultpen(fontsize(10pt)+linewidth(.8pt)); dotfactor=4; draw((0,4)--(18,4)--(18,-4)--(0,-4)--cycle); draw((6,4)--(6,0)--(12,0)--(12,-4)); label("$D$",(0,4),NW); label("$C$",(18,4),NE); label("$B$",(18,-4),SE); label("$A$",(0,-4),SW); label("$y$",(9,1)); [/asy]$ \textbf{(A) } 6\qquad \textbf{(B) } 7\qquad \textbf{(C) } 8\qquad \textbf{(D) } 9\qquad \textbf{(E) } 10$

Find the orders of the subgroups of the group of rotations of the cube, and find its normal subgroups.

The point $P(a,b)$ in the $xy$-plane is first rotated counterclockwise by $90^{\circ}$ around the point $(1,5)$ and then reflected about the line $y=-x$. The image of $P$ after these two transformations is at $(-6,3)$. What is $b-a$? $\textbf{(A) }1 \qquad \textbf{(B) }3 \qquad \textbf{(C) }5 \qquad \textbf{(D) }7 \qquad \textbf{(E) }9$

Five persons wearing badges with numbers $1, 2, 3, 4, 5$ are seated on $5$ chairs around a circular table. In how many ways can they be seated so that no two persons whose badges have consecutive numbers are seated next to each other? (Two arrangements obtained by rotation around the table are considered different)

Let $C: y=x^2+ax+b$ be a parabola passing through the point $(1,\ -1)$. Find the minimum volume of the figure enclosed by $C$ and the $x$ axis by a rotation about the $x$ axis. Proposed by kunny

Given a point $P_0$ in the plane of the triangle $A_1A_2A_3$. Define $A_s=A_{s-3}$ for all $s\ge4$. Construct a set of points $P_1,P_2,P_3,\ldots$ such that $P_{k+1}$ is the image of $P_k$ under a rotation center $A_{k+1}$ through an angle $120^o$ clockwise for $k=0,1,2,\ldots$. Prove that if $P_{1986}=P_0$, then the triangle $A_1A_2A_3$ is equilateral.

Let $ ABCD$ be a rectangle with center $ O$, $ AB\neq BC$. The perpendicular from $ O$ to $ BD$ cuts the lines $ AB$ and $ BC$ in $ E$ and $ F$ respectively. Let $ M,N$ be the midpoints of the segments $ CD,AD$ respectively. Prove that $ FM \perp EN$.

A unit square is rotated $45^\circ$ about its center. What is the area of the region swept out by the interior of the square? $ \textbf{(A)}\ 1-\frac{\sqrt2}2+\frac\pi4\qquad\textbf{(B)}\ \frac12+\frac\pi4\qquad\textbf{(C)}\ 2-\sqrt2+\frac\pi4\qquad\textbf{(D)}\ \frac{\sqrt2}2+\frac\pi4\qquad\textbf{(E)}\ 1+\frac{\sqrt2}4+\frac\pi8 $

Two circles $O_1$ and $O_2$ intersect each other at $M$ and $N$. The common tangent to two circles nearer to $M$ touch $O_1$ and $O_2$ at $A$ and $B$ respectively. Let $C$ and $D$ be the reflection of $A$ and $B$ respectively with respect to $M$. The circumcircle of the triangle $DCM$ intersect circles $O_1$ and $O_2$ respectively at points $E$ and $F$ (both distinct from $M$). Show that the circumcircles of triangles $MEF$ and $NEF$ have same radius length.

A $3\times3$ square is partitioned into $9$ unit squares. Each unit square is painted either white or black with each color being equally likely, chosen independently and at random. The square is the rotated $90^\circ$ clockwise about its center, and every white square in a position formerly occupied by a black square is painted black. The colors of all other squares are left unchanged. What is the probability that the grid is now entirely black? $ \textbf{(A)}\ \dfrac{49}{512} \qquad\textbf{(B)}\ \dfrac{7}{64} \qquad\textbf{(C)}\ \dfrac{121}{1024} \qquad\textbf{(D)}\ \dfrac{81}{512} \qquad\textbf{(E)}\ \dfrac{9}{32} $

In how many ways can you color the six sides of a cube in black or white? (Do note that the cube is unchanged when rotated?) A. 7 B. 10 C. 20 D. 30 E. 36

Twenty five of King Arthur's knights are seated at their customary round table. Three of them are chosen - all choices of three being equally likely - and are sent off to slay a troublesome dragon. Let $P$ be the probability that at least two of the three had been sitting next to each other. If $P$ is written as a fraction in lowest terms, what is the sum of the numerator and denominator?

A scanning code consists of a $7 \times 7$ grid of squares, with some of its squares colored black and the rest colored white. There must be at least one square of each color in this grid of $49$ squares. A scanning code is called [i]symmetric[/i] if its look does not change when the entire square is rotated by a multiple of $90 ^{\circ}$ counterclockwise around its center, nor when it is reflected across a line joining opposite corners or a line joining midpoints of opposite sides. What is the total number of possible symmetric scanning codes? $\textbf{(A)} \text{ 510} \qquad \textbf{(B)} \text{ 1022} \qquad \textbf{(C)} \text{ 8190} \qquad \textbf{(D)} \text{ 8192} \qquad \textbf{(E)} \text{ 65,534}$

A square-shaped quilt is divided into $16 = 4 \times 4$ equal squares. We say that the quilt is [i]UMD certified[/i] if each of these $16$ squares is colored red, yellow, or black, so that (i) all three colors are used at least once and (ii) the quilt looks the same when it is rotated $90, 180,$ or $270$ degrees about its center. How many distinct UMD certified quilts are there? \[\rm a. ~33\qquad \mathrm b. ~36 \qquad \mathrm c. ~45\qquad\mathrm d. ~54\qquad\mathrm e. ~81\]

Suppose a square of sidelengh $l$ is inside an unit square and does not contain its centre. Show that $l\le 1/2.$ [i]Marius Cavachi[/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]

A triangle with vertices $A(0, 2)$, $B(-3, 2)$, and $C(-3, 0)$ is reflected about the $x$-axis, then the image $\triangle A'B'C'$ is rotated counterclockwise about the origin by $90^{\circ}$ to produce $\triangle A''B''C''$. Which of the following transformations will return $\triangle A''B''C''$ to $\triangle ABC$? $\textbf{(A)}$ counterclockwise rotation about the origin by $90^{\circ}$. $\textbf{(B)}$ clockwise rotation about the origin by $90^{\circ}$. $\textbf{(C)}$ reflection about the $x$-axis $\textbf{(D)}$ reflection about the line $y = x$ $\textbf{(E)}$ reflection about the $y$-axis.

Given triangle $ABC$, squares $ABEF, BCGH, CAIJ$ are constructed externally on side $AB, BC, CA$, respectively. Let $AH \cap BJ = P_1$, $BJ \cap CF = Q_1$, $CF \cap AH = R_1$, $AG \cap CE = P_2$, $BI \cap AG = Q_2$, $CE \cap BI = R_2$. Prove that triangle $P_1 Q_1 R_1$ is congruent to triangle $P_2 Q_2 R_2$.