Let $A,B$ be adjacent vertices of a regular $n$-gon ($n\ge5$) with center $O$. A triangle $XYZ$, which is congruent to and initially coincides with $OAB$, moves in the plane in such a way that $Y$ and $Z$ each trace out the whole boundary of the polygon, with $X$ remaining inside the polygon. Find the locus of $X$.

A uniform disk, a thin hoop, and a uniform sphere, all with the same mass and same outer radius, are each free to rotate about a fixed axis through its center. Assume the hoop is connected to the rotation axis by light spokes. With the objects starting from rest, identical forces are simultaneously applied to the rims, as shown. Rank the objects according to their kinetic energies after a given time $t$, from least to greatest. [asy] size(225); pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); filldraw(circle((0,0),1),gray(.7)); draw((0,-1)--(2,-1),EndArrow); label("$\vec{F}$",(1, -1),S); label("Disk",(-1,0),W); filldraw(circle((5,0),1),gray(.7)); filldraw(circle((5,0),0.75),white); draw((5,-1)--(7,-1),EndArrow); label("$\vec{F}$",(6, -1),S); label("Hoop",(6,0),E); filldraw(circle((10,0),1),gray(.5)); draw((10,-1)--(12,-1),EndArrow); label("$\vec{F}$",(11, -1),S); label("Sphere",(11,0),E); [/asy] $ \textbf{(A)} \ \text{disk, hoop, sphere}$ $\textbf{(B)}\ \text{sphere, disk, hoop}$ $\textbf{(C)}\ \text{hoop, sphere, disk}$ $\textbf{(D)}\ \text{disk, sphere, hoop}$ $\textbf{(E)}\ \text{hoop, disk, sphere} $

Let $f(x)=\lim_{n\to\infty} (\cos ^ n x+\sin ^ n x)^{\frac{1}{n}}$ for $0\leq x\leq \frac{\pi}{2}.$ (1) Find $f(x).$ (2) Find the volume of the solid generated by a rotation of the figure bounded by the curve $y=f(x)$ and the line $y=1$ around the $y$-axis.

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.

Triangle $ OAB$ has $ O \equal{} (0,0)$, $ B \equal{} (5,0)$, and $ A$ in the first quadrant. In addition, $ \angle{ABO} \equal{} 90^\circ$ and $ \angle{AOB} \equal{} 30^\circ$. Suppose that $ \overline{OA}$ is rotated $ 90^\circ$ counterclockwise about $ O$. What are the coordinates of the image of $ A$? $ \textbf{(A)}\ \left( \minus{} \frac {10}{3}\sqrt {3},5\right) \qquad \textbf{(B)}\ \left( \minus{} \frac {5}{3}\sqrt {3},5\right) \qquad \textbf{(C)}\ \left(\sqrt {3},5\right) \qquad \textbf{(D)}\ \left(\frac {5}{3}\sqrt {3},5\right) \\ \textbf{(E)}\ \left(\frac {10}{3}\sqrt {3},5\right)$

A sled loaded with children starts from rest and slides down a snowy $25^\circ$ (with respect to the horizontal) incline traveling $85$ meters in $17$ seconds. Ignore air resistance. What is the coefficient of kinetic friction between the sled and the slope? $ \textbf {(A) } 0.36 \qquad \textbf {(B) } 0.40 \qquad \textbf {(C) } 0.43 \qquad \textbf {(D) } 1.00 \qquad \textbf {(E) } 2.01 $

Find all values of positive integers $a$ and $b$ such that it is possible to put $a$ ones and $b$ zeros in every of vertices in polygon with $a+b$ sides so it is possible to rotate numbers in those vertices with respect to primary position and after rotation one neighboring $0$ and $1$ switch places and in every other vertices other than those two numbers remain the same.

The rectangle with vertices $(0,0)$, $(0,3)$, $(2,0)$ and $(2,3)$ is rotated clockwise through a right angle about the point $(2,0)$, then about $(5,0)$, then about $(7,0$), and finally about $(10,0)$. The net effect is to translate it a distance $10$ along the $x$-axis. The point initially at $(1,1)$ traces out a curve. Find the area under this curve (in other words, the area of the region bounded by the curve, the $x$-axis and the lines parallel to the $y$-axis through $(1,0)$ and $(11,0)$).

Three convex polygons are given on a plane. Prove that there is no line cutting all the polygons if and only if each of the polygons can be separated from the other two by a line.

Given a triangle $OAB$ with the vetices $O(0,\ 0,\ 0),\ A(1,\ 0,\ 0),\ B(1,\ 1,\ 0)$ in the $xyz$ space. Let $V$ be the cone obtained by rotating the triangle around the $x$-axis. Find the volume of the solid obtained by rotating the cone $V$ around the $y$-axis.

A sequence of vertices $v_1,v_2,\ldots,v_k$ in a graph, where $v_i=v_j$ only if $i=j$ and $k$ can be any positive integer, is called a $\textit{cycle}$ if $v_1$ is attached by an edge to $v_2$, $v_2$ to $v_3$, and so on to $v_k$ connected to $v_1$. Rotations and reflections are distinct: $A,B,C$ is distinct from $A,C,B$ and $B,C,A$. Supposed a simple graph $G$ has $2013$ vertices and $3013$ edges. What is the minimal number of cycles possible in $G$?

Let $n \geq 6$ and $3 \leq p < n - p$ be two integers. The vertices of a regular $n$-gon are colored so that $p$ vertices are red and the others are black. Prove that there exist two congruent polygons with at least $[p/2] + 1$ vertices, one with all the vertices red and the other with all the vertices black.

Let there be given a polygon $P$ which is mapped onto itself by two rotations: $\rho_1$ with center $O_1$ and angle $\omega_1$, and $\rho_2$ with center $O_2$ and angle $\omega_2~(0<\omega_i<2\pi)$. Show that the ratio $\frac{\omega_1}{\omega_2}$ is rational.

Given $ n$ points on the plane, which are the vertices of a convex polygon, $ n > 3$. There exists $ k$ regular triangles with the side equal to $ 1$ and the vertices at the given points. [list][*] Prove that $ k < \frac {2}{3}n$. [*] Construct the configuration with $ k > 0.666n$.[/list]

Nine distinct positive integers are arranged in a circle such that the product of any two non-adjacent numbers in the circle is a multiple of $n$ and the product of any two adjacent numbers in the circle is not a multiple of $n$, where $n$ is a fixed positive integer. Find the smallest possible value for $n$.

A particle is located on the coordinate plane at $ (5,0)$. Define a [i]move[/i] for the particle as a counterclockwise rotation of $ \pi/4$ radians about the origin followed by a translation of $ 10$ units in the positive $ x$-direction. Given that the particle's position after $ 150$ moves is $ (p,q)$, find the greatest integer less than or equal to $ |p|\plus{}|q|$.

A uniform rod is partially in water with one end suspended, as shown in figure. The density of the rod is $5/9$ that of water. At equilibrium, what portion of the rod is above water? $\textbf{(A) } 0.25\\ \textbf{(B) } 0.33\\ \textbf{(C) } 0.5\\ \textbf{(D) } 0.67\\ \textbf{(E) } 0.75$

In the plane, there are two circles $\Gamma_1, \Gamma_2$ intersecting each other at two points $A$ and $B$. Tangents of $\Gamma_1$ at $A$ and $B$ meet each other at $K$. Let us consider an arbitrary point $M$ (which is different of $A$ and $B$) on $\Gamma_1$. The line $MA$ meets $\Gamma_2$ again at $P$. The line $MK$ meets $\Gamma_1$ again at $C$. The line $CA$ meets $\Gamma_2 $ again at $Q$. Show that the midpoint of $PQ$ lies on the line $MC$ and the line $PQ$ passes through a fixed point when $M$ moves on $\Gamma_1$. [color=red][Moderator edit: This problem was also discussed on http://www.mathlinks.ro/Forum/viewtopic.php?t=21414 .][/color]

Find the volume $V$ of the solid formed by a rotation of the region enclosed by the curve $y=2^{x}-1$ and two lines $x=0,\ y=1$ around the $y$ axis.

$AB$ is a diameter of a circle with center $O$. Let $C$ and $D$ be two different points on the circle on the same side of $AB$, and the lines tangent to the circle at points $C$ and $D$ meet at $E$. Segments $AD$ and $BC$ meet at $F$. Lines $EF$ and $AB$ meet at $M$. Prove that $E,C,M$ and $D$ are concyclic.

Take two points $A\ (-1,\ 0),\ B\ (1,\ 0)$ on the $xy$-plane. Let $F$ be the figure by which the whole points $P$ on the plane satisfies $\frac{\pi}{4}\leq \angle{APB}\leq \pi$ and the figure formed by $A,\ B$. Answer the following questions: (1) Illustrate $F$. (2) Find the volume of the solid generated by a rotation of $F$ around the $x$-axis.

For which of the following $ n$, $ n\times n$ chessboard cannot be covered using at most one unit square piece and many L-shaped pieces (an L-shaped piece is a 2x2 piece with one square removed)? $\textbf{(A)}\ 96 \qquad\textbf{(B)}\ 97 \qquad\textbf{(C)}\ 98 \qquad\textbf{(D)}\ 99 \qquad\textbf{(E)}\ 100$

Lisa and Bart are playing a game. A round table has $n$ lights evenly spaced around its circumference. Some of the lights are on and some of them off; the initial configuration is random. Lisa wins if she can get all of the lights turned on; Bart wins if he can prevent this from happening. On each turn, Lisa chooses the positions at which to flip the lights, but before the lights are flipped, Bart, knowing Lisa’s choices, can rotate the table to any position that he chooses (or he can leave the table as is). Then the lights in the positions that Lisa chose are flipped: those that are off are turned on and those that are on are turned off. Here is an example turn for $n = 5$ (a white circle indicates a light that is on, and a black circle indicates a light that is off): [asy] size(250); defaultpen(linewidth(1)); picture p = new picture; real r = 0.2; pair s1=(0,-4), s2=(0,-8); int[][] filled = {{1,2,3},{1,2,5},{2,3,4,5}}; draw(p,circle((0,0),1)); for(int i = 0; i < 5; ++i) { pair P = dir(90-72*i); filldraw(p,circle(P,r),white); label(p,string(i+1),P,2*P,fontsize(10)); } add(p); add(shift(s1)*p); add(shift(s2)*p); for(int j = 0; j < 3; ++j) for(int i = 0; i < filled[j].length; ++i) filldraw(circle(dir(90-72*(filled[j][i]-1))+j*s1,r)); label("$\parbox{15em}{Initial Position.}$", (-4.5,0)); label("$\parbox{15em}{Lisa says ``1,3,4.'' \\ Bart rotates the table one \\ position counterclockwise. }$", (-4.5,0)+s1); label("$\parbox{15em}{Lights in positions 1,3,4 are \\ flipped.}$", (-4.5,0)+s2);[/asy] Lisa can take as many turns as she needs to win, or she can give up if it becomes clear to her that Bart can prevent her from winning. (a) Show that if $n = 7$ and initially at least one light is on and at least one light is off, then Bart can always prevent Lisa from winning. (b) Show that if $n = 8$, then Lisa can always win in at most 8 turns.

Let $ ABC$ be a triangle, and $ D$ be a point where incircle touches side $ BC$. $ M$ is midpoint of $ BC$, and $ K$ is a point on $ BC$ such that $ AK\perp BC$. Let $ D'$ be a point on $ BC$ such that $ \frac{D'M}{D'K}=\frac{DM}{DK}$. Define $ \omega_{a}$ to be circle with diameter $ DD'$. We define $ \omega_{B},\omega_{C}$ similarly. Prove that every two of these circles are tangent.

Prove that the group of orientation-preserving symmetries of the cube is isomorphic to $S_4$ (the group of permutations of $\{1,2,3,4\}$).(20 points)