Polar coordinate equation of curve $C$ is $\rho=1+\cos\theta$. Polar coordinate of point $A$ is $(2,0)$. $C$ rotate around $A$ for a whole circle, the area of the figure that $C$ swept out by is________.

Let $P(z)=a_nz^n+a_{n-1}z^{n-1}+\ldots+a_mz^m$ be a polynomial with complex coefficients such that $a_m\neq 0, a_n\neq 0$ and $n>m$. Prove that \[\text{max}_{|z|=1}\{|P(z)|\}\ge\sqrt{2|a_ma_n|+\sum_{k=m}^{n} |a_k|^2}\]

Six chairs are evenly spaced around a circular table. One person is seated in each chair. Each person gets up and sits down in a chair that is not the same chair and is not adjacent to the chair he or she originally occupied, so that again one person is seated in each chair. In how many ways can this be done? $ \textbf{(A) }14\qquad\textbf{(B) }16\qquad\textbf{(C) }18\qquad\textbf{(D) }20\qquad\textbf{(E) }24 $

The figure below shows a right triangle $ \triangle ABC$. [asy]unitsize(15); pair A = (0, 4); pair B = (0, 0); pair C = (4, 0); draw(A -- B -- C -- cycle); pair D = (2, 0); real p = 7 - 3sqrt(3); real q = 4sqrt(3) - 6; pair E = p + (4 - p)*I; pair F = q*I; draw(D -- E -- F -- cycle); label("$A$", A, N); label("$B$", B, S); label("$C$", C, S); label("$D$", D, S); label("$E$", E, NE); label("$F$", F, W);[/asy] The legs $ \overline{AB}$ and $ \overline{BC}$ each have length $ 4$. An equilateral triangle $ \triangle DEF$ is inscribed in $ \triangle ABC$ as shown. Point $ D$ is the midpoint of $ \overline{BC}$. What is the area of $ \triangle DEF$?

Find the number of finite sequences $ \{a_1,a_2,\ldots,a_{2n\plus{}1}\}$, formed with nonnegative integers, for which $ a_1\equal{}a_{2n\plus{}1}\equal{}0$ and $ |a_k \minus{}a_{k\plus{}1}|\equal{}1$, for all $ k\in\{1,2,\ldots,2n\}$.

Circle is divided into $n$ arcs by $n$ marked points on the circle. After that circle rotate an angle $ 2\pi k/n $ (for some positive integer $ k $), marked points moved to $n$ [i] new points [/i], dividing the circle into $ n $ [i] new arcs[/i]. Prove that there is a new arc that lies entirely in the one of the old arсs. (It is believed that the endpoints of arcs belong to it.) [i]I. Mitrophanov[/i]

Three hoops are arranged concentrically as in the diagram. Each hoop is threaded with $ 20$ beads, $ 10$ of which are black and $ 10$ are white. On each hoop the positions of the beads are labelled $ 1$ through $ 20$ as shown. We say there is a match at position $ i$ if all three beads at position $ i$ have the same color. We are free to slide beads around a hoop, not breaking the hoop. Show that it is always possible to move them into a configuration involving no less than $ 5$ matches.

The graph of $ 2x^2 \plus{} xy \plus{} 3y^2 \minus{} 11x \minus{} 20y \plus{} 40 \equal{} 0$ is an ellipse in the first quadrant of the $ xy$-plane. Let $ a$ and $ b$ be the maximum and minimum values of $ \frac {y}{x}$ over all points $ (x, y)$ on the ellipse. What is the value of $ a \plus{} b$? $ \textbf{(A)}\ 3 \qquad \textbf{(B)}\ \sqrt {10} \qquad \textbf{(C)}\ \frac72 \qquad \textbf{(D)}\ \frac92 \qquad \textbf{(E)}\ 2\sqrt {14}$

An ice ballerina rotates at a constant angular velocity at one particular point. That is, she does not translationally move. Her arms are fully extended as she rotates. Her moment of inertia is $I$. Now, she pulls her arms in and her moment of inertia is now $\frac{7}{10}I$. What is the ratio of the new kinetic energy (arms in) to the initial kinetic energy (arms out)? $ \textbf {(A) } \dfrac {7}{10} \qquad \textbf {(B) } \dfrac {49}{100} \qquad \textbf {(C) } 1 \qquad \textbf {(C) } \dfrac {100}{49} \qquad \textbf {(E) } \dfrac {10}{7} $ [i]Problem proposed by Ahaan Rungta[/i]

A quadrilateral $ABCD$ is inscribed in a circle with center $O$. Let $A_1 B_1 C_1 D_1$ be the image of $ABCD$ after rotation with center $O$ and angle $\alpha \in (0,90^\circ)$. The points $P,Q,R$ and $S$ are intersections of $AB$ and $A_1 B_1$, $BC$ and $B_1 C_1$, $CD$ and $C_1 D_1$, and $DA$ and $D_1 A_1$. Prove that $PQRS$ is a parallelogram.

Let $ABC$ be a triangle with $\angle BAC=60^\circ$. Consider a point $P$ inside the triangle having $PA=1$, $PB=2$ and $PC=3$. Find the maximum possible area of the triangle $ABC$.

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.

In triangle $ ABC$ the medians $ \overline{AD}$ and $ \overline{CE}$ have lengths 18 and 27, respectively, and $ AB \equal{} 24$. Extend $ \overline{CE}$ to intersect the circumcircle of $ ABC$ at $ F$. The area of triangle $ AFB$ is $ m\sqrt {n}$, where $ m$ and $ n$ are positive integers and $ n$ is not divisible by the square of any prime. Find $ m \plus{} n$.

A [i]beautiful configuration[/i] of points is a set of $n$ colored points, such that if a triangle with vertices in the set has an angle of at least $120$ degrees, then exactly 2 of its vertices are colored with the same color. Determine the maximum possible value of $n$.

Which of the following represents the result when the figure shown below is rotated clockwise $120^\circ$ about its center? [asy] unitsize(6); draw(circle((0,0),5)); draw((-1,2.5)--(1,2.5)--(0,2.5+sqrt(3))--cycle); draw(circle((-2.5,-1.5),1)); draw((1.5,-1)--(3,0)--(4,-1.5)--(2.5,-2.5)--cycle); [/asy] [asy] unitsize(6); for (int i = 0; i < 5; ++i) { draw(circle((12*i,0),5)); } draw((-1,2.5)--(1,2.5)--(0,2.5+sqrt(3))--cycle); draw(circle((-2.5,-1.5),1)); draw((1.5,-1)--(3,0)--(4,-1.5)--(2.5,-2.5)--cycle); draw((14,-2)--(16,-2)--(15,-2+sqrt(3))--cycle); draw(circle((12,3),1)); draw((10.5,-1)--(9,0)--(8,-1.5)--(9.5,-2.5)--cycle); draw((22,-2)--(20,-2)--(21,-2+sqrt(3))--cycle); draw(circle((27,-1),1)); draw((24,1.5)--(22.75,2.75)--(24,4)--(25.25,2.75)--cycle); draw((35,2.5)--(37,2.5)--(36,2.5+sqrt(3))--cycle); draw(circle((39,-1),1)); draw((34.5,-1)--(33,0)--(32,-1.5)--(33.5,-2.5)--cycle); draw((50,-2)--(52,-2)--(51,-2+sqrt(3))--cycle); draw(circle((45.5,-1.5),1)); draw((48,1.5)--(46.75,2.75)--(48,4)--(49.25,2.75)--cycle); label("(A)",(0,5),N); label("(B)",(12,5),N); label("(C)",(24,5),N); label("(D)",(36,5),N); label("(E)",(48,5),N); [/asy]

For a given positive integer $n$, we wish to construct a circle of six numbers as shown below so that the circle has the following properties: (a) The six numbers are different three-digit numbers, none of whose digits is a 0. (b) Going around the circle clockwise, the first two digits of each number are the last two digits, in the same order, of the previous number. (c) All six numbers are divisible by $n$. The example above shows a successful circle for $n = 2$. For each of $n = 3, 4, 5, 6, 7, 8, 9$, either construct a circle that satisfies these properties, or prove that it is impossible to do so. [asy] pair a = (1,0); defaultpen(linewidth(0.7)); draw(a..-a..a); int[] num = {264,626,662,866,486,648}; for (int i=0;i<6;++i) { dot(a); label(format("$%d$",num[i]),a,a); a=dir(60*i+60); }[/asy]

A thin, uniform rod has mass $m$ and length $L$. Let the acceleration due to gravity be $g$. Let the rotational inertia of the rod about its center be $md^2$. The rod is suspended from a distance $kd$ from the center, and undergoes small oscillations with an angular frequency $\beta \sqrt{\frac{g}{d}}$. Find an expression for $\beta$ in terms of $k$. $ \textbf{(A)}\ 1+k^2$ $ \textbf{(B)}\ \sqrt{1+k^2}$ $ \textbf{(C)}\ \sqrt{\frac{k}{1+k}}$ $ \textbf{(D)}\ \sqrt{\frac{k^2}{1+k}}$ $ \textbf{(E)}\ \text{none of the above}$

When a right triangle is rotated about one leg, the volume of the cone produced is $800 \pi$ $\text{cm}^3$. When the triangle is rotated about the other leg, the volume of the cone produced is $1920 \pi$ $\text{cm}^3$. What is the length (in cm) of the hypotenuse of the triangle?

Circles $\Gamma_1$ and $\Gamma_2$ have centers $O_1$ and $O_2$ and intersect at $P$ and $Q$. A line through $P$ intersects $\Gamma_1$ and $\Gamma_2$ at $A$ and $B$, respectively, such that $AB$ is not perpendicular to $PQ$. Let $X$ be the point on $PQ$ such that $XA=XB$ and let $Y$ be the point within $AO_1 O_2 B$ such that $AYO_1$ and $BYO_2$ are similar. Prove that $2\angle{O_1 AY}=\angle{AXB}$. [i]Author: Matthew Brennan[/i]

Consider a right circular cylinder with radius $ r$ of the base, hight $ h$. Find the volume of the solid by revolving the cylinder about a diameter of the base.

Given a real number $\lambda > 1$, let $P$ be a point on the arc $BAC$ of the circumcircle of $\bigtriangleup ABC$. Extend $BP$ and $CP$ to $U$ and $V$ respectively such that $BU = \lambda BA$, $CV = \lambda CA$. Then extend $UV$ to $Q$ such that $UQ = \lambda UV$. Find the locus of point $Q$.

In the figure, triangle $ADE$ is produced from triangle $ABC$ by a rotation by $90^o$ about the point $A$. If angle $D$ is $60^o$ and angle $E$ is $40^o$, how large is then angle $u$? [img]https://1.bp.blogspot.com/-6Fq2WUcP-IA/Xzb9G7-H8jI/AAAAAAAAMWY/hfMEAQIsfTYVTdpd1Hfx15QPxHmfDLEkgCLcBGAsYHQ/s0/2009%2BMohr%2Bp1.png[/img]

A $(3n + 1) \times (3n + 1)$ table $(n \in \mathbb{N})$ is given. Prove that deleting any one of its squares yields a shape cuttable into pieces of the following form and its rotations: ''L" shape formed by cutting one square from a $2 \times 2$ squares.

For a function $ f(x)\equal{}(\ln x)^2\plus{}2\ln x$, let $ C$ be the curve $ y\equal{}f(x)$. Denote $ A(a,\ f(a)),\ B(b,\ f(b))\ (a<b)$ the points of tangency of two tangents drawn from the origin $ O$ to $ C$ and the curve $ C$. Answer the following questions. (1) Examine the increase and decrease, extremal value and inflection point , then draw the approximate garph of the curve $ C$. (2) Find the values of $ a,\ b$. (3) Find the volume by a rotation of the figure bounded by the part from the point $ A$ to the point $ B$ and line segments $ OA,\ OB$ around the $ y$-axis.

A square in the coordinate plane has vertices whose $y$-coordinates are $0$, $1$, $4$, and $5$. What is the area of the square? $ \textbf{(A)}\ 16\qquad\textbf{(B)}\ 17\qquad\textbf{(C)}\ 25\qquad\textbf{(D)}\ 26\qquad\textbf{(E)}\ 27 $