Let $C_{1}$, $C_{2}$ be circles intersecting in $X$, $Y$ . Let $A$, $D$ be points on $C_{1}$ and $B$, $C$ on $C_2$ such that $A$, $X$, $C$ are collinear and $D$, $X$, $B$ are collinear. The tangent to circle $C_{1}$ at $D$ intersects $BC$ and the tangent to $C_{2}$ at $B$ in $P$, $R$ respectively. The tangent to $C_2$ at $C$ intersects $AD$ and tangent to $C_1$ at $A$, in $Q$, $S$ respectively. Let $W$ be the intersection of $AD$ with the tangent to $C_{2}$ at $B$ and $Z$ the intersection of $BC$ with the tangent to $C_1$ at $A$. Prove that the circumcircles of triangles $YWZ$, $RSY$ and $PQY$ have two points in common, or are tangent in the same point. Proposed by Misiakos Panagiotis

We say that an integer $m$ is a perfect power if there are $a\in\mathbf{Z}$, $b\in\mathbf{N}$ with $b > 1$ such that $m = a^b$. Find all polynomials $P\in\mathbf{Z}[x]$ such that $P(n)$ is a perfect power for every $n\in\mathbf{N}$.

Sebastian, the traveling ant, walks on top of some square boards. He just walks horizontally or vertically through the squares of the boards and does not pass through the same square twice. On a board of $7\times 7$, in which squares can Sebastian start his journey so that he can pass through all the squares on the board?

Let $ I$ be incenter of triangle $ ABC$, $ M$ be midpoint of side $ BC$, and $ T$ be the intersection point of $ IM$ with incircle, in such a way that $ I$ is between $ M$ and $ T$. Prove that $ \angle BIM\minus{}\angle CIM\equal{}\frac{3}2(\angle B\minus{}\angle C)$, if and only if $ AT\perp BC$.

It is known that the general term $\{a_n\}$ of the sequence is $a_n =(\sqrt3 +\sqrt2)^{2n}$ ($n \in N*$), let $b_n= a_n +\frac{1}{a_n}$ . (1) Find the recurrence relation between $b_{n+2}$, $b_{n+1}$, $b_n$. (2) Find the unit digit of the integer part of $a_{2011}$.

In the table below, place the numbers 1--12 in the shaded cells. You start at the center cell (marked with $*$). You repeatedly move up, down, left, or right, chosen uniformly at random each time, until reaching a shaded cell. Your score is the number in the shaded cell that you end up at. Let $m$ be the least possible expected value of your score (based on how you placed the numbers), and $M$ be the greatest possible expected value of your score. Compute $m \cdot M$. [i]Proposed by Justin Hsieh[/i]

Prove that there exists a positive integer $n$ such that the last digits of $n^3$ are $...201320132013$.

Prove that $a=2$ is the greatest real number for which the inequality: $$ \frac{x_1}{x_n+x_2}+\frac{x_2}{x_1+x_3}+\dots+\frac{x_n}{x_{n-1}+x_1} \ge a $$ holds true for any $n \ge 4$ and any positive real numbers $x_1, x_2,\dots,x_n$.

Show that a function $ f(x)\equal{}\int_{\minus{}1}^1 (1\minus{}|\ t\ |)\cos (xt)\ dt$ is continuous at $ x\equal{}0$.

Find all continuous functions $f: \mathbb{R}\to\mathbb{R}$ such that for all real $x$ we have \[f(x)=f\left(x^{2}+\frac{x}{3}+\frac{1}{9}\right). \]

Each of integers from $1$ to $20$ are placed into the dots below. Two dots are [i]adjacent[/i], if below figure contains a line segment connecting them. Prove that how the numbers are arranged, it is possible to find an adjacent pair such that the difference between the numbers written on them is greater than $3$. [asy] real u=0.25cm; for(int i = 0; i < 4; ++i) { real v = u*(i+1); pair P1 = dir(90+0*72)*(0,v); pair P2 = dir(90+1*72)*(0,v); pair P3 = dir(90+2*72)*(0,v); pair P4 = dir(90+3*72)*(0,v); pair P5 = dir(90+4*72)*(0,v); dot(P1);dot(P2); dot(P3);dot(P4);dot(P5); path p = P1--P2--P3--P4--P5--cycle; draw(p); } [/asy]

As shown in figure , it is known that $ABCD$ is parallelogram, $A,B,C$ lie on circle $\odot O_1$, $AD$ and $BD$ intersect $\odot O$ at points $E$ and $F$ respectively, $C,D,F$ lie on circle $\odot O_2$, $AD$ intersects $\odot O_2$ at point $G$. If the radii of circles $\odot O_1$, $\odot O_2$ are $R_1, R_2$ respectively, prove that $\frac{EG}{AD}=\frac{R_2^2}{R_1^2}$. [img]https://cdn.artofproblemsolving.com/attachments/d/f/1d9925a77d4f3fe068bd24364fb396eaa9a27a.png[/img]

Can the set $\{1,2,...,33\}$ be partitioned into $11$ three-element sets, in each of which one element equals the sum of the other two?

Let $m,n\geq 2$ be integers. Let $f(x_1,\dots, x_n)$ be a polynomial with real coefficients such that $$f(x_1,\dots, x_n)=\left\lfloor \frac{x_1+\dots + x_n}{m} \right\rfloor\text{ for every } x_1,\dots, x_n\in \{0,1,\dots, m-1\}.$$ Prove that the total degree of $f$ is at least $n$.

Let $a_1, a_2,..., a_n$ be positive integers and $a$ positive integer greater than $1$ which is a multiple of the product $a_1a_2...a_n$. Prove that $a^{n+1} + a - 1$ is not divisible by $(a + a_1 -1)(a + a_2 - 1) ... (a + a_n -1)$.

Let $A_n$ be the number of ways to tile a $4 \times n$ rectangle using $2 \times 1$ tiles. Prove that $A_n$ is divisible by 2 if and only if $A_n$ is divisible by 3.

Let $a_1,a_2,\cdots,a_n,b_1,b_2,\cdots,b_n$ are real numbers in $[1,2]$. If $\sum_{i=1}^{n}a_i^2=\sum_{i=1}^{n}b_i^2$, prove that $$\sum_{i=1}^{n}\frac{a_i^3}{b_i}\leq\frac{17}{10}\sum_{i=1}^{n}a_i^2.$$ Find when the equality holds.

A simple pendulum of length $L$ is constructed from a point object of mass $m$ suspended by a massless string attached to a fixed pivot point. A small peg is placed a distance $2L/3$ directly below the fixed pivot point so that the pendulum would swing as shown in the figure below. The mass is displaced $5$ degrees from the vertical and released. How long does it take to return to its starting position? [asy] // Code by riben size(275); draw(circle((0,0),1),linewidth(2)); filldraw(circle((0,0),1),gray); draw((0,0)--(0,-70.8)); draw(circle((0,-71.8),3)); filldraw(circle((0,-71.8),3),gray); draw(circle((0,-45),1)); filldraw(circle((0,-45),1),gray); filldraw(circle((15,-70),3),gray,linewidth(0.2)); filldraw(circle((-15,-67),3),gray,linewidth(0.2)); draw((0,0)--(14.5,-66.5),dashed); draw((0,-45)--(-13,-65),dashed); // Labels label("Fixed Pivot Point",(0,0),4*E); label("Small Peg",(0,-45),12*E); label("Point Object of mass m",(0,-70),17*E); draw((-40,1)--(-40,-76.8),EndArrow(size=5)); draw((-40,-76.8)--(-40,1),EndArrow(size=5)); label("L",(-40,-37.9),E*2); [/asy] (A) $\pi \sqrt{\frac{L}{g}} \left(1+\sqrt{\frac{2}{3}}\right)$ (B) $\pi \sqrt{\frac{L}{g}} \left(2+\frac{2}{\sqrt{3}}\right)$ (C) $\pi \sqrt{\frac{L}{g}} \left(1+\frac{1}{3}\right)$ (D) $\pi \sqrt{\frac{L}{g}} \left(1+\sqrt{3}\right)$ (E) $\pi \sqrt{\frac{L}{g}} \left(1+\frac{1}{\sqrt{3}}\right)$

Suppose that $A = \left(\frac12, \sqrt3 \right)$. Suppose that $B, C, D$ are chosen on the ellipse $x^2 + (y/2)^2 = 1$ such that the area of $ABCD$ is maximized. Assume that $A, B, C, D$ lie on the ellipse going counterclockwise. What are all possible values of $B$?

A set $S=\{ s_1,s_2,...,s_k\}$ of positive real numbers is "polygonal" if $k\geq 3$ and there is a non-degenerate planar $k-$gon whose side lengths are exactly $s_1,s_2,...,s_k$; the set $S$ is multipolygonal if in every partition of $S$ into two subsets,each of which has at least three elements, exactly one of these two subsets in polygonal. Fix an integer $n\geq 7$. (a) Does there exist an $n-$element multipolygonal set, removal of whose maximal element leaves a multipolygonal set? (b) Is it possible that every $(n-1)-$element subset of an $n-$element set of positive real numbers be multipolygonal?

Given the true statement: If a quadrilateral is a square, then it is a rectangle. It follows that, of the converse and the inverse of this true statement is: $ \textbf{(A)}\ \text{only the converse is true} \qquad\textbf{(B)}\ \text{only the inverse is true }\qquad \textbf{(C)}\ \text{both are true} \qquad$ $\textbf{(D)}\ \text{neither is true} \qquad\textbf{(E)}\ \text{the inverse is true, but the converse is sometimes true} $

Let $m$ be a fixed integer greater than $1$. The sequence $x_0$, $x_1$, $x_2$, $\ldots$ is defined as follows: \[x_i = \begin{cases}2^i&\text{if }0\leq i \leq m - 1;\\\sum_{j=1}^mx_{i-j}&\text{if }i\geq m.\end{cases}\] Find the greatest $k$ for which the sequence contains $k$ consecutive terms divisible by $m$ . [i]Proposed by Marcin Kuczma, Poland[/i]

We define an operation $\oplus$ on the set $\{0, 1\}$ by \[ 0 \oplus 0 = 0 \,, 0 \oplus 1 = 1 \,, 1 \oplus 0 = 1 \,, 1 \oplus 1 = 0 \,.\] For two natural numbers $a$ and $b$, which are written in base $2$ as $a = (a_1a_2 \ldots a_k)_2$ and $b = (b_1b_2 \ldots b_k)_2$ (possibly with leading 0's), we define $a \oplus b = c$ where $c$ written in base $2$ is $(c_1c_2 \ldots c_k)_2$ with $c_i = a_i \oplus b_i$, for $1 \le i \le k$. For example, we have $7 \oplus 3 = 4$ since $ 7 = (111)_2$ and $3 = (011)_2$. For a natural number $n$, let $f(n) = n \oplus \left[ n/2 \right]$, where $\left[ x \right]$ denotes the largest integer less than or equal to $x$. Prove that $f$ is a bijection on the set of natural numbers.

Let $p = 10009$ be a prime number. Determine the number of ordered pairs of integers $(x,y)$ such that $1\le x,y \le p$ and $x^3-3xy+y^3+1$ is divisible by $p$.

Find all polynomials of degree 3, such that for each $x,y\geq 0$: \[p(x+y)\geq p(x)+p(y)\]