Let $P(X)=X^n+a_1X^{n-1}+\ldots+a_n$ be a plynomial with real roots $x_1. x_2,\ldots,x_n$. Denote $E_k=x_1^k+x_2^k+\ldots+x_n^k, \forall k\in\mathbb{N}$. There exists an $m\in\mathbb{N}$ such that $E_m=E_{m+1}=E_{m+2}=1$. Find $\max\{P(-2),P(2)\}$.

For a given a positive integer $n$, find all integers $x_1, x_2,... , x_n$ subject to $0 < x_1 < x_2 < ...< x_n < x_{n+1}$ and $$x_nx_{n+1} \le 2(x_1 + x_2 + ... + x_n).$$

Let be a real number $ a, $ and a function $ f:\mathbb{R}_{>0 }\longrightarrow\mathbb{R}_{>0 } . $ Show that the following relations are equivalent. $ \text{(i)}\quad\varepsilon\in\mathbb{R}_{>0 } \implies\left( \lim_{x\to\infty } \frac{f(x)}{x^{a+\varepsilon }} =0\wedge \lim_{x\to\infty } \frac{f(x)}{x^{a-\varepsilon }} =\infty \right) $ $ \text{(ii)}\quad\lim_{x\to\infty } \frac{\ln f(x)}{\ln x } =a $

Let $\omega$ be a circle with center $O$ and let $A$ be a point outside $\omega$. The tangents from $A$ touch $\omega$ at points $B$, and $C$. Let $D$ be the point at which the line $AO$ intersects the circle such that $O$ is between $A$ and $D$. Denote by $X$ the orthogonal projection of $B$ onto $CD$, by $Y$ the midpoint of the segment $BX$ and by $Z$ the second point of intersection of the line $DY$ with $\omega$. Prove that $ZA$ and $ZC$ are perpendicular to each other.

Prove that for a positive number $ r>1, $ there is a nondecreasing sequence of positive numbers $ \left( a_v\right)_{v\ge 1} $ such that $$ r=\lim_{n\to\infty }\sum_{i=1}^n \frac{a_i}{a_{i+1}} . $$

Are there positive integers $a, b, c$ greater than $2011$ such that: $(a+ \sqrt{b})^c=...2010,2011...$?

A sum of $335$ pairwise distinct positive integers equals $100000$. a) What is the least number of uneven integers in that sum? b) What is the greatest number of uneven integers in that sum?

Triangle $ABC$ has $AC = 3$, $BC = 5$, $AB = 7$. A circle is drawn internally tangent to the circumcircle of $ABC$ at $C$, and tangent to $AB$. Let $D$ be its point of tangency with $AB$. Find $BD - DA$. [asy] /* File unicodetex not found. */ /* Geogebra to Asymptote conversion, documentation at artofproblemsolving.com/Wiki, go to User:Azjps/geogebra */ import graph; size(6cm); real labelscalefactor = 2.5; /* changes label-to-point distance */ pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); /* default pen style */ pen dotstyle = black; /* point style */ real xmin = -4.5, xmax = 7.01, ymin = -3, ymax = 8.02; /* image dimensions */ /* draw figures */ draw(circle((1.37,2.54), 5.17)); draw((-2.62,-0.76)--(-3.53,4.2)); draw((-3.53,4.2)--(5.6,-0.44)); draw((5.6,-0.44)--(-2.62,-0.76)); draw(circle((-0.9,0.48), 2.12)); /* dots and labels */ dot((-2.62,-0.76),dotstyle); label("$C$", (-2.46,-0.51), SW * labelscalefactor); dot((-3.53,4.2),dotstyle); label("$A$", (-3.36,4.46), NW * labelscalefactor); dot((5.6,-0.44),dotstyle); label("$B$", (5.77,-0.17), SE * labelscalefactor); dot((0.08,2.37),dotstyle); label("$D$", (0.24,2.61), SW * labelscalefactor); clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); label("$7$",(-3.36,4.46)--(5.77,-0.17), NE * labelscalefactor); label("$3$",(-3.36,4.46)--(-2.46,-0.51),SW * labelscalefactor); label("$5$",(-2.46,-0.51)--(5.77,-0.17), SE * labelscalefactor); /* end of picture */ [/asy]

Find all prime numbers $p$ and $q$ such that $$1 + \frac{p^q - q^p}{p + q}$$ is a prime number. [i]Proposed by Dorlir Ahmeti, Albania[/i]

A median of an acute-angled triangle dissects it into two triangles. Prove that each of them can be covered by a semidisc congruent to a half of the circumdisc of the initial triangle.

Find the number of triples of integers $(a, b, c)$ where $1 \le a < b < c \le 20$ and $a$, $b$, $c$ form the sides of a non-degenerate triangle. [i]Team #3[/i]

In convex quadrilateral $ ABCD$, $ AB \equal{} BC$ and $ AD \equal{} DC$. Point $ E$ lies on segment $ AB$ and point $ F$ lies on segment $ AD$ such that $ B$, $ E$, $ F$, $ D$ lie on a circle. Point $ P$ is such that triangles $ DPE$ and $ ADC$ are similar and the corresponding vertices are in the same orientation (clockwise or counterclockwise). Point $ Q$ is such that triangles $ BQF$ and $ ABC$ are similar and the corresponding vertices are in the same orientation. Prove that points $ A$, $ P$, $ Q$ are collinear.

On the 8×8 chessboard there are two identical markers in the squares a1 and c3. Alice and Bob in turn make the following moves (the first move is Alice’s): a player picks any marker and moves it horizontally to the right or vertically upwards through any number of squares. The aim of each player is to get tothe square h8. Which player has a winning strategy no matter what does his opponent? (There may be only one marker on a square,the markers may not go through each other.) The 8x8 chessboard consists of columns lettered a to h from left to right and rows numbered 1-8 from bottom to top [img]https://cdn.artofproblemsolving.com/attachments/1/f/4c5548f606fda915e0a50a8cf886ff93e1f86d.png[/img]

Let $v_1,v_2,\ldots,v_n$ vectors in $\mathbb{R}^2$ such that $|v_i|\leq 1$ for $1 \leq i \leq n$ and $\sum_{i=1}^n v_i=0$. Prove that there exists a permutation $\sigma$ of $(1,2,\ldots,n)$ such that $\left|\sum_{j=1}^k v_{\sigma(j)}\right| \leq\sqrt 5$ for every $k$, $1\leq k \leq n$. [i]Remark[/i]: If $v = (x,y)\in \mathbb{R}^2$, $|v| = \sqrt{x^2 + y^2}$.

Let $A$ be a corner of a cube. Let $B$ and $C$ the midpoints of two edges in the positions shown on the figure below: [center][img]http://i.imgur.com/tEODnV0.png[/img][/center] The intersection of the cube and the plane containing $A,B,$ and $C$ is some polygon, $P$. [list=a] [*] How many sides does $P$ have? Justify your answer. [*] Find the ratio of the area of $P$ to the area of $\triangle{ABC}$ and prove that your answer is correct.

2008 white stones and 1 black stone are in a row. An 'action' means the following: select one black stone and change the color of neighboring stone(s). Find all possible initial position of the black stone, to make all stones black by finite actions.

A line parallel to the side $AC$ of triangle $ABC$ cuts the side $AB$ at $K$ and the side $BC$ at $M$. $O$ is the intersection point of $AM$ and $CK$. If $AK = AO$ and $KM = MC$, prove that $AM = KB$.

Determine if there exists pairwise distinct positive integers $a_1$, $a_2$,$ ...$, $a_{101}$, $b_1$, $b_2$,$ ...$, $b_{101}$ satisfying the following property: for each non-empty subset $S$ of $\{1, 2, ..., 101\}$ the sum $\sum_{i \in S} a_i$ divides $100! + \sum_{i \in S} b_i$.

[i]Second Test, May 16[/i] Let $a$, $b$, and $c$ be positive real numbers such that $abc = 1$. Prove that $\frac{ab}{a^{5}+b^{5}+ab}+\frac{bc}{b^{5}+c^{5}+bc}+\frac{ca}{c^{5}+a^{5}+ca}\le 1$ . When does equality hold?

Let $f : R^+ \to R$ be a strictly increasing function such that $f\left(\frac{x+y}{2}\right) < \frac{f(x)+ f(y)}{2}$ for all $x,y > 0$. Prove that the sequence $a_n = f(n)$ ($n \in N$) does not contain an infinite arithmetic progression.

After each Goober ride, the driver rates the passenger as $1$, $2$, $3$, $4$, or $5$ stars. The passenger's overall rating is determined as the average of all of the ratings given to him or her by drivers so far. Noah had been on several rides, and his rating was neither $1$ nor $5$. Then he got a $1$ star on a ride because he barfed on the driver. Show that the number of $5$ stars that Noah needs in order to climb back to at least his overall rating before barng is independent of the number of rides that he had taken.

Koshchey opened an account at the bank. Initially, it had 0 rubles. On the first day, Koshchey puts $k>0$ rubles in, and every next day adds one ruble more there than the day before. Each time after Koshchey deposits money into the account, the total amount in the account is divided by two by the bank. Find all such $k{}$ for which the amount on the account will always be an integer number of rubles. [i]Proposed by S. Berlov[/i]

Compute the base 10 value of $14641_{99}$

Let $A$ be an $n\times n$ matrix with strictly positive elements and two vectors $u,v\in\mathbb{R}^n$, also with strictly positive elements, such that $$Au=v\text{ and }Av=u.$$ Prove that $u=v$.

Find the largest positive integer $n$ with the property that $n+6(p^3+1)$ is prime whenever $p$ is a prime number such that $2\le p<n$. Justify your answer.