Find maximum value of number $a$ such that for any arrangement of numbers $1,2,\ldots ,10$ on a circle, we can find three consecutive numbers such their sum bigger or equal than $a$.

There are $10001$ students at an university. Some students join together to form several clubs (a student may belong to different clubs). Some clubs join together to form several societies (a club may belong to different societies). There are a total of $k$ societies. Suppose that the following conditions hold: [i]i.)[/i] Each pair of students are in exactly one club. [i]ii.)[/i] For each student and each society, the student is in exactly one club of the society. [i]iii.)[/i] Each club has an odd number of students. In addition, a club with ${2m+1}$ students ($m$ is a positive integer) is in exactly $m$ societies. Find all possible values of $k$. [i]Proposed by Guihua Gong, Puerto Rico[/i]

A number $N$ has $2009$ positive factors. What is the maximum number of positive factors that $N^2$ could have?

If the angles $ a,b,c,d,$ and $ e$ are known, what does the angle $ u$ equal? [img]http://i250.photobucket.com/albums/gg265/geometry101/GeometryImage1.jpg[/img] A. $ 108^\circ$ B. $ 2a$ C. $ 3a$ D. $ c\plus{}d$ E. $ a\plus{}c\plus{}d$

Let $ABTCD$ be a convex pentagon with area $22$ such that $AB = CD$ and the circumcircles of triangles $TAB$ and $TCD$ are internally tangent. Given that $\angle{ATD} = 90^{\circ}, \angle{BTC} = 120^{\circ}, BT = 4,$ and $CT = 5$, compute the area of triangle $TAD$.

Two perpendicular chords of a circle, $AM, BN$ , which intersect at point $K$, define on the circle four arcs with pairwise different length, with $AB$ being the smallest of them. We draw the chords $AD, BC$ with $AD // BC$ and $C, D$ different from $N, M$ . If $L$ is the intersection point of $DN, M C$ and $T$ the intersection point of $DC, KL,$ prove that $\angle KTC = \angle KNL$.

Determine all functions $f:\mathbb R\to\mathbb R$ that satisfy equation: $$ f(x^3+y^3) =f(x^3) + 3x^2f(x)f(y) + 3f(x)f(y)^2 + y^6f(y) $$ for all reals $x,y$

Nick is a runner, and his goal is to complete four laps around a circuit at an average speed of $10$ mph. If he completes the first three laps at a constant speed of only $9$ mph, what speed does he need to maintain in miles per hour on the fourth lap to achieve his goal?

Let $n \ge 3$ be a fixed integer. A game is played by $n$ players sitting in a circle. Initially, each player draws three cards from a shuffled deck of $3n$ cards numbered $1, 2, \dots, 3n$. Then, on each turn, every player simultaneously passes the smallest-numbered card in their hand one place clockwise and the largest-numbered card in their hand one place counterclockwise, while keeping the middle card. Let $T_r$ denote the configuration after $r$ turns (so $T_0$ is the initial configuration). Show that $T_r$ is eventually periodic with period $n$, and find the smallest integer $m$ for which, regardless of the initial configuration, $T_m=T_{m+n}$. [i]Proposed by Carl Schildkraut and Colin Tang[/i]

Incircle of triangle $ ABC$ touches $ AB,AC$ at $ P,Q$. $ BI, CI$ intersect with $ PQ$ at $ K,L$. Prove that circumcircle of $ ILK$ is tangent to incircle of $ ABC$ if and only if $ AB\plus{}AC\equal{}3BC$.

Inside of the square $ABCD$ the point $P$ is given such that $|PA|:|PB|:|PC|=1:2:3$. Find $\angle APB$.

At a summer school there are $7$ courses. Each participant was a student in at least one course, and each course was taken by exactly $40$ students. It is known that for each $2$ courses there were at most $9$ students who took them both. Prove that at least $120$ students participated at this summer school.

Find all polynomials $P(x) = x^n +a_1x^{n-1} +...+a_n$ whose zeros (with their multiplicities) are exactly $a_1,a_2,...,a_n$.

Let $ABC$ be a triangle with incenter $I$, incircle $\gamma$ and circumcircle $\Gamma$. Let $M,N,P$ be the midpoints of sides $\overline{BC}$, $\overline{CA}$, $\overline{AB}$ and let $E,F$ be the tangency points of $\gamma$ with $\overline{CA}$ and $\overline{AB}$, respectively. Let $U,V$ be the intersections of line $EF$ with line $MN$ and line $MP$, respectively, and let $X$ be the midpoint of arc $BAC$ of $\Gamma$. (a) Prove that $I$ lies on ray $CV$. (b) Prove that line $XI$ bisects $\overline{UV}$.

Find all functions $f : R \to R$ satisfying $f(x)f(y) = f(x + y) + xy$ for all $x, y \in R$.

Let $n$ be a positive integer with exactly twelve positive divisors $1=d_1 < \cdots < d_{12}=n$. We say $n$ is [i]trite[/i] if \[ 5 + d_6(d_6+d_4) = d_7d_4. \] Compute the sum of the two smallest trite positive integers. [i]Proposed by Brandon Wang[/i]

Let $ x_{1},x_{2},\cdots,x_{m},y_{1},y_{2},\cdots,y_{n}$ be positive real numbers. Denote by $ X \equal{} \sum_{i \equal{} 1}^{m}x,Y \equal{} \sum_{j \equal{} 1}^{n}y.$ Prove that $ 2XY\sum_{i \equal{} 1}^{m}\sum_{j \equal{} 1}^{n}|x_{i} \minus{} y_{j}|\ge X^2\sum_{j \equal{} 1}^{n}\sum_{l \equal{} 1}^{n}|y_{i} \minus{} y_{l}| \plus{} Y^2\sum_{i \equal{} 1}^{m}\sum_{k \equal{} 1}^{m}|x_{i} \minus{} x_{k}|$

The incircle of $\triangle{ABC}$ touches $BC$, $CA$, $AB$ at $D$, $E$, $F$, respectively. Point $P$ is chosen on $EF$ such that $AP$ is parallel to $BC$, and $AD$ intersects the incircle of $\triangle{ABC}$ again at $G$. Show that $\angle AGP = 90^{\circ}$.

Given an isosceles triangle $ABC$ with base $BC$. On the sides $BC$, $AC$ and $AB$ points $X,Y$ and $Z$ are chosen respectively such that triangles $ABC$ and $YXZ$ are similar. Point $W$ is symmetric to point $X$ with respect to the midpoint of $BC$. Prove that points $X,Y,Z$ and $W$ lie on a circle.

Let $n \geq 1$ be an integer. What is the maximum number of disjoint pairs of elements of the set $\{ 1,2,\ldots , n \}$ such that the sums of the different pairs are different integers not exceeding $n$?

Tyler has two calculators, both of which initially display zero. The first calculators has only two buttons, $[+1]$ and $[\times 2]$. The second has only the buttons $[+1]$ and $[\times 4]$. Both calculators update their displays immediately after each keystroke. A positive integer $n$ is called [i]ambivalent[/i] if the minimum number of keystrokes needed to display $n$ on the first calculator equals the minimum number of keystrokes needed to display $n$ on the second calculator. Find the sum of all ambivalent integers between $256$ and $1024$ inclusive. [i]Proposed by Joshua Xiong[/i]

Consider all binary sequences (sequences consisting of 0’s and 1’s). In such a sequence the following four types of operation are allowed: (a) $010 \rightarrow 1$, (b) $1 \rightarrow 010$, (c) $110 \rightarrow 0$, and (d) $0 \rightarrow 110$. Determine if it is possible to obtain the sequence $100...0$ (with $2003$ zeroes) from the sequence $0...01$ (with $2003$ zeroes).

Let $n\geq 2$ be an integer. For each natural $m$ and each integer sequence $0<k_1<k_2<\cdots <k_m$ for which $k_1+\cdots+k_m=n$, Michael wrote down the number $\frac{1}{k_1\cdot k_2\cdots k_m} $ on the board. Prove that the sum of the numbers on the board is less than $1$.

Point $I$ is incenter of triangle $ABC$ in which $AB \neq AC$. Lines $BI$ and $CI$ intersect sides $AC$ and $AB$ in points $D$ and $E$, respectively. Determine all measures of angle $BAC$, for which may be $DI = EI$.

If $ \left (a\plus{}\frac{1}{a} \right )^2\equal{}3$, then $ a^3\plus{}\frac{1}{a^3}$ equals: $ \textbf{(A)}\ \frac{10\sqrt{3}}{3} \qquad \textbf{(B)}\ 3\sqrt{3} \qquad \textbf{(C)}\ 0 \qquad \textbf{(D)}\ 7\sqrt{7} \qquad \textbf{(E)}\ 6\sqrt{3}$