Consider, on a plane, the triangle $ ABC, $ vectors $ \vec x,\vec y,\vec z, $ real variable $ \lambda >0 $ and $ M,N,P $ such that $$ \left\{\begin{matrix} \overrightarrow{AM}=\lambda\cdot\vec x\\\overrightarrow{AN}=\lambda\cdot\vec y \\\overrightarrow{AP}=\lambda\cdot\vec z \end{matrix}\right. . $$ Find the locus of the center of mass of $ MNP. $ [i]Dan Brânzei and Gheorghe Iurea[/i]

Player $A$ places an odd number of boxes around a circle and distributes $2013$ balls into some of these boxes. Then the player $B$ chooses one of these boxes and takes the balls in it. After that the player $A$ chooses half of the remaining boxes such that none of two are consecutive and take the balls in them. If player $A$ guarantees to take $k$ balls, find the maximum possible value of $k$.

Two right triangles, $T_1$ and $T_2$, have areas of 1 and 2, respectively. One side length of one triangle is congruent to a different side length in the other, and another side length of the first triangle is congruent to yet another side length in the other. What is the square of the product of the third side lengths of $T_1$ and $T_2$? $\textbf{(A) }\frac{28}3\qquad\textbf{(B) }10\qquad\textbf{(C) }\frac{32}3\qquad\textbf{(D) }\frac{34}3\qquad\textbf{(E) }12$

Two arbitrary distinct lattice points are selected on the coordinate plane within the square marked by the points $(0,0)$, $(3,0)$, $(0,3)$, and $(3,3)$ (the lattice points may lie on a side or a corner of the square). What is the probability that the distance between the two points is at most $\sqrt{2}$?

In an acute triangle $ABC$, let $ \overline {AK}, \overline {BL}, \overline {CM} $ be the altitudes of the triangle concurrent at the point $ H $ and let $ P $ the midpoint of $ \overline {AH} $. Let's define $ S = \overline {BH} \cap \overline {MK} $ and $ T = \overline {LP} \cap \overline {AB} $. Show that $ \overline {TS} \perp \overline {BC} $

Let $k,m,$ and $n$ be natural numbers such that $m+k+1$ is a prime greater than $n+1$. Let $c_{s}=s(s+1).$ Prove that the product \[(c_{m+1}-c_{k})(c_{m+2}-c_{k})\cdots (c_{m+n}-c_{k})\] is divisible by the product $c_{1}c_{2}\cdots c_{n}$.

A sequence of integers $a_1,a_2,\ldots $ is such that $a_1=1,a_2=2$ and for $n\ge 1$, \[a_{n+2}=\left\{\begin{array}{cl}5a_{n+1}-3a_{n}, &\text{if}\ a_n\cdot a_{n+1}\ \text{is even},\\ a_{n+1}-a_{n}, &\text{if}\ a_n\cdot a_{n+1}\ \text{is odd},\end{array}\right. \] Prove that $a_n\not= 0$ for all $n$.

A circle concentric with the inscribed circle of $ABC$ intersects the sides of the triangle at six points forming a convex hexagon $A_1A_2B_1B_2C_1C_2$ (points $C_1$ and $C_2$ on the $AB$ side, $A_1$ and $A_2$ on $BC$, $B_1$ and $B_2$ on $AC$). Prove that if line $A_1B_1$ is parallel to the bisector of angle $B$, then line $A_2C_2$ is parallel to the bisector of angle $C$.

Graph $G$ is defined in 3d space. It has $e$ edges and every vertex are connected if the distance between them is $1.$ Given that there exists the Hamilton cycle, prove that for $e>1,$ we have $$\min d(v)\le 1+2\left(\frac{e}{2}\right)^{0.4}.$$

For any positive integer $n$ define $$E(n)=n(n+1)(2n+1)(3n+1)\cdots (10n+1).$$ Find the greatest common divisor of $E(1),E(2),E(3),\dots ,E(2009).$

The six-pointed star in the figure is regular: all interior angles of the small triangles are equal. Each of the thirteen marked points is assigned a color, green or red. Prove that there are always three points of the same color, which are the vertices of an equilateral triangle.

The scores of this problem were: one time 17/20 (by the runner-up) one time 4/20 (by Andrei Negut) one time 1/20 (by the winner) the rest had zero... just to give an idea of the difficulty. Let $A_{i},B_{i},S_{i}$ ($i=1,2,3$) be invertible real $2\times 2$ matrices such that [list][*]not all $A_{i}$ have a common real eigenvector, [*]$A_{i}=S_{i}^{-1}B_{i}S_{i}$ for $i=1,2,3$, [*]$A_{1}A_{2}A_{3}=B_{1}B_{2}B_{3}=I$.[/list] Prove that there is an invertible $2\times 2$ matrix $S$ such that $A_{i}=S^{-1}B_{i}S$ for all $i=1,2,3$.

Let $a,b,c$ be positive reals satisfying $a+b+c = \sqrt[7]{a} + \sqrt[7]{b} + \sqrt[7]{c}$. Prove that $a^a b^b c^c \ge 1$. [i]Proposed by Evan Chen[/i]

Let $\{a_n\}^\infty_{n=0}$ be the sequence of integers such that $a_0=1$, $a_1=1$, $a_{n+2}=2a_{n+1}-2a_n$. Decide whether $$a_n=\sum_{k=0}^{\left\lfloor\frac n2\right\rfloor}\binom n{2k}(-1)^k.$$

Let be a natural number $ n\ge 2 $ and a real number $ r>1. $ Determine the natural numbers $ k $ having the property that the affixes of $ r^ke^{\pi ki/n} ,r^{k+1}e^{\pi (k+1)i/n} ,r^{k+n}e^{\pi (k+n)i/n} ,r^{k+n+1}e^{\pi (k+n+1) i/n} $ in the complex plane represent the vertices of a trapezoid.

In the plane, there are $n \geqslant 6$ pairwise disjoint disks $D_{1}, D_{2}, \ldots, D_{n}$ with radii $R_{1} \geqslant R_{2} \geqslant \ldots \geqslant R_{n}$. For every $i=1,2, \ldots, n$, a point $P_{i}$ is chosen in disk $D_{i}$. Let $O$ be an arbitrary point in the plane. Prove that \[O P_{1}+O P_{2}+\ldots+O P_{n} \geqslant R_{6}+R_{7}+\ldots+R_{n}.\] (A disk is assumed to contain its boundary.)

Find all the integer pairs $ (x,y)$ such that: \[ x^2\minus{}y^4\equal{}2009\]

There are distinct points $O, A, B, K_1, . . . , K_n, L_1, . . . , L_n$ on a plane such that no three points are collinear. The open line segments $K_1L_1, . . . , K_nL_n$ are coloured red, other points on the plane are left uncoloured. An allowed path from point $O$ to point $X$ is a polygonal chain with first and last vertices at points $O$ and $X$, containing no red points. For example, for $n = 1$, and $K_1 = (-1, 0)$, $L_1 = (1, 0)$, $O = (0,-1)$, and $X = (0,1)$, $OK_1X$ and $OL_1X$ are examples of allowed paths from $O$ to $X$, there are no shorter allowed paths. Find the least positive integer n such that it is possible that the first vertex that is not $O$ on any shortest possible allowed path from $O$ to $A$ is closer to $B$ than to $A$, and the first vertex that is not $O$ on any shortest possible allowed path from $O$ to $B$ is closer to $A$ than to $B$.

Let $b$ and $c$ be real numbers not both equal to $1$ such that $1,b,c$ is an arithmetic progression and $1,c,b$ is a geometric progression. What is $100(b-c)$? [i]Proposed by Noah Kravitz[/i]

Let $I$ be a point on the bisector of angle $BAC$ of a triangle $ABC$. Points $M,N$ are taken on the respective sides $AB$ and $AC$ so that $\angle ABI=\angle NIC$ and $\angle ACI=\angle MIB$. Show that $I$ is the incenter of triangle $ABC$ if and only if points $M,N$ and $I$ are collinear.

Given an ellipse that is not a circle. (1) Prove that the rhombus tangent to the ellipse at all four of its sides with minimum area is unique. (2) Construct this rhombus using a compass and a straight edge.

Find all nondecreasing functions $f:\mathbb R\to \mathbb R$ such that, for all $x,y\in \mathbb R$, $$f(f(x))+f(y)=f(x+f(y))+1.$$ [i]Proposed by Carl Schildkraut[/i]

Define the function $\xi : \mathbb Z^2 \to \mathbb Z$ by $\xi(n,k) = 1$ when $n \le k$ and $\xi(n,k) = -1$ when $n > k$, and construct the polynomial \[ P(x_1, \dots, x_{1000}) = \prod_{n=1}^{1000} \left( \sum_{k=1}^{1000} \xi(n,k)x_k \right). \] (a) Determine the coefficient of $x_1x_2 \dots x_{1000}$ in $P$. (b) Show that if $x_1, x_2, \dots, x_{1000} \in \left\{ -1,1 \right\}$ then $P(x_1,x_2,\dots,x_{1000}) = 0$. [i]Proposed by Evan Chen[/i]

For $a>2$, let $f(t)=\frac{\sin ^ 2 at+t^2}{at\sin at},\ g(t)=\frac{\sin ^ 2 at-t^2}{at\sin at}\ \left(0<|t|<\frac{\pi}{2a}\right)$ and let $C: x^2-y^2=\frac{4}{a^2}\ \left(x\geq \frac{2}{a}\right).$ Answer the questions as follows. (1) Show that the point $(f(t),\ g(t))$ lies on the curve $C$. (2) Find the normal line of the curve $C$ at the point $\left(\lim_{t\rightarrow 0} f(t),\ \lim_{t\rightarrow 0} g(t)\right).$ (3) Let $V(a)$ be the volume of the solid generated by a rotation of the part enclosed by the curve $C$, the nornal line found in (2) and the $x$-axis. Express $V(a)$ in terms of $a$, then find $\lim_{a\to\infty} V(a)$.

A simple graph is called "divisibility", if it's possible to put distinct numbers on its vertices such that there is an edge between two vertices if and only if number of one of its vertices is divisible by another one. A simple graph is called "permutationary", if it's possible to put numbers $1,2,...,n$ on its vertices and there is a permutation $ \pi $ such that there is an edge between vertices $i,j$ if and only if $i>j$ and $\pi(i)< \pi(j)$ (it's not directed!) Prove that a simple graph is permutationary if and only if its complement and itself are divisibility. [i]Proposed by Morteza Saghafian[/i] .