In a finite, simple, connected graph $G$ we play the following game: initially we color all the vertices with a different color. In each step we choose a vertex randomly (with uniform distribution), and then choose one of its neighbors randomly (also with uniform distribution), and color it to the the same color as the originally chosen vertex (if the two chosen vertices already have the same color, we do nothing). The game ends when all the vertices have the same color. Knowing graph $G$ find the probability for each vertex that the game ends with all vertices having the same color as the chosen vertex.

Find all pairs of positive integers \( n \) and \( x \) such that \[ 1^n + 2^n + 3^n + \cdots + n^n = x! \] [i](Petko Lazarov, Bulgaria)[/i]

Let $O$ is a point in a plane $P$ and let $[OX,[OY,[OZ$ is distinct ray in $P$. Prove that, if $A \in [OX$ , $B \in [OY$ and $C \in [OZ$ points such that $\triangle OAB$ , $\triangle OBC$ and $\triangle OCA$ 's perimeter is 2, there is only one $(A,B,C)$ triple

Compute the number of ordered triples of positive integers $(a,b,c)$ such that $a + b + c + ab + bc + ac = abc + 1$.

Find all pairs of nonzero rational numbers $x, y$, such that every positive rational number can be written as $\frac{\{rx\}} {\{ry\}}$ for some positive rational $r$.

Solve the system of equations:$$\left\{ \begin{array}{l} ({x^2} + y)\sqrt {y - 2x} - 4 = 2{x^2} + 2x + y\\ {x^3} - {x^2} - y + 6 = 4\sqrt {x + 1} + 2\sqrt {y - 1} \end{array} \right.(x,y \in \mathbb{R}).$$

Call a polynomial $f$ with positive integer coefficients \textit{triangle-compatible} if any three coefficients of $f$ satisfy the triangle inequality. For instance, $3x^3 + 4x^2 + 6x + 5$ is triangle-compatible, but $3x^3 + 3x^2 + 6x + 5$ is not. Given that $f$ is a degree $20$ triangle-compatible polynomial with $-20$ as a root, what is the least possible value of $f(1)$? [i] Note: this problem is also Premier #3 [/i]

Two circles $\Gamma_1$ and $\Gamma_2$are given with centres $O_1$ and $O_2$ and common exterior tangents $\ell_1$ and $\ell_2$. The line $\ell_1$ intersects $\Gamma_1$ in $A$ and $\Gamma_2$ in $B$. Let $X$ be a point on segment $O_1O_2$, not lying on $\Gamma_1$ or $\Gamma_2$. The segment $AX$ intersects $\Gamma_1$ in $Y \ne A$ and the segment $BX$ intersects $\Gamma_2$ in $Z \ne B$. Prove that the line through $Y$ tangent to $\Gamma_1$ and the line through $Z$ tangent to $\Gamma_2$ intersect each other on $\ell_2$.

A acute triangle $\triangle ABC$ has circumcenter $O$. The circumcircle of $OAB$, called $O_1$, and the circumcircle of $OAC$, called $O_2$, meets $BC$ again at $D ( \not=B )$ and $E ( \not= C )$ respectively. The perpendicular bisector of $BC$ hits $AC$ again at $F$. Prove that the circumcenter of $\triangle ADE$ lies on $AC$ if and only if the centers of $O_1, O_2$ and $F$ are colinear.

For how many numbers $n$ ranging from $1$ to $10$, inclusive, is $5n + 1$ a prime number?

Hui is an avid reader. She bought a copy of the best seller [i]Math is Beautiful[/i]. On the first day, she read $1/5$ of the pages plus $12$ more, and on the second day she read $1/4$ of the remaining pages plus $15$ more. On the third day she read $1/3$ of the remaining pages plus $18$ more. She then realizes she has $62$ pages left, which she finishes the next day. How many pages are in this book? $ \textbf{(A)}\ 120 \qquad\textbf{(B)}\ 180\qquad\textbf{(C)}\ 240\qquad\textbf{(D)}\ 300\qquad\textbf{(E)}\ 360 $

[color=darkred]Let $ m$ and $ n$ be two nonzero natural numbers. In every cell $ 1 \times 1$ of the rectangular table $ 2m \times 2n$ are put signs $ \plus{}$ or $ \minus{}$. We call [i]cross[/i] an union of all cells which are situated in a line and in a column of the table. Cell, which is situated at the intersection of these line and column is called [i]center of the cross[/i]. A transformation is defined in the following way: firstly we mark all points with the sign $ \minus{}$. Then consecutively, for every marked cell we change the signs in the cross, whose center is the choosen cell. We call a table [i]accesible[/i] if it can be obtained from another table after one transformation. Find the number of all [i]accesible[/i] tables.[/color]

How many pairs $(a,b)$ of non-zero real numbers satisfy the equation \[ \frac{1}{a} + \frac{1}{b} = \frac{1}{a+b}? \] $\text{(A)} \ \text{none} \qquad \text{(B)} \ 1 \qquad \text{(C)} \ 2 \qquad \text{(D)} \ \text{one pair for each} ~b \neq 0$ $\text{(E)} \ \text{two pairs for each} ~b \neq 0$

Let $k$ be a circle with center $M.$ There is another circle $k_1$ whose center $M_1$ lies on $k,$ and we denote the line through $M$ and $M_1$ by $g.$ Let $T$ be a point on $k_1$ and inside $k.$ The tangent $t$ to $k_1$ at $T$ intersects $k$ in two points $A$ and $B.$ Denote the tangents (diifferent from $t$) to $k_1$ passing through $A$ and $B$ by $a$ and $b$, respectively. Prove that the lines $a,b,$ and $g$ are either concurrent or parallel.

Let $ABC$ be an isosceles triangle with $AB = AC$, $P$ be the midpoint of the minor arc $AB$ of its circumcircle, and $Q$ be the midpoint of $AC$. A circumcircle of triangle $APQ$ centered at $O$ meets $AB$ for the second time at point $K$. Prove that lines $PO$ and $KQ$ meet on the bisector of angle $ABC$.

Two circles $S_1$ and $S_2$ touch externally at $F$. their external common tangent touches $S_1$ at $A$ and $S_2$ at $B$. A line, parallel to $AB$ and tangent to $S_2$ at $C$, intersects $S_1$ at $D$ and $E$. Prove that points $A,F,C$ are collinear. (A. Kalinin)

Triangle $ABC$ has sides $a = |BC| > b = |AC|$. The points $K$ and $H$ on the segment $BC$ satisfy $|CH| = (a + b)/3$ and $|CK| = (a - b)/3$. If $G$ is the centroid of triangle $ABC$, prove that $\angle KGH = 90^o$.

Let $p$ be a prime number. Prove that from a $p^2\times p^2$ array of squares, we can select $p^3$ of the squares such that the centers of any four of the selected squares are not the vertices of a rectangle with sides parallel to the edges of the array.

Let $\{x_n\}$ be a sequence of natural numbers such that \[(a) 1 = x_1 < x_2 < x_3 < \ldots; \quad (b) x_{2n+1} \leq 2n \quad \forall n.\] Prove that, for every natural number $k$, there exist terms $x_r$ and $x_s$ such that $x_r - x_s = k.$

Let $A$ be a set of positive integers containing the number $1$ and at least one more element. Given that for any two different elements $m, n$ of A the number $ \frac{m+1 }{(m+1,n+1) }$ is also an element of $A$, prove that $A$ coincides with the set of positive integers.

Prove that if the angles $\alpha$ and $\beta$ satisfy $\sin(\alpha + \beta) = 2 \sin \alpha$, Then $$\alpha < \beta$$

You are given a checkered square, the side of which is $n – 1$ long and contains $n \geq 10$ nodes. A non-return path is a path along edges, the intersection of which with any horizontal or vertical line is a segment, point or empty set, and which does not pass along any edge more than once. What is the smallest number of non-return paths that can cover all the edges? (An edge is a unit segment between adjacent nodes.)

Find all natural numbers which can be written in the form $\frac{(a+b+c)^2}{abc}$ , where $a,b,c \in N$.

In the game of Yahtzee , players have to achieve various combinations of values with $5$ dice. In a round, a player can roll the dice three times. At the second and third rolls, he can choose which dice to re-roll and which to keep. What is the probability that a player achieves at least four $6$’s in a round, given that he plays with the optimal strategy to maximise this probability? Writing the answer as $p/q$ where $p$ and $q$ are coprime, you should submit the sum of all prime factors of $p$, counted with multiplicity. So for example if you obtained $\frac{p}{q} = \frac{3^4 \cdot 11}{ 2^5 \cdot 5}$ then the submitted answer should be $4 \cdot 3 + 11 = 23$.

Given triangle $ABC$ and its incircle $\omega$ prove you can use just a ruler and drawing at most 8 lines to construct points$A',B',C'$ on $\omega$ such that $A,B',C'$ and $B,C',A'$ and $C,A',B'$ are collinear.