The tourist has come to the Moscow by train. All-day-long he wandered randomly through the streets. Than he had a supper in the cafe on the square and decided to return to the station only through the streets that he has passed an odd number of times. Prove that he is always able to do that.

$\omega$ is the circumcirle of an acute triangle $ABC$. The tangent line passing through $A$ intersects the tangent lines passing through points $B$ and $C$ at points $K$ and $L$, respectively. The line parallel to $AB$ through $K$ and the line parallel to $AC$ through $L$ intersect at point $P$. Prove that $BP=CP$. (Author: P. Kozhevnikov)

Mr. DoBa has a bag of markers. There are 2 blue, 3 red, 4 green, and 5 yellow markers. Mr. DoBa randomly takes out two markers from the bag. The probability that these two markers are different colors can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $m+n$. [i]Proposed by Raina Yang[/i]

Let $V$ be the set of vertices of a regular $21$-gon. Given a non-empty subset $U$ of $V$ , let $m(U)$ be the number of distinct lengths that occur between two distinct vertices in $U$. What is the maximum value of $\frac{m(U)}{|U|}$ as $U$ varies over all non-empty subsets of $V$ ?

Suppose $m>n>1$ are positive integers such that there exist $n$ complex numbers $x_1, x_2, \ldots, x_n$ for which [list] [*]$x_1^k+x_2^k+\ldots+x_n^k=1$ for $k=1, 2, \ldots, n-1;$ [*]$x_1^n+x_2^n+\ldots+x_n^n=2;$ and [*]$x_1^m+x_2^m+\ldots+x_n^m= 4.$ [/list] Compute the smallest possible value of $m+n.$

The points $A, B, C, D$ are marked on the straight line in this order. Circle $\omega_1$ passes through points $A$ and $C$, and the circle $\omega_2$ passes through points $B$ and $D$. On the circle $\omega_2$, the point $E$ is marked so that $AB = BE$, and on the circle $\omega_1$, the point $F$ is marked so that $CD = CF$. The line $AE$ intersects the circle $\omega_2$ a second time at point $X$, and the line $DF$ intersects the circle $\omega_1$ at point $Y$. Prove that the $XY$ lines and $AD$ is perpendicular. [i]A.D.Tereshin[/i]

The following sequence lists all the positive rational numbers that do not exceed $\frac12$ by first listing the fraction with denominator 2, followed by the one with denominator 3, followed by the two fractions with denominator 4 in increasing order, and so forth so that the sequence is \[ \frac12,\frac13,\frac14,\frac24,\frac15,\frac25,\frac16,\frac26,\frac36,\frac17,\frac27,\frac37,\cdots. \] Let $m$ and $n$ be relatively prime positive integers so that the $2012^{\text{th}}$ fraction in the list is equal to $\frac{m}{n}$. Find $m+n$.

Let $a + bi$ and $c + di$ be two roots of the equation $x^n = 1990$, where $n \geq 3$ is an integer and $a,b,c,d \in \mathbb R$. Under the linear transformation $f =\left(\begin{array}{cc}a&c\\b &d\end{array}\right)$, we have $(2, 1) \to (1, 2)$. Denote $r$ to be the distance from the image of $(2, 2)$ to the origin. Find the range of $r.$

Suppose $x$ is a real number such that $x^2=10x+7$. Find the unique ordered pair of integers $(m,n)$ such that $x^3=mx+n$.

A triangle with sides $a, b, c$ is folded along a line $\ell$ so that a vertex $C$ is on side $c$. Find the segments on which point $C$ divides $c$, given that the angles adjacent to $\ell$ are equal. [i](2 points)[/i]

Nondegenerate $ \triangle ABC$ has integer side lengths, $ BD$ is an angle bisector, $ AD \equal{} 3$, and $ DC \equal{} 8$. What is the smallest possible value of the perimeter? $ \textbf{(A)}\ 30 \qquad \textbf{(B)}\ 33 \qquad \textbf{(C)}\ 35 \qquad \textbf{(D)}\ 36 \qquad \textbf{(E)}\ 37$

$120$ pirates distribute $119$ gold pieces among themselves. Then the captain checks if any pirate has $15$ or more gold pieces. If he finds the first one, he must give all his gold pieces to other pirates, whereby he may not give more than one gold piece to anyone. This control is repeated as long as there is any pirate with $15$ or more gold pieces. Does this process end after a lot of checks?

Let $ ABC$ be a triangle with angle $ A <$ angle $ C < 90^\circ <$ angle $ B$. Consider the bisectors of the external angles at $ A$ and $ B$, each measured from the vertex to the opposoite side (extended). Suppose both of these line-segments are equal to $ AB$. Compute the angle $ A$.

The lengths of all sides and both diagonals of a quadrilateral are less than $1$ metre. Prove that it may be placed in a circle of radius $0.9$ metres.

For every pair of reals $0 < a < b < 1$, we define sequences $\{x_n\}_{n \ge 0}$ and $\{y_n\}_{n \ge 0}$ by $x_0 = 0$, $y_0 = 1$, and for each integer $n \ge 1$: \begin{align*} x_n & = (1 - a) x_{n - 1} + a y_{n - 1}, \\ y_n & = (1 - b) x_{n - 1} + b y_{n - 1}. \end{align*} The [i]supermean[/i] of $a$ and $b$ is the limit of $\{x_n\}$ as $n$ approaches infinity. Over all pairs of real numbers $(p, q)$ satisfying $\left (p - \textstyle\frac{1}{2} \right)^2 + \left (q - \textstyle\frac{1}{2} \right)^2 \le \left(\textstyle\frac{1}{10}\right)^2$, the minimum possible value of the supermean of $p$ and $q$ can be expressed as $\textstyle\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $100m + n$. [i]Proposed by Lewis Chen[/i]

The set $S$ consists of five integers. If pairs of distinct elements of $S$ are added, the following ten sums are obtained: 1967,1972,1973,1974,1975,1980,1983,1984,1989,1991. What are the elements of $S$?

Given that $x$ satisfies $2^{4x} \cdot 2^{4x} \cdot 8^{4x} = 16^5$, find the value of $x$. $\textbf{(A) }1\qquad\textbf{(B) }2\qquad\textbf{(C) }4\qquad\textbf{(D) }5\qquad\textbf{(E) }10$

Let $X={1,2,3,4,5,6,7,8,9,10}$ and $A={1,2,3,4}$. Find the number of $4$-element subsets $Y$ of $X$ such that $10\in Y$ and the intersection of $Y$ and $A$ is not empty.

Let $P$ be a point inside a triangle $ABC$ such that \[\angle P AB = \angle P BC = \angle P CA\] Suppose $AP, BP, CP$ meet the circumcircles of triangles $P BC, P CA, P AB$ at $X, Y, Z$ respectively $(\neq P)$ . Prove that \[[XBC] + [Y CA] + [ZAB] \ge 3[ABC]\]

The vertices of a convex $2550$-gon are colored black and white as follows: black, white, two black, two white, three black, three white, ..., 50 black, 50 white. Dania divides the polygon into quadrilaterals with diagonals that have no common points. Prove that there exists a quadrilateral among these, in which two adjacent vertices are black and the other two are white. [i]D. Rudenko[/i]

Show that the sequence \begin{align*} a_n = \left \lfloor \left( \sqrt[3]{n-2} + \sqrt[3]{n+3} \right)^3 \right \rfloor \end{align*} contains infinitely many terms of the form $a_n^{a_n}$

In the triangle $ABC$ the altitude $BD$ and $CT$ are drawn, they intersect at the point $H$. The point $Q$ is the foot of the perpendicular drawn from the point $H$ on the bisector of the angle $A$. Prove that the bisector of the external angle $A$ of the triangle $ABC$, the bisector of the angle $BHC$ and the line $QM$, where $M$ is the midpoint of the segment $DT$, intersect at one point. (Matvsh Kursky)

Let $N = 2^{\left(2^2\right)}$ and $x$ be a real number such that $N^{\left(N^N\right)} = 2^{(2^x)}$. Find $x$.

Fedja used matches to put down the equally long sides of a parallelogram whose vertices are not on a common line. He figures out that exactly 7 or 9 matches, respectively, fit into the diagonals. How many matches compose the parallelogram's perimeter?

$(POL 2)$ Given two segments $AB$ and $CD$ not in the same plane, find the locus of points $M$ such that $MA^2 +MB^2 = MC^2 +MD^2.$