Baron Munсhausen discovered the following theorem: "For any positive integers $a$ and $b$ there exists a positive integer $n$ such that $an$ is a perfect square, while $bn$ is a perfect cube". Determine if the statement of Baron’s theorem is correct.

a) Given real numbers $a_1,a_2,b_1,b_2$ and positive $p_1,p_2,q_1,q_2$. Prove that in the table $2\times 2$ $$(a_1 + b_1)/(p_1 + q_1) , (a_1 + b_2)/(p_1 + q_2) $$ $$(a_2 + b_1)/(p_2 + q_1) , (a_2 + b_2)/(p_2 + q_2)$$ there is a number in the table, that is not less than another number in the same row and is not greater than another number in the same column (a saddle point). b) Given real numbers $a_1, a_2, ... , a_n, b_1, b_2, ... , b_n$ and positive $p_1, p_2, ... , p_n, q_1, q_2, ... , q_n$. We construct the table $n\times n$, with the numbers ($0 < i,j \le n$) $$(a_i + b_j)/(p_i + q_j)$$ in the intersection of the $i$-th row and $j$-th column. Prove that there is a number in the table, that is not less than arbitrary number in the same row and is not greater than arbitrary number in the same column (a saddle point).

In a sequence $a_1, a_2, ..$ of real numbers the product $a_1a_2$ is negative, and to define $a_n$ for $n > 2$ one pair $(i, j)$ is chosen among all the pairs $(i, j), 1 \le i < j < n$, not chosen before, so that $a_i +a_j$ has minimum absolute value, and then $a_n$ is set equal to $a_i + a_j$ . Prove that $|a_i| < 1$ for some $i$.

The bisectors of angles $A$, $B$, and $C$ of triangle $ABC$ meet for the second time its circumcircle at points $A_1$, $B_1$, $C_1$ respectively. Let $A_2$, $B_2$, $C_2$ be the midpoints of segments $AA_1$, $BB_1$, $CC_1$ respectively. Prove that the triangles $A_1B_1C_1$ and $A_2B_2C_2$ are similar.

What is the smallest prime number $p$ such that $p^3+4p^2+4p$ has exactly $30$ positive divisors ?

We call a natural number $b$ [i]lucky [/i] if for any natural number $a$ such that $a^5$ is divisible by $b^2$, the number $a^2$ is divisible by $b$. Find the number of [i]lucky [/i] natural numbers less than $2010$.

Let $k$ be a constant such that exactly three real values of $x$ satisfy $$x-|x^2-4x+3| = k$$ The sum of all possible values of $k$ can be expressed in the form $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers, find $m+n$. [i]Proposed by Andy Xu[/i]

The incircle of a triangle $ABC$ centered at $I$ touches the sides $BC, CA$, and $AB$ at points $A_1, B_1, $ and $C_1$ respectively. The excircle centered at $J$ touches the side $AC$ at point $B_2$ and touches the extensions of $AB, BC$ at points $C_2, A_2$ respectively. Let the lines $IB_2$ and $JB_1$ meet at point $X$, the lines $IC_2$ and $JC_1$ meet at point $Y$, the lines $IA_2$ and $JA_1$ meet at point $Z$. Prove that if one of points $X, Y, Z$ lies on the incircle then two remaining points also lie on it.

The smallest value of the function $f(x) =|x| +\left|\frac{1 - 2013x}{2013 - x}\right|$ where $x \in [-1, 1] $ is: (A): $\frac{1}{2012}$, (B): $\frac{1}{2013}$, (C): $\frac{1}{2014}$, (D): $\frac{1}{2015}$, (E): None of the above.

Given positive integer $ n \ge 5 $ and a convex polygon $P$, namely $ A_1A_2...A_n $. No diagonals of $P$ are concurrent. Proof that it is possible to choose a point inside every quadrilateral $ A_iA_jA_kA_l (1\le i<j<k<l\le n) $ not on diagonals of $P$, such that the $ \tbinom{n}{4} $ points chosen are distinct, and any segment connecting these points intersect with some diagonal of P.

Let $ \{a_k\}^{\infty}_1$ be a sequence of non-negative real numbers such that: \[ a_k \minus{} 2 a_{k \plus{} 1} \plus{} a_{k \plus{} 2} \geq 0 \] and $ \sum^k_{j \equal{} 1} a_j \leq 1$ for all $ k \equal{} 1,2, \ldots$. Prove that: \[ 0 \leq a_{k} \minus{} a_{k \plus{} 1} < \frac {2}{k^2} \] for all $ k \equal{} 1,2, \ldots$.

Felix is playing a card-flipping game. $n$ face-down cards are randomly colored, each with equal probability of being black or red. Felix starts at the $1$st card. When Felix is at the $k$th card, he guesses its color and then flips it over. For $k < n$, if he guesses correctly, he moves onto the $(k + 1)$-th card. If he guesses incorrectly, he gains $k$ penalty points, the cards are replaced with newly randomized face-down cards, and he moves back to card $1$ to continue guessing. If Felix guesses the $n$th card correctly, the game ends. What is the expected number of penalty points Felix earns by the end of the game?

How many functions $f:\mathbb N \to \mathbb N$ are there such that $f(0)=2011$, $f(1) = 111$, and $$f\left(\max \{x + y + 2, xy\}\right) = \min \{f (x + y), f (xy + 2)\}$$ for all non-negative integers $x$, $y$?

At noon on a certain day, Minneapolis is $N$ degrees warmer than St. Louis. At $4{:}00$ the temperature in Minneapolis has fallen by $5$ degrees while the temperature in St. Louis has risen by $3$ degrees, at which time the temperatures in the two cities differ by $2$ degrees. What is the product of all possible values of $N?$ $(\textbf{A})\: 10\qquad(\textbf{B}) \: 30\qquad(\textbf{C}) \: 60\qquad(\textbf{D}) \: 100\qquad(\textbf{E}) \: 120$

Can an infinite set of natural numbers be found, such that for all triplets $(a,b,c)$ of it we have $abc + 1 $ perfect square? (20 points )

Baron Munchausen plays billiards on a table with the shape of an equilateral triangle. He claims to have shot a ball from one of the sides of this table so that it passed through a certain point three times in three different directions and then returned to the original point on the side. Can that be true, assuming that the usual law of reflection holds? (Μ Evdokimov)

Suppose $f(x)$ is a twice continuously differentiable real valued function defined for all real numbers $x$ and satisfying $$|f(x)| \leq 1$$ for all x and $$(f(0))^2+(f'(0))^2=4.$$ Prove that there exists a real number $x_0$ such that $$f(x_0)+f''(x_0)=0.$$

Find all solutions $(x, y) \in \mathbb Z^2$ of the equation \[x^3 - y^3 = 2xy + 8.\]

Two isosceles trapezoids are inscribed in a circle in such a way that each side of each trapezoid is parallel to a certain side of the other trapezoid . Prove that the diagonals of one trapezoid are equal to the diagonals of the other.

In the regular hexagon $ABCDEF$, a point $X$ was marked on the diagonal $AD$ such that $\angle AEX = 65^\circ$. What is the degree measure of the angle $\angle XCD$? [i]A.V.Smirnov, I.A.Efremov[/i]

Alex has a piece of cheese. He chooses a positive number $a\neq 1$ and cuts the piece into several pieces one by one. Every time he chooses a piece and cuts it in the same ratio $1:a.$ His goal is to divide the cheese into two piles of equal masses. Can he do it?

Sequence $\{ a_n \}$ satisfies: $a_1=3$, $a_2=7$, $a_n^2+5=a_{n-1}a_{n+1}$, $n \geq 2$. If $a_n+(-1)^n$ is prime, prove that there exists a nonnegative integer $m$ such that $n=3^m$.

Let $n$ be an even positive integer. On a board n real numbers are written. In a single move we can erase any two numbers from the board and replace each of them with their product. Prove that for every $n$ initial numbers one can in finite number of moves obtain $n$ equal numbers on the board.

Let $ C_1$ be a circle and $ P$ be a fixed point outside the circle $ C_1$. Quadrilateral $ ABCD$ lies on the circle $ C_1$ such that rays $ AB$ and $ CD$ intersect at $ P$. Let $ E$ be the intersection of $ AC$ and $ BD$. (a) Prove that the circumcircle of triangle $ ADE$ and the circumcircle of triangle $ BEC$ pass through a fixed point. (b) Find the the locus of point $ E$.

Let $ABC$ be a triangle with $AB<AC<BC$.Let $O$ be the center of it's circumcircle and $D$ be the center of minor arc $\overarc{AB}$.Line $AD$ intersects $BC$ at $E$ and the circumcircle of $BDE$ intersects $AB$ at $Z$ ,($Z\not=B$).The circumcircle of $ADZ$ intersects $AC$ at $H$ ,($H\not=A$),prove that $BE=AH$.