Let $ABC$ be an acute triangle, $\Gamma$ is its circumcircle and $O$ is its circumcenter. Let $F$ be the point on $AC$ such that the $\angle COF=\angle ACB$, such that $F$ and $B$ lie in opposite sides with respect to $CO$. The line $FO$ cuts $BC$ at $G$. The line parallel to $BC$ through $A$ intersects $\Gamma$ again at $M$. The lines $CO$ and $MG$ meet at $K$. Show that the circumcircles of the triangles $BGK$ and $AOK$ meet on $AB$.

The integers $1$, $2$, $3$, $4$, and $5$ are written on a blackboard. It is allowed to wipe out two integers $a$ and $b$ and replace them with $a + b$ and $ab$. Is it possible, by repeating this procedure, to reach a situation where three of the five integers on the blackboard are $2009$?

Determine all functions $f$ defined on the set of all positive integers and taking non-negative integer values, satisfying the three conditions: [list] [*] $(i)$ $f(n) \neq 0$ for at least one $n$; [*] $(ii)$ $f(x y)=f(x)+f(y)$ for every positive integers $x$ and $y$; [*] $(iii)$ there are infinitely many positive integers $n$ such that $f(k)=f(n-k)$ for all $k<n$. [/list]

Eight politicians stranded on a desert island on January 1st, 1991, decided to establish a parliament. They decided on the following rules of attendance: (a) There should always be at least one person present on each day. (b) On no two days should the same subset attend. (c) The members present on day $N$ should include for each $K<N$, $(K\ge 1)$ at least one member who was present on day $K$. For how many days can the parliament sit before one of the rules is broken?

Find the smallest positive integer whose remainder when divided by $5$ is $3$, when divided by $7$ is $5$, and when divided by $9$ is $1$.

Bob chose six of the nine digits from $1$ to $9$ and wrote the list, ordered from smallest to largest, of all three-digit numbers that can be formed using the digits you chose. At Bob's list, the number $317$ appears at position $22$. What number appears at position $60$ in the list from Bob? Find all possibilities.

There is a city with $n$ citizens. The city wants to buy [i]sceptervirus[/i] tests with which it is possible to analyze the samples of several people at the same time. The result of a test can be the following: [list] [*][i]Virus positive[/i]: there is at least one currently infected person among the people whose samples were analyzed, and none of them were healed from an earlier infection. [*][i]Antibody positive[/i]: there is at least one person who was healed from an earlier infection among the people whose samples were analyzed, and none of them are infected now. [*][i]Neutral[/i]: either all of the people whose samples were analyzed are not infected, or there is at least one currently infected person and one person who was healed from an earlier infection. (Viruses and antibodies in samples completely neutralize each other.) [/list] What is the smallest number of tests to buy if we would like to know if the sceptervirus is present now or it has been present earlier? (The samples are taken from the people at the same time. The people are either infected now, have been infected earlier, or haven't contacted the virus yet.) [i]Proposed by Csongor Beke, Cambridge[/i]

A bag contains $5$ identical blue marbles and $5$ identical green marbles. In how many ways can $5$ marbles from the bag be arranged in a row if each blue marble must be adjacent to at least $1$ green marble?

For any set $A = \{x_1, x_2, x_3, x_4, x_5\}$ of five distinct positive integers denote by $S_A$ the sum of its elements, and denote by $T_A$ the number of triples $(i, j, k)$ with $1 \le i < j < k \le 5$ for which $x_i + x_j + x_k$ divides $S_A$. Find the largest possible value of $T_A$.

Prove that $(\sqrt{2}-1)^n$ $\forall n\in \mathbb{Z}^{+}$ can be represented as $\sqrt{m}-\sqrt{m-1}$ for some $m\in \mathbb{Z}^{+}$.

A square is divided into several (greater than one) convex polygons with mutually different numbers of sides. Prove that one of these polygons is a triangle. (A.Zaslavsky)

Given positive integers $ a,b,$ find all positive integers $ x,y$ satisfying the equation: $ x^{a\plus{}b}\plus{}y\equal{}x^a y^b$.

Let $ABCD$ and $A'B'C'D'$ be two convex quadrilaterals whose corresponding sides are equal, i.e., $AB = A'B', BC = B'C'$, etc. Prove that if $\angle A > \angle A'$, then $\angle B < \angle B', \angle C > \angle C', \angle D < \angle D'$.

Thirteen children are sitting at a round table, each holding two cards. Each card has one of the numbers $1, 2, ..., 13$ written on it, and each number is written on exactly two cards. On a signal, each child gives the card with the lower number to his neighbor on the right (and at the same time receives his card with the lower number from the neighbor on the left). Prove that after a finite number of such exchanges, a situation arises when at least one of the children will have two cards with the same number.

A set of $n$ standard dice are shaken and randomly placed in a straight line. If $n < 2r$ and $r < s$, then the probability that there will be a string of at least $r$, but not more than $s$, consecutive $1$'s can be written as $\frac{P}{6^{s+2}}$. Find an explicit expression for $P$.

On the side $AB$ of the triangle $ABC$ mark the points $M$ and $N$, such that $BM = BC$ and $AN = AC$. Then on the sides $BC$ and $AC$ mark the points$ P$ and $Q$, respectively, such that $BP = BN$ and $AQ = AM$. Prove that the points $C, Q, M, N$ and $P$ lie on the same circle.

Let the real numbers $a$, $b$, and $c$ satisfy the equations $(a+b)(b+c)(c+a)=abc$ and $(a^9+b^9)(b^9+c^9)(c^9+a^9)=(abc)^9$. Prove that at least one of $a$, $b$, and $c$ equals $0$.

A charity sells 140 benefit tickets for a total of $ \$2001$. Some tickets sell for full price (a whole dollar amount), and the rest sells for half price. How much money is raised by the full-price tickets? $ \textbf{(A)} \ \$782 \qquad \textbf{(B)} \ \$986 \qquad \textbf{(C)} \ \$1158 \qquad \textbf{(D)} \ \$1219 \qquad \textbf{(E)} \ \$1449$

For a positive integer $k\ge 2$ define $\mathcal{T}_k=\{(x,y)\mid x,y=0,1,\ldots, k-1\}$ to be a collection of $k^2$ lattice points on the cartesian coordinate plane. Let $d_1(k)>d_2(k)>\cdots$ be the decreasing sequence of the distinct distances between any two points in $T_k$. Suppose $S_i(k)$ be the number of distances equal to $d_i(k)$. Prove that for any three positive integers $m>n>i$ we have $S_i(m)=S_i(n)$.

Alex distributes some cookies into several boxes and records the number of cookies in each box. If the same number appears more than once, it is recorded only once. Serge takes one cookie from each box and puts them on the first plate. Then he takes one cookie from each box that is still non-empty and puts the cookies on the second plate. He continues until all the boxes are empty. Then Serge records the number of cookies on each plate. Again, if the same number appears more than once, it is recorded only once. Prove that Alex's record contains the same number of numbers as Serge's record.

A complete set of the Encyclopedia of Mathematics has $10$ volumes. There are ten mathematicians in Mathemagic Land, and each of them owns two volumes of the Encyclopedia. Together they own two complete sets. Show that there is a way for each mathematician to donate one book to the library such that the library receives a complete set.

You have a $4$-digit whole number that is a perfect square. Another number is built adding $ 1$ to the unit's digit, subtracting $ 1$ from the ten's digit, adding $ 1$ to the hundred's digit and subtracting $ 1$ from the ones digit of one thousand. If the number you get is also a perfect square, find the original number. It's unique?

Let $A=\{(x,y): 0\le x,y < 1\}.$ For $(x,y)\in A,$ let \[S(x,y)=\sum_{\frac12\le\frac mn\le2}x^my^n,\] where the sum ranges over all pairs $(m,n)$ of positive integers satisfying the indicated inequalities. Evaluate \[\lim_{(x,y)\to(1,1),(x,y)\in A}(1-xy^2)(1-x^2y)S(x,y).\]

In a zoo, there are five more foxes than rabbits, and three more foxes than pandas. Are there more pandas or rabbits, and by how much more? $\textbf{(A) }\text{Pandas, }1\qquad\textbf{(B) }\text{Pandas, }2\qquad\textbf{(C) }\text{Rabbits, }1\qquad\textbf{(D) }\text{Rabbits, }2\qquad\textbf{(E) }\text{Rabbits, }3$

Determine the largest positive integer $n$ for which there exist pairwise different sets $\mathbb{S}_1 , ..., \mathbb{S}_n$ with the following properties: $1$) $|\mathbb{S}_i \cup \mathbb{S}_j | \leq 2004$ for any two indices $1 \leq i, j\leq n$, and $2$) $\mathbb{S}_i \cup \mathbb{S}_j \cup \mathbb{S}_k = \{ 1,2,...,2008 \}$ for any $1 \leq i < j < k \leq n$ [i]Proposed by Ivan Matic[/i]