In a row of $2009$ weights, the weight of each weight is an integer grams and does not exceed $1$ kg. The weights of any two adjacent weights differ by exactly $1$ g, and the total weight of all weights in grams is an even number. Prove that weights can be separated into two piles, the sums of the weights in which are equal.

In a football league, there are $n\geq 6$ teams. Each team has a homecourt jersey and a road jersey with different color. When two teams play, the home team always wear homecourt jersey and the road team wear their homecourt jersey if the color is different from the home team's homecourt jersey, or otherwise the road team shall wear their road jersey. It is required that in any two games with 4 different teams, the 4 teams' jerseys have at least 3 different color. Find the least number of color that the $n$ teams' $2n$ jerseys may use.

Cities $A,B,C,D$ are positioned in such a way that $A$ is closer to $C$ than to $D$, and $B$ is closer to $C$ than to $D$. Prove that every point on the straight road from $A$ to $B$ is closer to $C$ than to $D$.

On a tournament with $ n \ge 3$ participants, every two participants played exactly one match and there were no draws. A three-element set of participants is called a [i]draw-triple[/i] if they can be enumerated so that the first defeated the second, the second defeated the third, and the third defeated the first. Determine the largest possible number of draw-triples on such a tournament.

On a table, there is an empty bag and a chessboard containing exactly one token on each square. Next to the table is a large pile that contains an unlimited supply of tokens. Using only the following types of moves what is the maximum possible number of tokens that can be in the bag? $\bullet$ Type 1: Choose a non-empty square on the chessboard that is not in the rightmost column. Take a token from this square and place it, along with one token from the pile, on the square immediately to its right. $\bullet$ Type 2: Choose a non-empty square on the chessboard that is not in the bottommost row. Take a token from this square and place it, along with one token from the pile, on the square immediately below it. $\bullet$ Type 3: Choose two adjacent non-empty squares. Remove a token from each and put them both into the bag.

Let $G$ be a weighted bipartite graph $A \cup B$, with $|A| = |B| = n$. In other words, each edge in the graph is assigned a positive integer value, called its [i]weight.[/i] Also, define the weight of a perfect matching in $G$ to be the sum of the weights of the edges in the matching. Let $G'$ be the graph with vertex set $A \cup B$, and (which) contains the edge $e$ if and only if $e$ is part of some minimum weight perfect matching in $G$. Show that all perfect matchings in $G'$ have the same weight.

Let $A$ be the number of all points with integer coordinates in a three-dimensional coordinate system. We assume that nine arbitrary points in $A$ will be colored blue. Show that we can always find two blue dots so that the line segment between them contains at least one point from $A$.

Two persons, $X$ and $Y$ , play with a die. $X$ wins a game if the outcome is $1$ or $2$; $Y$ wins in the other cases. A player wins a match if he wins two consecutive games. For each player determine the probability of winning a match within $5$ games. Determine the probabilities of winning in an unlimited number of games. If $X$ bets $1$, how much must $Y$ bet for the game to be fair ?

Let $ S$ be a set with $ n$ elements, and $ F$ be a family of subsets of $ S$ with $ 2^{n\minus{}1}$ elements, such that for each $ A,B,C\in F$, $ A\cap B\cap C$ is not empty. Prove that the intersection of all of the elements of $ F$ is not empty.

In a plane are given $n > 2$ distinct points. Some pairs of these points are connected by segments so that no two of the segments intersect. Prove that there are at most $3n-6$ segments.

For an integer $n\geq 3$, place $n$ points on the plane in such a way that all the distances between the points are at most one and exactly $n$ of the pairs of points have the distance one.

Let $P$ be the probability that the product of $2020$ real numbers chosen independently and uniformly at random from the interval $[-1, 2]$ is positive. The value of $2P - 1$ can be written in the form $\left(\frac{m}{n} \right)^b$, where $m$, $n$ and $b$ are positive integers such that $m$ and $n$ are relatively prime and $b$ is as large as possible. Compute $m + n + b$.

Two symbols $A$ and $B$ obey the rule $ABBB = B$. Given a word $x_1x_2\ldots x_{3n+1}$ consisting of $n$ letters $A$ and $2n+1$ letters $B$, show that there is a unique cyclic permutation of this word which reduces to $B$.

The circle is divided into $49$ areas so that no three areas touch at one point. The resulting “map” is colored in three colors so that no two adjacent areas have the same color. The border of two areas is considered to be colored in both colors. Prove that on the circle there are two diametrically opposite points, colored in one color.

Given is a natural number $n>4$. There are $n$ points marked on the plane, no three of which lie on the same line. Vasily draws one by one all the segments connecting pairs of marked points. At each step, drawing the next segment $S$, Vasily marks it with the smallest natural number, which hasn't appeared on a drawn segment that has a common end with $S$. Find the maximal value of $k$, for which Vasily can act in such a way that he can mark some segment with the number $k$?

In the simple and connected graph $G$ let $x_i$ be the number of vertices with degree $i$. Let $d>3$ be the biggest degree in the graph $G$. Prove that if : $$x_d \ge x_{d-1} + 2x_{d-2}+... +(d-1)x_1$$ Then there exists a vertex with degree $d$ such that after removing that vertex the graph $G$ is still connected. Proposed by [i]Ali Mirzaie[/i]

In a convex $n$-gon, several diagonals are drawn. Among these diagonals, a diagonal is called [i]good[/i] if it intersects exactly one other diagonal drawn (in the interior of the $n$-gon). Find the maximum number of good diagonals.

We call a rectangle of the size $1 \times 2$ a domino. Rectangle of the $2 \times 3$ removing two opposite (under center of rectangle) corners we call tetramino. These figures can be rotated. It requires to tile rectangle of size $2008 \times 2010$ by using dominoes and tetraminoes. What is the minimal number of dominoes should be used?

Some towns in a country are connected by two–way roads, so that for any two towns there is a unique path along the roads connecting them. It is known that there is exactly 100 towns which are directly connected to only one town. Prove that we can construct 50 new roads in order to obtain a net in which every two towns will be connected even if one road gets closed.

We consider the set of expressions that can be written with real numbers, $\pm$, $+$, $\times$, and parenthesis, such that if each $\pm$ is independently replaced with either $+$ or $-$, we are left with a valid arithmetic expression. For example, this includes: \[0\pm 1, 1 \pm 2, 1+2\times (1+2\pm 3), (1 \pm 2) \times (3 \pm 4).\] We define the [i]range[/i] of an expression of this form to be the set of all of the possible values when replacing each $\pm$ with either a $+$ or a $-$. For example, [list] [*] $1 \pm 2$ has range $\{-1,3\}$, since $1-2=-1$ and $1+2=3$. [*] $(1 \pm 1) \times (1 \pm 1)$ has range $\{0,4\}$, since $(1-1)(1-1)=(1-1)(1+1)=(1+1)(1-1)=0$ and $(1+1)(1+1)=4.$ [*] $(1 \pm 2)(3\pm 4)$ has range $\{-7,-3,1,21\}$, since $(1-2)(3+4)=-7$, $(1+2)(3-4)=-3$, $(1-2)(3-4)=1$, and $(1+2)(3+4)=21$. [/list] We will prove that every finite nonempty set of real numbers is the range of some expression of this form. Call a nonempty set of real numbers [i]good[/i] if it is the range of some expression of this form. (a) For each of the following sets, find an expression with a range equal to the given set. You do not need to justify the expression. [list=i] [*] $\{1\}$ [*] $\{1,3\}$ [*] $\{-1,0,1\}$ [/list] (b) Prove that if $S$ and $T$ are good sets, the product set $S \cdot T = \{ xy \mid x \in S, y \in T \}$ (the set of product of elements of $S$ with elements of $T$) is good. (c) Prove that if a set $S$ not containing $0$ is good, the set $S \cup \{ 0 \}$ (obtained upon adding $0$ to $S$) is good. (d) Prove that every finite nonempty set of real numbers is good.

Let $n\geqslant 3$ be a positive integer. A circular necklace is called [i]fun[/i] if it has $n{}$ black beads and $n{}$ white beads. A move consists of cutting out a segment of consecutive beads and reattaching it in reverse. Prove that it is possible to change any fun necklace into any other fun necklace using at most $(n-1)$ moves. [i]Note:[/i] Necklaces related by rotations or reflections are considered to be the same. [i]Proposed by Dylan Toh[/i]

For non-coplanar points are given in space. A plane $\pi$ is called [i]equalizing [/i] if all four points have the same distance from $\pi$. Find the number of equilizing planes.

There are $38$ people in the California Baseball League (CBL). The CBL cannot start playing games until people are split into teams of exactly $9$ people (with each person in exactly one team). Moreover, there must be an even number of teams. What is the fewest number of people who must join the CBL such that the CBL can start playing games? The CBL may not revoke membership of the $38$ people already in the CBL.

At a big New Year's Eve party, each guest receives two hats: one red and one blue. At the beginning of the party, all the guests put on the red hat. Several times throughout the evening, the announcer announces the name of one of the guests and, at that moment, the named and each of his friends change the hat they are wearing for the other color. Show that the announcer can make it so that all the guests are wearing the blue hat when the party is over. Note: All guests remain at the party from start to finish.

$N$ in natural. There are natural numbers from $N^3$ to $N^3+N$ on the board. $a$ numbers was colored in red, $b$ numbers was colored in blue. Sum of red numbers in divisible by sum of blue numbers. Prove, that $b|a$