Let $n\geqslant 3$ be an integer. Prove that there exists a set $S$ of $2n$ positive integers satisfying the following property: For every $m=2,3,...,n$ the set $S$ can be partitioned into two subsets with equal sums of elements, with one of subsets of cardinality $m$.

Edges of a planar graph $G$ are colored either with blue or red. Prove that there is a vertex like $v$ such that when we go around $v$ through a complete cycle, edges with the endpoint at $v$ change their color at most two times. Clarifications for complete cycle: If all the edges with one endpoint at $v$ are $(v,u_1),(v,u_2),\ldots,(v,u_k)$ such that $u_1,u_2,\ldots,u_k$ are clockwise with respect to $v$ then in the sequence of $(v,u_1),(v,u_2),\ldots,(v,u_k),(v,u_1)$ there are at most two $j$ such that colours of $(v,u_j),(v,u_{j+1})$ ($j \mod k$) differ.

You have $2$ columns of $11$ squares in the middle, in the right and in the left you have columns of $9$ squares (centered on the ones of $11$ squares), then columns of $7,5,3,1$ squares. (This is the way it was explained in the original thread, http://www.artofproblemsolving.com/Forum/viewtopic.php?t=44430 ; anyway, i think you can understand how it looks) Several rooks stand on the table and beat all the squares ( a rook beats the square it stands in, too). Prove that one can remove several rooks such that not more than $11$ rooks are left and still beat all the table. [i]Proposed by D. Rostovsky, based on folklore[/i]

What is the greatest number of parts into which the plane can be cut by the edges of $ n $ squares?

Suppose that $X$ is a set with $n$ elements and $\mathcal F\subseteq X^{(k)}$ and $X_1,X_2,...,X_s$ is a partition of $X$. We know that for every $A,B\in \mathcal F$ and every $1\le j\le s$, $E=B\cap (\bigcup_{i=1}^{j}X_i)\neq A\cap (\bigcup_{i=1}^{j} X_i)=F$ shows that none of $E,F$ contains the other one. Prove that \[|\mathcal F|\le \max_{\sum\limits_{i=1}^{S}w_i=k}\prod_{j=1}^{s}\binom{|X_j|}{w_j}\] (15 points) Exam time was 5 hours and 20 minutes.

A prism is called [i]binary [/i] if it can be assigned to each of its vertices a number from the set $\{-1, 1\}$, such that the product of the numbers assigned to the vertices of each face is equal to $-1$. a) Prove that the number of vertices of the binary prisms is divisible for $8$. b) Prove that a prism with $2000$ vertices is binary.

There are $17$ people at a party, and each has a reputation that is either $1$, $2$, $3$, $4$, or $5$. Some of them split into pairs under the condition that within each pair, the two people’s reputations differ by at most $1$. Compute the largest value of $k$ such that no matter what the reputations of these people are, they are able to form $k$ pairs

Let $T$ be the answer to question $17$, and let $N =\frac{24}{T}$. Leanne flips a fair coin $N$ times. Let $X$ be the number of times that within a series of three consecutive flips, there were exactly two heads or two tails. What is the expected value of $X$?

A cake has the form of an $ n$ x $ n$ square composed of $ n^{2}$ unit squares. Strawberries lie on some of the unit squares so that each row or column contains exactly one strawberry; call this arrangement $\mathcal{A}$. Let $\mathcal{B}$ be another such arrangement. Suppose that every grid rectangle with one vertex at the top left corner of the cake contains no fewer strawberries of arrangement $\mathcal{B}$ than of arrangement $\mathcal{A}$. Prove that arrangement $\mathcal{B}$ can be obtained from $ \mathcal{A}$ by performing a number of switches, defined as follows: A switch consists in selecting a grid rectangle with only two strawberries, situated at its top right corner and bottom left corner, and moving these two strawberries to the other two corners of that rectangle.

Given $n$ different positive numbers $a_1,a_2,...,a_n$. We construct all the possible sums (from $1$ to $n$ terms). Prove that among those sums there are at least $n(n+1)/2$ different ones.

Six teams participate in a hockey tournament. Each team plays once against every other team. In each game, a team is awarded $3$ points for a win, $1$ point for a draw, and none for a loss. After the tournament the teams are ranked by total points. No two teams have the same total points. Each team (except the bottom team) has $2$ points more than the team ranking one place lower. Prove that the team that finished fourth has won two games and lost three games.

Misha came to country with $n$ cities, and every $2$ cities are connected by the road. Misha want visit some cities, but he doesn`t visit one city two time. Every time, when Misha goes from city $A$ to city $B$, president of country destroy $k$ roads from city $B$(president can`t destroy road, where Misha goes). What maximal number of cities Misha can visit, no matter how president does?

[b]p1.[/b] What is $7\%$ of one half of $11\%$ of $20000$ ? [b]p2.[/b] Three circles centered at $A, B$, and $C$ are tangent to each other. Given that $AB = 8$, $AC = 10$, and $BC = 12$, find the radius of circle $ A$. [b]p3. [/b]How many positive integer values of $x$ less than $2012$ are there such that there exists an integer $y$ for which $\frac{1}{x} +\frac{2}{2y+1} =\frac{1}{y}$ ? [b]p4. [/b]The positive difference between $ 8$ and twice $x$ is equal to $11$ more than $x$. What are all possible values of $x$? [b]p5.[/b] A region in the coordinate plane is bounded by the equations $x = 0$, $x = 6$, $y = 0$, and $y = 8$. A line through $(3, 4)$ with slope $4$ cuts the region in half. Another line going through the same point cuts the region into fourths, each with the same area. What is the slope of this line? [b]p6.[/b] A polygon is composed of only angles of degrees $138$ and $150$, with at least one angle of each degree. How many sides does the polygon have? [b]p7.[/b] $M, A, T, H$, and $L$ are all not necessarily distinct digits, with $M \ne 0$ and $L \ne 0$. Given that the sum $MATH +LMT$, where each letter represents a digit, equals $2012$, what is the average of all possible values of the three-digit integer $LMT$? [b]p8. [/b]A square with side length $\sqrt{10}$ and two squares with side length $\sqrt{7}$ share the same center. The smaller squares are rotated so that all of their vertices are touching the sides of the larger square at distinct points. What is the distance between two such points that are on the same side of the larger square? [b]p9.[/b] Consider the sequence $2012, 12012, 20120, 20121, ...$. This sequence is the increasing sequence of all integers that contain “$2012$”. What is the $30$th term in this sequence? [b]p10.[/b] What is the coefficient of the $x^5$ term in the simplified expansion of $(x +\sqrt{x} +\sqrt[3]{x})^{10}$ ? PS. You had better use hide for answers.

On each day, more than half of the inhabitants of Évora eats [i]sericaia[/i] as dessert. Show that there is a group of 10 inhabitants of Évora such that, on each of the last 2010 days, at least one of the inhabitants ate [i]sericaia[/i] as dessert.

Let $ n\ge 5$ be a positive integer. Prove that the set $ \{1,2,\ldots,n\}$ can be partitioned into two non-zero subsets $ S_n$ and $ P_n$ such that the sum of elements in $ S_n$ is equal to the product of elements in $ P_n$.

A strip of width $w$ is the set of all points which lie on, or between, two parallel lines distance $w$ apart. Let $S$ be a set of $n$ ($n \ge 3$) points on the plane such that any three different points of $S$ can be covered by a strip of width $1$. Prove that $S$ can be covered by a strip of width $2$.

$(GDR 4)$ Find the maximal number of regions into which a sphere can be partitioned by $n$ circles.

An apartment building consists of 20 rooms number 1, 2,..., 20 arranged clockwise in a circle. To move from one room to another, one can either walk to the next room clockwise (i.e. from room $i$ to room $(i+1)\pmod{20}$) or walk across the center to the opposite room (i.e. from room $i$ to room $(i+10)\pmod{20}$). Find the number of ways to move from room 10 to room 20 without visiting the same room twice.

A table $2 \times 2010$ is divided to unit cells. Ivan and Peter are playing the following game. Ivan starts, and puts horizontal $2 \times 1$ domino that covers exactly two unit table cells. Then Peter puts vertical $1 \times 2$ domino that covers exactly two unit table cells. Then Ivan puts horizontal domino. Then Peter puts vertical domino, etc. The person who cannot put his domino will lose the game. Find who have winning strategy.

Let $A=\{1,2,\ldots,2n\}$ and let the function $g:A\to A$ be defined by $g(k)=2n-k+1$. Does there exist a function $f:A\to A$ such that $f(k)\ne g(k)$ and $f(f(f(k)))=g(k)$ for all $k\in A$, if (a) $n=999$; (b) $n=1000$?

Four equal right-angled triangles are given. We are allowed to cut any triangle into two new ones along the altitude dropped on to the hypotenuse. This operation may be repeated with any of the triangles from the new set. Prove that after any number of such operations there will be congruent triangles among those obtained. (AV Shapovalov)

Let $n$ be a positive integer. Find the number of sequences $x_{1},x_{2},\ldots x_{2n-1},x_{2n}$, where $x_{i}\in\{-1,1\}$ for each $i$, satisfying the following condition: for any integer $k$ and $m$ such that $1\le k\le m\le n$ then the following inequality holds \[\left|\sum_{i=2k-1}^{2m}x_{i}\right|\le\ 2\]

It is known that in a set of five coins three are genuine (and have the same weight) while two coins are fakes, each of which has a different weight from a genuine coin. What is the smallest number of weighings on a scale with two cups that is needed to locate one genuine coin?

On a long enough highway, a passenger in a bus observes the traffic. He notes that, during an hour, the bus going with a constant velocity overpasses $a$ cars and gets overpassed by $b$ cars, while $c$ cars pass in the opposite direction. Assuming that the traffic is the same in both directions, is it possible to determine the number of cars that pass along the highway per hour? (You may assume that the velocity of a car can take only two values.)

Can you mark the cube's vertices with the three-digit binary numbers in such a way, that the numbers at all the possible couples of neighbouring vertices differ in at least two digits?