Let $n\ge 2$ be an integer, and let $a_1, a_2, \cdots , a_n$ be positive integers. Show that there exist positive integers $b_1, b_2, \cdots, b_n$ satisfying the following three conditions: $\text{(A)} \ a_i\le b_i$ for $i=1, 2, \cdots , n;$ $\text{(B)} \ $ the remainders of $b_1, b_2, \cdots, b_n$ on division by $n$ are pairwise different; and $\text{(C)} \ $ $b_1+b_2+\cdots b_n \le n\left(\frac{n-1}{2}+\left\lfloor \frac{a_1+a_2+\cdots a_n}{n}\right \rfloor \right)$ (Here, $\lfloor x \rfloor$ denotes the integer part of real number $x$, that is, the largest integer that does not exceed $x$.)

Positive integers $n$, $k>1$ are given. Pasha and Vova play a game on a board $n\times k$. Pasha begins, and further they alternate the following moves. On each move a player should place a border of length 1 between two adjacent cells. The player loses if after his move there is no way from the bottom left cell to the top right without crossing any order. Determine who of the players has a winning strategy.

The transportation ministry has just decided to pay $80$ companies to repair $2400$ roads. These roads connects $100$ cities. Each road is between two cities and there is at most one road between any two cities. Each company must repair exactly $30$ roads, and each road is repaired by exactly one company. For a company to repair a road, it is necessary for the company to set up stations at the both cities on its endpoints. Prove that there are at least $8$ companies stationed at one city.

Let $ P$ be a point inside a regular tetrahedron $ T$ of unit volume. The four planes passing through $ P$ and parallel to the faces of $ T$ partition $ T$ into 14 pieces. Let $ f(P)$ be the joint volume of those pieces that are neither a tetrahedron nor a parallelepiped (i.e., pieces adjacent to an edge but not to a vertex). Find the exact bounds for $ f(P)$ as $ P$ varies over $ T.$

In two wooden boxes, there are $1994$ and $2024$ marbles, respectively. Spiro and Cvetko play the following game: alternately, each player takes a turn and removes some marbles from one of the boxes, so that the number of removed marbles in that turn is a divisor of the current number of marbles in the other box. The winner of the game is the one after whose turn both boxes are empty. Spiro takes the first turn. Which of the players has a winning strategy?

A $10 \times 10$ square on a grid is split by $80$ unit grid segments into $20$ polygons of equal area (no one of these segments belongs to the boundary of the square). Prove that all polygons are congruent. [i]($6$ points)[/i]

There is $207$ boxes on the table which numbered $1,2, \dots , 207$ respectively. Firstly Aslı puts a red ball in each of the $100$ boxes that she chooses and puts a white ball in each of the remaining ones. After that Zehra, writes a pair $(i,j)$ on the blackboard such that $1\leq i \leq j \leq 207$. Finally, Aslı tells Zehra that for every pair; whether the color of the balls which is inside the box which numbered by these numbers are the same or not. Find the least possible value of $N$ such that Zehra can guarantee finding all colors that has been painted to balls in each of the boxes with writing $N$ pairs on the blackboard.

In a group of $n$ people, every weekend someone organizes a party in which he invites all of his acquaintances. Those who meet at a party become acquainted. After each of the $n$ people has organized a party, there still are two people not knowing each other. Show that these two will never get to know each other at such a party.

[u]Round 1[/u] [b]p1.[/b] Three snails – Alice, Bobby, and Cindy – were racing down a road. Whenever one snail passed another, it waved at the snail it passed. During the race, Alice waved $3$ times and was waved at twice. Bobby waved $4$ times and was waved at $3$ times. Cindy waved $5$ times. How many times was she waved at? [b]p2.[/b] Sherlock and Mycroft are playing Battleship on a $4\times 4$ grid. Mycroft hides a single $3\times 1$ cruiser somewhere on the board. Sherlock can pick squares on the grid and fire upon them. What is the smallest number of shots Sherlock has to fire to guarantee at least one hit on the cruiser? [b]p3.[/b] Thirty girls – $13$ of them in red dresses and $17$ in blue dresses – were dancing in a circle, hand-in-hand. Afterwards, each girl was asked if the girl to her right was in a blue dress. Only the girls who had both neighbors in red dresses or both in blue dresses told the truth. How many girls could have answered “Yes”? [b]p4.[/b] Herman and Alex play a game on a $5\times 5$ board. On his turn, a player can claim any open square as his territory. Once all the squares are claimed, the winner is the player whose territory has the longer border. Herman goes first. If both play their best, who will win, or will the game end in a draw? [img]https://cdn.artofproblemsolving.com/attachments/5/7/113d54f2217a39bac622899d3d3eb51ec34f1f.png[/img] [b]p5.[/b] Is it possible to find $2014$ distinct positive integers whose sum is divisible by each of them? [u]Round 2[/u] [b]p6.[/b] Hermione and Ron play a game that starts with 129 hats arranged in a circle. They take turns magically transforming the hats into animals. On each turn, a player picks a hat and chooses whether to change it into a badger or into a raven. A player loses if after his or her turn there are two animals of the same species right next to each other. Hermione goes first. Who loses? [b]p7.[/b] Three warring states control the corner provinces of the island whose map is shown below. [img]https://cdn.artofproblemsolving.com/attachments/e/a/4e2f436be1dcd3f899aa34145356f8c66cda82.png[/img] As a result of war, each of the remaining $18$ provinces was occupied by one of the states. None of the states was able to occupy any province on the coast opposite their corner. The states would like to sign a peace treaty. To do this, they each must send ambassadors to a place where three provinces, one controlled by each state, come together. Prove that they can always find such a place to meet. For example, if the provinces are occupied as shown here, the squares mark possible meeting spots. [img]https://cdn.artofproblemsolving.com/attachments/e/b/81de9187951822120fc26024c1c1fbe2138737.png[/img] PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Natural numbers are placed in an infinite grid. Such that the number in each cell is equal to the number of its adjacent cells having the same number. Find the most distinct numbers this infinite grid can have. (Two cells of the grid are adjacent if they have a common vertex)

An integer partition, is a way of writing n as a sum of positive integers. Two sums that differ only in the order of their summands are considered the same partition. [quote]For example, 4 can be partitioned in five distinct ways: 4 3 + 1 2 + 2 2 + 1 + 1 1 + 1 + 1 + 1[/quote] The number of partitions of n is given by the partition function $p\left ( n \right )$. So $p\left ( 4 \right ) = 5$ . Determine all the positive integers so that $p\left ( n \right )+p\left ( n+4 \right )=p\left ( n+2 \right )+p\left ( n+3 \right )$.

Find maximal numbers of planes, such there are $6$ points and 1) $4$ or more points lies on every plane. 2) No one line passes through $4$ points.

There is a set of $50$ cards. Each card on both sides is colored in one of three colors — red, blue or white, and for each card its two sides are colored in different colors. The cards were laid out on the table. The card [i]lies beautifully[/i] if at least one of two conditions is met: its upper side — red; its underside is blue. It turned out that exactly $25$ cards are lying beautifully. Then all the cards were turned over. Now some of the cards are lying beautifully on the table. How many of them can there be? [i]K.A.Sukhov[/i]

Prove that the numbers $1,2,...,1998$ cannot be separated into three classes whose sums of elements are divisible by $2000,3999$, and $5998$, respectively.

Let $n$ be a positive integer relatively prime to $6$. We paint the vertices of a regular $n$-gon with three colours so that there is an odd number of vertices of each colour. Show that there exists an isosceles triangle whose three vertices are of different colours.

A $(3n + 1) \times (3n + 1)$ table $(n \in \mathbb{N})$ is given. Prove that deleting any one of its squares yields a shape cuttable into pieces of the following form and its rotations: ''L" shape formed by cutting one square from a $2 \times 2$ squares.

[url=http://www.mathlinks.ro/Forum/viewtopic.php?t=107262]IMO 2007 HKTST 1[/url] Problem 4 (a) Given five points on a plane such that no three of the points are collinear. Show that among the triangles which are drwan using any three of these five points as vertices, at least three of the triangles formed are not acute-angled triangles. (An acute-angled triangle is one in which all the three interior angles are acute angles.) (b) Given any 100 points on a plane such that no three of the points are collinear. SHow that among the triangles which are drawn using any three of these 100 points as vertices, at least 30% of the trinagles are not acute-angled triangles.

There are some boys and girls that study in a school. A group of boys is called [i]sociable[/i], if each girl knows at least one of the boys in the group. A group of girls is called [i]sociable[/i], if each boy knows at least one of the girls in the group. If the number of [i]sociable[/i] groups of boys is odd, prove that the number of [i]sociable[/i] groups of girls is also odd.

$n\in{Z^{+}}$ and $A={1,\ldots ,n}$. $f: N\rightarrow N$ and $\sigma: N\rightarrow N$ are two permutations, if there is one $k\in A$ such that $(f\circ\sigma)(1),\ldots ,(f\circ\sigma)(k)$ is increasing and $(f\circ\sigma)(k),\ldots ,(f\circ\sigma)(n)$ is decreasing sequences we say that $f$ is good for $\sigma$. $S_\sigma$ shows the set of good functions for $\sigma$. a) Prove that, $S_\sigma$ has got $2^{n-1}$ elements for every $\sigma$ permutation. b)$n\geq 4$, prove that there are permutations $\sigma$ and $\tau$ such that, $S_{\sigma}\cap S_{\tau}=\phi$ .

If $S$ is a set which consists of $12$ elements, what is the maximum number of pairs $(a,b)$ such that $a, b\in S$ and $\frac{b}{a}$ is a prime number?

Let $n,k \in \mathbb{Z}^{+}$ with $k \leq n$ and let $S$ be a set containing $n$ distinct real numbers. Let $T$ be a set of all real numbers of the form $x_1 + x_2 + \ldots + x_k$ where $x_1, x_2, \ldots, x_k$ are distinct elements of $S.$ Prove that $T$ contains at least $k(n-k)+1$ distinct elements.

Let $n > 1$ be a fixed integer. On an infinite row of squares, there are $n$ stones on square $1$ and no stones on squares $2$, $3$, $4$, $\ldots$. Curious George plays a game in which a [i]move[/i] consists of taking two adjacent piles of sizes $a$ and $b$, where $a-b$ is a nonzero even integer, and transferring stones to equalize the piles (so that both piles have $\frac{a+b}{2}$ stones). The game ends when no more moves can be made. George wants to analyze the number of moves it takes to end the game. (a) Suppose George wants to end the game as quickly as possible. How many moves will it take him? (b) Suppose George wants to end the game as slowly as possible. Show that for all $n > 2$, the game will end after at most $\frac{3}{16}n^2$ moves. [i]Scoring note:[/i] For part (b), partial credit will be awarded for correct proofs of weaker bounds, eg. $\frac{1}{4}n^2$, $n^k$, or $k^n$ (for some $k \geq 2$).

As shown in the figure, chess pieces are placed at the intersection points of the $64$ grid lines of the $7\times 7$ grid table. At most $1$ piece is placed at each point, and a total of $k$ left chess pieces are placed. No matter how they are placed, there will always be $4$ chess pieces, and the grid in which they are located the points form the four vertices of a rectangle (the sides of the rectangle are parallel to the grid lines). Try to find the minimum value of $k$. [img]https://cdn.artofproblemsolving.com/attachments/5/b/23a79f43d3f4c9aade1ba9eaa7a282c3b3b86f.png[/img]

Starting with any $n$-tuple $R_0$, $n\ge 1$, of symbols from $A,B,C$, we define a sequence $R_0, R_1, R_2,\ldots,$ according to the following rule: If $R_j= (x_1,x_2,\ldots,x_n)$, then $R_{j+1}= (y_1,y_2,\ldots,y_n)$, where $y_i=x_i$ if $x_i=x_{i+1}$ (taking $x_{n+1}=x_1$) and $y_i$ is the symbol other than $x_i, x_{i+1}$ if $x_i\neq x_{i+1}$. Find all positive integers $n>1$ for which there exists some integer $m>0$ such that $R_m=R_0$.

[b]p1.[/b] Consider this statement: An equilateral polygon circumscribed about a circle is also equiangular. Decide whether this statement is a true or false proposition in euclidean geometry. If it is true, prove it; if false, produce a counterexample. [b]p2.[/b] Show that the fraction $\frac{x^2-3x+1}{x-3}$ has no value between $1$ and $5$, for any real value of $x$. [b]p3.[/b] A man walked a total of $5$ hours, first along a level road and then up a hill, after which he turned around and walked back to his starting point along the same route. He walks $4$ miles per hour on the level, three miles per hour uphill, and $r$ miles per hour downhill. For what values of $r$ will this information uniquely determine his total walking distance? [b]p4.[/b] A point $P$ is so located in the interior of a rectangle that the distance of $P$ from one comer is $5$ yards, from the opposite comer is $14$ yards, and from a third comer is $10$ yards. What is the distance from $P$ to the fourth comer? [b]p5.[/b] Each small square in the $5$ by $5$ checkerboard shown has in it an integer according to the following rules: $\begin{tabular}{|l|l|l|l|l|} \hline & & & & \\ \hline & & & & \\ \hline & & & & \\ \hline & & & & \\ \hline & & & & \\ \hline \end{tabular}$ i. Each row consists of the integers $1, 2, 3, 4, 5$ in some order. ii. Тhе order of the integers down the first column has the same as the order of the integers, from left to right, across the first row and similarly fог any other column and the corresponding row. Prove that the diagonal squares running from the upper left to the lower right contain the numbers $1, 2, 3, 4, 5$ in some order. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].