Grisha wrote $n$ different natural numbers, the sum of which does not exceed $S$. The saboteur added to each of them a number from the half-interval $[0, 1)$. The sabotage is successful if there exists two subsets, the sums of the numbers in which differ by no more than $1$. At what minimum $S$ can Grisha ensure that the sabotage will definitely not be succeeded?

The four bottom corners of a cube are colored red, green, blue, and purple. How many ways are there to color the top four corners of the cube so that every face has four different colored corners? Prove that your answer is correct.

There is a positive integer $A$. Two operations are allowed: increasing this number by $9$ and deleting a digit equal to $1$ from any position. Is it always possible to obtain $A+1$ by applying these operations several times?

Jenny is going to attend a sports camp for $7$ days. Each day, she will play exactly one of three sports: hockey, tennis or camogie. The only restriction is that in any period of $4$ consecutive days, she must play all three sports. Find, with proof, the number of possible sports schedules for Jennys week.

p1. Find an integer that has the following properties: a) Every two adjacent digits in the number are prime. b) All prime numbers referred to in item (a) above are different. p2. Determine all integers up to $\sqrt{50+\sqrt{n}}+\sqrt{50-\sqrt{n}}$ p3. The following figure shows the path to form a series of letters and numbers “OSN2015”. Determine as many different paths as possible to form the series of letters and numbers by following the arrows. [img]https://cdn.artofproblemsolving.com/attachments/6/b/490a751457871184a506c2966f8355f20cebbd.png[/img] p4. Given an acute triangle $ABC$ with $L$ as the circumcircle. From point $A$, a perpendicular line is drawn on the line segment $BC$ so that it intersects the circle $L$ at point $X$. In a similar way, a perpendicular line is made from point $B$ and point $C$ so that it intersects the circle $L$, at point $Y$ and point $Z$, respectively. Is arc length $AY$ = arc length $AZ$ ? p5. The students of class VII.3 were divided into five groups: $A, B, C, D$ and $E$. Each group conducted five science experiments for five weeks. Each week each group performs an experiment that is different from the experiments conducted by other groups. Determine at least two possible trial schedules in week five, based on the following information: $\bullet$ In the first week, group$ D$ did experiment $4$. $\bullet$ In the second week, group $C$ did the experiment $5$. $\bullet$ In the third week, group $E$ did the experiment $5$. $\bullet$ In the fourth week, group $A$ did experiment $4$ and group $D$ did experiment $2$.

For any given triangle $A_0B_0C_0$ consider a sequence of triangles constructed as follows: a new triangle $A_1B_1C_1$ (if any) has its sides (in cm) that equal to the angles of $A_0B_0C_0$ (in radians). Then for $\vartriangle A_1B_1C_1$ consider a new triangle $A_2B_2C_2$ (if any) constructed in the similar พay, i.e., $\vartriangle A_2B_2C_2$ has its sides (in cm) that equal to the angles of $A_1B_1C_1$ (in radians), and so on. Determine for which initial triangles $A_0B_0C_0$ the sequence never terminates.

Prove that each of the numbers $1, 2, 3, ..., 2^n$ can be written in one of two colors (red and blue) such that no non-constant $2n$-term arithmetic sequence chosen from these numbers is monochromatic .

Let $ S \equal{}\{1,2,3, \ldots, 2n\}$ ($ n \in \mathbb{Z}^\plus{}$). Ddetermine the number of subsets $ T$ of $ S$ such that there are no 2 element in $ T$ $ a,b$ such that $ |a\minus{}b|\equal{}\{1,n\}$

Luffy is playing with some magic boxes and a machine. Each box has a value(number) inside. Opening a box, Luffy sees the value, adds the value to his score(if the box value is negative, Luffy loses points) and destroys the box. Putting a box of value $X$ in the machine, this box vanishes and it is replaced by two new boxes of values $X+1$ and $X-1$(it's [b]not[/b] known which one has the respective value, but he can identify the new boxes). At the beginning, Luffy has $0$ points and has a box whose value is known(it is zero). a) Prove that Luffy can reach at least $1000$ points b) Is it possible that Luffy reaches at least $1000000$ points, [b]without[/b] have less than $-42$ points in any moment?

Let $n \ge 3$ be a natural number. Determine the number $a_n$ of all subsets of $\{1, 2,...,n\}$ consisting of three elements such that one of them is the arithmetic mean of the other two. [i]Proposed by Walther Janous[/i]

Let $a$ and $b$ be two positive integers satifying $gcd(a, b) = 1$. Consider a pawn standing on the grid point $(x, y)$. A step of type A consists of moving the pawn to one of the following grid points: $(x+a, y+a),(x+a,y-a), (x-a, y + a)$ or $(x - a, y - a)$. A step of type B consists of moving the pawn to $(x + b,y + b),(x + b,y - b), (x - b,y + b)$ or $(x - b,y - b)$. Now put a pawn on $(0, 0)$. You can make a (nite) number of steps, alternatingly of type A and type B, starting with a step of type A. You can make an even or odd number of steps, i.e., the last step could be of either type A or type B. Determine the set of all grid points $(x,y)$ that you can reach with such a series of steps.

On a plane, there are $7$ seats. Each is assigned to a passenger. The passengers walk on the plane one at a time. The first passenger sits in the wrong seat (someone else's). For all the following people, they either sit in their assigned seat, or if it is full, randomly pick another. You are the last person to board the plane. What is the probability that you sit in your own seat?

The children sitting around a circle are playing the game as follows. At first the teacher gives each child an even number of candies (bigger than $ 0$, may be equal, maybe different). A certain child gives half of his candies to his neighbor on the right. Then the child who has just received candies does the same if he has an even number of candies, otherwise he gets one candy from the teacher and then does the job; and so on. Prove that after several steps there will be a child who will be able, giving the teacher half of his candies, to make the numbers of candies of all the children equal.

A team contest between $n$ participants is to be held. Each of these participants has exactly $k$ enemies among the other participants. (If $A$ is an enemy of $B$, then $B$ is an enemy of $A$. Nobody is his own enemy.) Assume that there are no three participants such that every two of them are enemies of each other. A [i]subversive enmity[/i] will mean an (un-ordered) pair of two participants which are enemies of each other and which belong to one and the same team. Show that one can divide the participants into two teams such that the number of subversive enmities is $\leq\frac{k\left(n-2k\right)}{2}$. (The teams need not be of equal size.) [i]Note.[/i] The actual source of this problem is: Glenn Hopkins, William Staton, [i]Maximal Bipartite Subgraphs[/i], Ars Combinatoria 13 (1982), pp. 223-226. It should be noticed that $\frac{k\left(n-2k\right)}{2}\leq\frac{n^{2}}{16}$, so the bound in the problem can be replaced by $\frac{n^{2}}{16}$ (which makes it weaker, but independent of $k$). darij

In each square of a $4\times 4$ board, we are to write an integer in such a way that the sums of the numbers in each column and in each row are nonnegative integral powers of $2$. Is it possible to do this in such a way that every two of these eight sums are different? Justify your answer.

Numbers $1$ to $22$ are written on a board. A "move" is a procedure of picking two numbers $a,b$ on the board such that $b \geq a+2$, then erasing $a$ and $b$ to be replaced with $a+1$ and $b-1$. Determine the maximum possible number of moves that can be done on the board.

We define a beehive of order $n$ as follows: a beehive of order 1 is one hexagon To construct a beehive of order $n$, take a beehive of order $n-1$ and draw a layer of hexagons in the exterior of these hexagons. See diagram for examples of $n=2,3$ Initially some hexagons are infected by a virus. If a hexagon has been infected, it will always be infected. Otherwise, it will be infected if at least 5 out of the 6 neighbours are infected. Let $f(n)$ be the minimum number of infected hexagons in the beginning so that after a finite time, all hexagons become infected. Find $f(n)$.

A hydra consists of several heads and several necks, where each neck joins two heads. When a hydra's head $A$ is hit by a sword, all the necks from head $A$ disappear, but new necks grow up to connect head $A$ to all the heads which weren't connected to $A$. Heracle defeats a hydra by cutting it into two parts which are no joined. Find the minimum $N$ for which Heracle can defeat any hydra with $100$ necks by no more than $N$ hits.

In a group of mathematicians everybody has at least one friend (friendship is a symmetric relation). Show that there is a mathematician all of whose friends have average number of friends not smaller than the average number of friends in the whole group.

Prove that among any $ 16 $ numbers smaller than $ 101 $ there are four of them that have the property that the sum of two of them is equal to the sum of the other two.

Given an integer $n>1$. The board $n\times n$ is colored white and black in a chess-like manner. We call any non-empty set of different cells of the board as a [i]figure[/i]. We call figures $F_1$ and $F_2$ [i]similar[/i], if $F_1$ can be obtained from $F_2$ by a rotation with respect to the center of the board by an angle multiple of $90^\circ$ and a parallel transfer. (Any figure is similar to itself.) We call a figure $F$ [i]connected[/i] if for any cells $a,b\in F$ there is a sequence of cells $c_1,\ldots,c_m\in F$ such that $c_1 = a$, $c_m = b$, and also $c_i$ and $c_{i+1}$ have a common side for each $1\le i\le m - 1$. Find the largest possible value of $k$ such that for any connected figure $F$ consisting of $k$ cells, there are figures $F_1,F_2$ similar to $F$ such that $F_1$ has more white cells than black cells and $F_2$ has more black cells than white cells in it.

For which natural number $n$ is it possible to draw $n$ line segments between vertices of a regular $2n$-gon so that every vertex is an endpoint for exactly one segment and these segments have pairwise different lengths?

The plane is partitioned into congruent equilateral triangles such that any two of them which are not disjoint have either a common vertex or a common side. Is there a circle containing exactly 1988 points in its interior?

Wanda the Worm likes to eat Pascal's triangle. One day, she starts at the top of the triangle and eats $\textstyle\binom{0}{0}=1$. Each move, she travels to an adjacent positive integer and eats it, but she can never return to a spot that she has previously eaten. If Wanda can never eat numbers $a,b,c$ such that $a+b=c$, prove that it is possible for her to eat 100,000 numbers in the first 2011 rows given that she is not restricted to traveling only in the first 2011 rows. (Here, the $n+1$st row of Pascal's triangle consists of entries of the form $\textstyle\binom{n}{k}$ for integers $0\le k\le n$. Thus, the entry $\textstyle\binom{n}{k}$ is considered adjacent to the entries $\textstyle\binom{n-1}{k-1}$, $\textstyle\binom{n-1}{k}$, $\textstyle\binom{n}{k-1}$, $\textstyle\binom{n}{k+1}$, $\textstyle\binom{n+1}{k}$, $\textstyle\binom{n+1}{k+1}$.) [i]Linus Hamilton.[/i]

Let $S$ be a square with the side length 20 and let $M$ be the set of points formed with the vertices of $S$ and another 1999 points lying inside $S$. Prove that there exists a triangle with vertices in $M$ and with area at most equal with $\frac 1{10}$. [i]Yugoslavia[/i]