On a circle there are some black and white points (there are at least $12$ points). Each point has $10$ neighbors ($5$ left and $5$ right neighboring points), $5$ being black and $5$ white. Prove that the number of points on the circle is divisible by $4$.

Gus is an inhabitant on an $11$ by $11$ grid of squares. He can walk from one square to an adjacent square (vertically or horizontally) in $1$ unit of time. There are also two vents on the grid, one at the top left and one at the bottom right. If Gus is at one vent, he can teleport to the other vent in $0.5$ units of time. Let an ordered pair of squares $(a,b)$ on the grid be [i]sus [/i] if the fastest path from $a$ to $b$ requires Gus to teleport between vents. Walking on top of a vent does not count as teleporting between vents. What is the total number of ordered pairs of squares that are [i]sus[/i]? Note that the pairs $(a_1,b_1)$ and $(a_2,b_2)$ are considered distinct if and only if $a_1 \ne a_2$ or $b_1 \ne b_2$.

Pythagoras drew some points in the plane and and connected some of these with segments. Now Tortillagoras wants to write a positive integer next to every point, such that one number divides another number if and only if these numbers are written next to points that Pythagoras has connected.Can Tortillagoras do this for the following drawings? [i]In part b), vertices in the same row or column but not adjacent are not connected.[/i] [img]https://cdn.artofproblemsolving.com/attachments/1/e/7356e39e44e45e3263275292af3719595e2dd2.png[/img]

Joe, who is already feared by all bandits in the Wild West, would like to officially become a sheriff. To do that, he has to take a special exam where he has to demonstrate his talent in three different areas: tracking, shooting and lasso throwing. He successfully completes each task with a given probability, independently of each other. He passes the exam if he can complete at least two of the tasks successfully. Joe calculated that in case he starts with tracking and completes it successfully, his chance of passing the exam is $32\%$. If he starts with successful shooting, the chance of passing is $49\%$, whereas if he starts with successful lasso throwing, he passes with probability $52\%$. The overall probability of passing (calculated before the start of the exam) is $X/1000$ . What is the value of $X$?

There are two non-empty stacks of $n$ and $m$ coins on a table. The following operations are allowed: $\bullet$ The same number of coins are removed from both stacks. $\bullet$ The number of coins in a stack is tripled. For which pairs $(n, m)$ is it possible that after finitely many operations, no coins are more available?

Is it possible to fill all the cells of the table $9 \times 2002$ with natural numbers so that the sum of the numbers in any column and the sum of the numbers in any string would be prime numbers?

Located on the plane $\left[ \frac43 n \right]$ rectangles with sides parallel to the coordinate axes. It is known that any rectangle intersects at least n rectangles. Prove that exists a rectangle that intersects all rectangles.

Let $S=\{1,2,\ldots,10\}$. Three numbers are chosen with replacement from $S$. If the chosen numbers denote the lengths of sides of a triangle, then the probability that they will form a triangle is: $\textbf{(A)}~\frac{101}{200}$ $\textbf{(B)}~\frac{99}{200}$ $\textbf{(C)}~\frac12$ $\textbf{(D)}~\frac{110}{200}$

The points $S(i, j)$ with integer Cartesian coordinates $0 < i \leq n, 0 < j \leq m, m \leq n$, form a lattice. Find the number of: [b](a)[/b] rectangles with vertices on the lattice and sides parallel to the coordinate axes; [b](b)[/b] squares with vertices on the lattice and sides parallel to the coordinate axes; [b](c)[/b] squares in total, with vertices on the lattice.

[b]p1.[/b] The edges of a tetrahedron are all tangent to a sphere. Prove that the sum of the lengths of any pair of opposite edges equals the sum of the lengths of any other pair of opposite edges. (Two edges of a tetrahedron are said to be opposite if they do not have a vertex in common.) [b]p2.[/b] Find the simplest formula possible for the product of the following $2n - 2$ factors: $$\left(1+\frac12 \right),\left(1-\frac12 \right), \left(1+\frac13 \right) , \left(1-\frac13 \right),...,\left(1+\frac{1}{n} \right), \left(1-\frac{1}{n} \right)$$. Prove that your formula is correct. [b]p3.[/b] Solve $$\frac{(x + 1)^2+1}{x + 1} + \frac{(x + 4)^2+4}{x + 4}=\frac{(x + 2)^2+2}{x + 2}+\frac{(x + 3)^2+3}{x + 3}$$ [b]p4.[/b] Triangle $ABC$ is inscribed in a circle, $BD$ is tangent to this circle and $CD$ is perpendicular to $BD$. $BH$ is the altitude from $B$ to $AC$. Prove that the line $DH$ is parallel to $AB$. [img]https://cdn.artofproblemsolving.com/attachments/e/9/4d0b136dca4a9b68104f00300951837adef84c.png[/img] [b]p5.[/b] Consider the picture below as a section of a city street map. There are several paths from $A$ to $B$, and if one always walks along the street, the shortest paths are $15$ blocks in length. Find the number of paths of this length between $A$ and $B$. [img]https://cdn.artofproblemsolving.com/attachments/8/d/60c426ea71db98775399cfa5ea80e94d2ea9d2.png[/img] [b]p6.[/b] A [u]finite [/u] [u]graph [/u] is a set of points, called [u]vertices[/u], together with a set of arcs, called [u]edges[/u]. Each edge connects two of the vertices (it is not necessary that every pair of vertices be connected by an edge). The [u]order [/u] of a vertex in a finite graph is the number of edges attached to that vertex. [u]Example[/u] The figure at the right is a finite graph with $4$ vertices and $7$ edges. [img]https://cdn.artofproblemsolving.com/attachments/5/9/84d479c5dbd0a6f61a66970e46ab15830d8fba.png[/img] One vertex has order $5$ and the other vertices order $3$. Define a finite graph to be [u]heterogeneous [/u] if no two vertices have the same order. Prove that no graph with two or more vertices is heterogeneous. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

A rabbit initially stands at the position $0$, and repeatedly jumps on the real line. In each jump, the rabbit can jump to any position corresponds to an integer but it cannot stand still. Let $N(a)$ be the number of ways to jump with a total distance of $2019$ and stop at the position $a$. Determine all integers $a$ such that $N(a)$ is odd.

$m$ boys and $n$ girls ($m>n$) sat across a round table, supervised by a teacher, and they did a game, which went like this. At first, the teacher pointed a boy to start the game. The chosen boy put a coin on the table. Then, consecutively in a clockwise order, everyone did his turn. If the next person is a boy, he will put a coin to the existing pile of coins. If the next person is a girl, she will take a coin from the existing pile of coins. If there is no coin on the table, the game ends. Notice that depending on the chosen boy, the game could end early, or it could go for a full turn. If the teacher wants the game to go for at least a full turn, how many possible boys could be chosen? [i]Hendrata Dharmawan, Boston, USA[/i]

It is known that in a group of $2024$ students each student has at least $1011$ acquaintances among the remaining members of the group. What is more, there exists a student that has at least $1012$ acquaintances in the group. Prove that for every pair of students $X, Y$, there exist students $X_0 = X, X_1, ..., X_{n - 1}, X_n = Y$ in the group such that for every index $i = 0, ..., n - 1$, the students $X_i$ and $X_{i + 1}$ are acquaintances. [i]Proposed by Mirko Petruševski[/i]

There are $ n$ football teams in a round-robin competition where every 2 teams meet once. The winner of each match receives 3 points while the loser receives 0 points. In the case of a draw, both teams receive 1 point each. Let $ k$ be as follows: $ 2 \leq k \leq n \minus{} 1$. At least how many points must a certain team get in the competition so as to ensure that there are at most $ k \minus{} 1$ teams whose scores are not less than that particular team's score?

Given a $ n \times n$ grid board . How many ways can an $X$ and an $O$ be placed in such a way that they are not in adjacent squares?

What minimum number of colours is sufficient to colour all positive real numbers so that every two numbers whose ratio is 4 or 8 have different colours?

$25$ lightbulbs are distributed in the following way: the first $24$ are placed on a circumference, placing a bulb at each vertex of a regular $24$-gon, and the remaining bulb is placed on the center of said circumference. At any time, the following operations may be applied: [list] [*] Take two vertices on the circumference with an odd amount of vertices between them, and change the state of the bulbs on those vertices and the center bulb. [*] Take three vertices on the circumference that form an equilateral triangle, change the state of the bulbs on those vertices and the center bulb.[/list] Prove from any starting configuration of on and off lightbulbs, it is always possible to reach a configuration where all the bulbs are on.

Determine the greatest possible value of n that satisfies the following condition: for any choice of $n$ subsets $M_1, ...,M_n$ of the set $M = \{1,2,...,n\}$ satisfying the conditions i) $i \in M_i$ and ii) $i \in M_j \Leftrightarrow j \notin M_i$ for all $i \ne j$, there exist $M_k$ and $M_l$ such that $M_k \cup M_l = M$. (Moscow regional olympiad,adopted)

On a square table of $2011$ by $2011$ cells we place a finite number of napkins that each cover a square of $52$ by $52$ cells. In each cell we write the number of napkins covering it, and we record the maximal number $k$ of cells that all contain the same nonzero number. Considering all possible napkin configurations, what is the largest value of $k$? [i]Proposed by Ilya Bogdanov and Rustem Zhenodarov, Russia[/i]

The country of Harpland has three types of coins: green, white and orange. The unit of currency in Harpland is the shilling. Any coin is worth a positive integer number of shillings, but coins of the same color may be worth different amounts. A set of coins is stacked in the form of an equilateral triangle of side $n$ coins, as shown below for the case of $n=6$. [asy] size(100); for (int j=0; j<6; ++j) { for (int i=0; i<6-j; ++i) { draw(Circle((i+j/2,0.866j),0.5)); } } [/asy] The stacking has the following properties: (a) no coin touches another coin of the same color; (b) the total worth, in shillings, of the coins lying on any line parallel to one of the sides of the triangle is divisible by by three. Prove that the total worth in shillings of the [i]green[/i] coins in the triangle is divisible by three.

What is the smallest $r$ such that three disks of radius $r$ can completely cover up a unit disk?

Consider all products by $2, 4, 6, ..., 2000$ of the elements of the set $A =\left\{\frac12, \frac13, \frac14,...,\frac{1}{2000},\frac{1}{2001}\right\}$ . Find the sum of all these products.

Carrie, Miranda, Charlotte, and Samantha are sitting at a table with $5$ numbered chairs (numbered $1$ through $5$). One chair is left open for Big, should he decide to join the four for lunch. In how many distinct ways can the four women occupy the table?

Suppose that $1000$ students are standing in a circle. Prove that there exists an integer $k$ with $100 \leq k \leq 300$ such that in this circle there exists a contiguous group of $2k$ students, for which the first half contains the same number of girls as the second half. [i]Proposed by Gerhard Wöginger, Austria[/i]

Points $A_1, A_2,\ldots, A_n$ are found on a circle in this order. Each point $A_i$ has exactly $i$ coins. A move consists in taking two coins from two points (may be the same point) and moving them to adjacent points (one move clockwise and another counter-clockwise). Find all possible values of $n$ for which it is possible after a finite number of moves to obtain a configuration with each point $A_i$ having $n+1-i$ coins.