Let $m,n,k$ be pairwise relatively prime positive integers greater than $3$. Find the minimal possible number of points on the plane with the following property: there are $x$ of them which are the vertices of a regular $x$-gon for $x = m, x = n, x = k$. (E.Piryutko)

There are 2018 points marked on a sphere. A zebra wants to paint each point white or black and, perhaps, connect some pairs of points of different colors with a segment. Find the residue modulo 5 of the number of ways to do this.

There are $n$ instruments in the laboratory, each two of them can be connected with a wire. Moreover, if four devices $A, B, C, D$, are such that wires of $AB$, $BC$ and $CD$ are connected but there is no connected pair between $CA$, $AD$ and $DB$, a collapse occurs. A professor invented a wiring diagram that does not collapse. Coming to the laboratory, he found that the collapse has not yet occurred, but the devices are connected not according to his scheme. Prove that he can implement his scheme, each time connecting or disconnecting a pair of devices, so that the collapse won’t happen anytime.

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.

On the circumference of a circle are $21$ points. Prove that among the arcs which join any two of these points, at least $100$ of them must subtend an angle at the centre of the circle not exceeding $120^o$ . ( A . F . Sidorenko)

In an island there are $n$ castles, and each castle is in country $A$ or $B$. There is one commander per castle, and each commander belongs to the same country as the castle he's initially in. There are some (two-way) roads between castles (there may be roads between castles of different countries), and call two castles adjacent if there is a road between them. Prove that the following two statements are equivalent: (1) If some commanders from country $B$ move to attack an adjacent castle in country $A$, some commanders from country $A$ could appropriately move in defense to adjacent castles in country $A$ so that in every castle of country $A$, the number of country $A$'s commanders defending that castle is not less than the number of country $B$'s commanders attacking that castle. (Each commander can defend or attack only one castle at a time.) (2) For any arbitrary set $X$ of castles in country $A$, the number of country $A$'s castles that are in $X$ or adjacent to at least one of the castle in $X$ is not less than the number of country $B$'s castles that are adjacent to at least one of the castles in $X$.

Alice and Bob play a game. First, Alice secretly picks a finite set $S$ of lattice points in the Cartesian plane. Then, for every line $\ell$ in the plane which is horizontal, vertical, or has slope $+1$ or $-1$, she tells Bob the number of points of $S$ that lie on $\ell$. Bob wins if he can determine the set $S$. Prove that if Alice picks $S$ to be of the form \[S = \{(x, y) \in \mathbb{Z}^2 \mid m \le x^2 + y^2 \le n\}\] for some positive integers $m$ and $n$, then Bob can win. (Bob does not know in advance that $S$ is of this form.) [i]Proposed by Mark Sellke[/i]

Let $a$ and $b$ be two positive integers and $A$ be a subset of $\{1, 2,…, a + b \}$ that has more than $ \frac {a + b} {2}$ elements. Show that there are two numbers in $A$ whose difference is $a$ or $b$.

Two vertices of a cube are $A,O$ such that $AO$ is the diagonal of one its faces. A $n-$run is a sequence of $n+1$ vertices of the cube such that each $2$ consecutive vertices in the sequence are $2$ ends of one side of the cube. Is the $1386-$runs from $O$ to itself less than $1386-$runs from $O$ to $A$ or more than it?

Call a positive integer [i]convenient[/i] if its digits can be partitioned into two collections of contiguous digits whose element sums are $7$ and $11$. For example, $3456$ is convenient, but $4247$ is not. Compute the number of convenient positive integers less than or equal to $10^5$.

Let $n \geqslant 100$ be an integer. Ivan writes the numbers $n, n+1, \ldots, 2 n$ each on different cards. He then shuffles these $n+1$ cards, and divides them into two piles. Prove that at least one of the piles contains two cards such that the sum of their numbers is a perfect square.

The numbers $1,2,...,64 $ are written on the unit squares of a table $8 \times 8$. The two smallest numbers of every row are marked black and the two smaller numbers of every comlumn are marked white. Prove or disprove that there are at least k numbers on the table that are marked both black and white when: a) $k=3$ b) $k=4$ c) $k=5$ .

[b]p1. [/b] In basketball, teams can score $1, 2$, or $3$ points each time. Suppose that Duke basketball have scored $8$ points so far. What is the total number of possible ways (ordered) that they have scored? For example, $(1, 2, 2, 2, 1)$,$(1, 1, 2, 2, 2)$ are two different ways. [b]p2.[/b] All the positive integers that are coprime to $2021$ are grouped in increasing order, such that the nth group contains $2n - 1$ numbers. Hence the first three groups are $\{1\}$, $\{2, 3, 4\}$, $\{5, 6, 7, 8, 9\}$. Suppose that $2022$ belongs to the $k$th group. Find $k$. [b]p3.[/b] Let $A = (0, 0)$ and $B = (3, 0)$ be points in the Cartesian plane. If $R$ is the set of all points $X$ such that $\angle AXB \ge 60^o$ (all angles are between $0^o$ and $180^o$), find the integer that is closest to the area of $R$. [b]p4.[/b] What is the smallest positive integer greater than $9$ such that when its left-most digit is erased, the resulting number is one twenty-ninth of the original number? [b]p5. [/b] Jonathan is operating a projector in the cartesian plane. He sets up $2$ infinitely long mirrors represented by the lines $y = \tan(15^o)x$ and $y = 0$, and he places the projector at $(1, 0)$ pointed perpendicularly to the $x$-axis in the positive $y$ direction. Jonathan furthermore places a screen on one of the mirrors such that light from the projector reflects off the mirrors a total of three times before hitting the screen. Suppose that the coordinates of the screen is $(a, b)$. Find $10a^2 + 5b^2$. [b]p6.[/b] Dr Kraines has a cube of size $5 \times 5 \times 5$, which is made from $5^3$ unit cubes. He then decides to choose $m$ unit cubes that have an outside face such that any two different cubes don’t share a common vertex. What is the maximum value of $m$? [b]p7.[/b] Let $a_n = \tan^{-1}(n)$ for all positive integers $n$. Suppose that $$\sum_{k=4}^{\infty}(-1)^{\lfloor \frac{k}{2} \rfloor +1} \tan(2a_k)$$ is equals to $a/b$ , where $a, b$ are relatively prime. Find $a + b$. [b]p8.[/b] Rishabh needs to settle some debts. He owes $90$ people and he must pay \$ $(101050 + n)$ to the $n$th person where $1 \le n \le 90$. Rishabh can withdraw from his account as many coins of values \$ $2021$ and \$ $x$ for some fixed positive integer $x$ as is necessary to pay these debts. Find the sum of the four least values of $x$ so that there exists a person to whom Rishabh is unable to pay the exact amount owed using coins. [b]p9.[/b] A frog starts at $(1, 1)$. Every second, if the frog is at point $(x, y)$, it moves to $(x + 1, y)$ with probability $\frac{x}{x+y}$ and moves to $(x, y + 1)$ with probability $\frac{y}{x+y}$ . The frog stops moving when its $y$ coordinate is $10$. Suppose the probability that when the frog stops its $x$-coordinate is strictly less than $16$, is given by $m/n$ where $m, n$ are positive integers that are relatively prime. Find $m + n.$ [b]p10.[/b] In the triangle $ABC$, $AB = 585$, $BC = 520$, $CA = 455$. Define $X, Y$ to be points on the segment $BC$. Let $Z \ne A$ be the intersection of $AY$ with the circumcircle of $ABC$. Suppose that $XZ$ is parallel to $AC$ and the circumcircle of $XYZ$ is tangent to the circumcircle of $ABC$ at $Z$. Find the length of $XY$ . PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

The numbers from $1$ to $64$ are written on the squares of a chessboard (from $1$ to $8$ from left to right on the first row , from $9$ to $16$ from left to right on the second row , and so on). Pluses are written before some of the numbers, and minuses are written before the remaining numbers in such a way that there are $4$ pluses and $4$ minuses in each row and in each column . Prove that the sum of the written numbers is equal to zero.

$64$ non-negative numbers whose sum equals $1956$ are arranged in a square table, eight numbers in each row and each column. The sum of the numbers on the two longest diagonals is equal to $112$. The numbers situated symmetrically with respect to any of the longest diagonals are equal. (a) Prove that the sum of numbers in any column is less than $1035$. (b) Prove that the sum of numbers in any row is less than $518$.

For a sequence $x_1,x_2,\ldots,x_n$ of real numbers, we define its $\textit{price}$ as \[\max_{1\le i\le n}|x_1+\cdots +x_i|.\] Given $n$ real numbers, Dave and George want to arrange them into a sequence with a low price. Diligent Dave checks all possible ways and finds the minimum possible price $D$. Greedy George, on the other hand, chooses $x_1$ such that $|x_1 |$ is as small as possible; among the remaining numbers, he chooses $x_2$ such that $|x_1 + x_2 |$ is as small as possible, and so on. Thus, in the $i$-th step he chooses $x_i$ among the remaining numbers so as to minimise the value of $|x_1 + x_2 + \cdots x_i |$. In each step, if several numbers provide the same value, George chooses one at random. Finally he gets a sequence with price $G$. Find the least possible constant $c$ such that for every positive integer $n$, for every collection of $n$ real numbers, and for every possible sequence that George might obtain, the resulting values satisfy the inequality $G\le cD$. [i]Proposed by Georgia[/i]

A dance with 2018 couples takes place in Havana. For the dance, 2018 distinct points labeled $0, 1,\ldots, 2017$ are marked in a circumference and each couple is placed on a different point. For $i\geq1$, let $s_i=i\ (\textrm{mod}\ 2018)$ and $r_i=2i\ (\textrm{mod}\ 2018)$. The dance begins at minute $0$. On the $i$-th minute, the couple at point $s_i$ (if there's any) moves to point $r_i$, the couple on point $r_i$ (if there's any) drops out, and the dance continues with the remaining couples. The dance ends after $2018^2$ minutes. Determine how many couples remain at the end. Note: If $r_i=s_i$, the couple on $s_i$ stays there and does not drop out.

A polygon with $2n+1$ vertices is given. Show that it is possible to assign numbers $1,2,\ldots ,4n+2$ to the vertices and midpoints of the sides of the polygon so that for each side the sum of the three numbers assigned to it is the same.

A spider is walking on the boundary of equilateral triangle $\triangle{ABC}$ (vertices labelled in counterclockwise order), starting at vertex $A$. Each second, she moves to one of her two adjacent vertices with equal probability. The windiness of a path that starts and ends at $A$ is the net number of counterclockwise revolutions made. For example, the windiness of the path $ABCA$ is $1,$ and the windiness of the path $ABCACBACBA$ is $-1$. What is the remainder modulo $1000$ of the sum of the squares of the windiness values taken over all possible paths that end back at vertex $A$ after $2025$ seconds?

A class consists of 26 students with two students sitting on each desk. Suddenly, the students decide to change seats, such that every two students that were previously sitting together are now apart. Find the maximum value of positive integer $N$ such that, regardless of the students' sitting positions, at the end there is a set $S$ consisting of $N$ students satisfying the following property: every two of them have never been sitting together.

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.

A class has $24$ students. Each group consisting of three of the students meet, and choose one of the other $21$ students, A, to make him a gift. In this case, A considers each member of the group that offered him a gift as being his friend. Prove that there is a student that has at least $10$ friends.

In the garden of Wonderland, there are $2016$ apples, $2017$ bananas and $2018$ oranges.Two monkeys Adu and Bakar play the following game: alternatively each of them takes and eats one fruit of any kind except for the one that he took in previous turn (in the first turn, each of them can take a fruit of any kind). Who can not take a fruit is the loser. Which monkey has the winning strategy if Adu plays first?

There are $9$ vertical columns in a row. In some places, horizontal sticks are inserted between adjacent columns, no two are at the same height. The beetle crawls from the bottom up; when he meets the wand, he crawls over it to the next column and continues to crawl up. It is known that if a beetle starts at the bottom of the first (leftmost) column, then it will end its journey on the ninth (rightmost) column. Is it always possible to remove one stick so that the beetle ends up at the top of the fifth column? (For example, if the sticks are arranged as in picture, the beetle will crawl along a solid line. If you remove the third one A stick in the path of the beetle, then it will crawl along the dotted line.) [i] Proposed by G. Karavaev[/i]

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]