A configuration of $4027$ points in the plane is called Colombian if it consists of $2013$ red points and $2014$ blue points, and no three of the points of the configuration are collinear. By drawing some lines, the plane is divided into several regions. An arrangement of lines is good for a Colombian configuration if the following two conditions are satisfied: i) No line passes through any point of the configuration. ii) No region contains points of both colors. Find the least value of $k$ such that for any Colombian configuration of $4027$ points, there is a good arrangement of $k$ lines. Proposed by [i]Ivan Guo[/i] from [i]Australia.[/i]

In a tournament, every team plays exactly once against every other team. One won match earns $3$ points for the winner and $0$ for the loser. With a draw both teams receive $1$ point each. At the end of the tournament it appears that all teams together have achieved $15$ points. The last team on the final list scored exactly $1$ point. The second to last team has not lost a match. a) How many teams participated in the tournament? b) How many points did the team score in second place in the final ranking?

In a round robin chess tournament each player plays every other player exactly once. The winner of each game gets $ 1$ point and the loser gets $ 0$ points. If the game is tied, each player gets $ 0.5$ points. Given a positive integer $ m$, a tournament is said to have property $ P(m)$ if the following holds: among every set $ S$ of $ m$ players, there is one player who won all her games against the other $ m\minus{}1$ players in $ S$ and one player who lost all her games against the other $ m \minus{} 1$ players in $ S$. For a given integer $ m \ge 4$, determine the minimum value of $ n$ (as a function of $ m$) such that the following holds: in every $ n$-player round robin chess tournament with property $ P(m)$, the final scores of the $ n$ players are all distinct.

155 birds $ P_1, \ldots, P_{155}$ are sitting down on the boundary of a circle $ C.$ Two birds $ P_i, P_j$ are mutually visible if the angle at centre $ m(\cdot)$ of their positions $ m(P_iP_j) \leq 10^{\circ}.$ Find the smallest number of mutually visible pairs of birds, i.e. minimal set of pairs $ \{x,y\}$ of mutually visible pairs of birds with $ x,y \in \{P_1, \ldots, P_{155}\}.$ One assumes that a position (point) on $ C$ can be occupied simultaneously by several birds, e.g. all possible birds.

For a given positive integer $n$, there are a total of $5n$ balls labeled with numbers $1$, $2$, $3$, $\cdots$, $n$, with 5 balls for each number. The balls are put into $n$ boxes, with $5$ balls in each box. Show that you can color two balls red and one ball blue in each box so that the sum of the numbers on the red balls is twice the sum of the numbers on the blue balls.

Three friends Archie, Billie, and Charlie play a game. At the beginning of the game, each of them has a pile of $2024$ pebbles. Archie makes the first move, Billie makes the second, Charlie makes the third and they continue to make moves in the same order. In each move, the player making the move must choose a positive integer $n$ greater than any previously chosen number by any player, take $2n$ pebbles from his pile and distribute them equally to the other two players. If a player cannot make a move, the game ends and that player loses the game. $\hspace{5px}$ Determine all the players who have a strategy such that, regardless of how the other two players play, they will not lose the game. [i]Proposed by Ilija Jovčeski, Macedonia[/i]

Determine the maximal length $L$ of a sequence $a_1,\dots,a_L$ of positive integers satisfying both the following properties: [list=disc] [*]every term in the sequence is less than or equal to $2^{2023}$, and [*]there does not exist a consecutive subsequence $a_i,a_{i+1},\dots,a_j$ (where $1\le i\le j\le L$) with a choice of signs $s_i,s_{i+1},\dots,s_j\in\{1,-1\}$ for which \[s_ia_i+s_{i+1}a_{i+1}+\dots+s_ja_j=0.\] [/list]

The Magician and his Assistant show trick. The Viewer writes on the board the sequence of $N$ digits. Then the Assistant covers some pair of adjacent digits so that they become invisible. Finally, the Magician enters the show, looks at the board and guesses the covered digits and their order. Find the minimal $N$ such that the Magician and his Assistant can agree in advance so that the Magician always guesses right

On a circle are given the points $A_1, B_1, A_2, B_2, \cdots, A_9, B_9$ in this order. All the segments $A_iB_j (i, j=1, 2, \cdots, 9$ must be colored in one of $k$ colors, so that no two segments from the same color intersect (inside the circle) and for every $i$ there is a color, such that no segments with an end $A_i$, nor $B_i$ is colored such. What is the least possible $k$?

Find the number of positive integers $k \le 2018$ for which there exist integers $m$ and $n$ so that $k = 2^m + 2^n$. For example, $64 = 2^5 + 2^5$, $65 = 2^0 + 2^6$, and $66 = 2^1 + 2^6$.

In the coordinate plane consider the set $ S$ of all points with integer coordinates. For a positive integer $ k$, two distinct points $A$, $ B\in S$ will be called $ k$-[i]friends[/i] if there is a point $ C\in S$ such that the area of the triangle $ ABC$ is equal to $ k$. A set $ T\subset S$ will be called $ k$-[i]clique[/i] if every two points in $ T$ are $ k$-friends. Find the least positive integer $ k$ for which there exits a $ k$-clique with more than 200 elements. [i]Proposed by Jorge Tipe, Peru[/i]

A grasshopper is in the left above corner of a $10\times 10$ square. At each step he can jump a square below or a square to the right. Also, he can also fly from a cell of the bottom row to a cell of the above row, and from a cell of the rightmost column to a cell of the leftmost column. Prove that the grasshopper has to do at leat $9$ flies in order to visit each cell of the square at least once. [I]Proposed by N. Vlasova[/I]

Given a positive integer $k,$ prove that for any integer $n \geq 20k,$ there exist $n - k$ pairwise distinct positive integers whose squares add up to $n(n + 1)(2n + 1)/6.$ [i]The Problem Selection Committee[/i]

A finite sequence of natural numbers $a_1, a_2, \dots, a_n$ is given. A sub-sequence $a_{k+1}, a_{k+2}, \dots, a_l$ will be called a [i]repetition[/i] if there exists a natural number $p\leq \frac{l-k}2$ such that $a_i=a_{i+p}$ for $k+1\leq i\leq l-p$, but $a_i\neq a_{i+p}$ for $i=k$ (if $k>0$) and $i=l-p+1$ (if $l<n$). Show that the sequence contains less than $n$ repetitions.

Let $ABCDEF$ be a regular hexagon of side length $2$. Let us construct parallels to its sides passing through its vertices and midpoints, which divide the hexagon into $24$ congruent equilateral triangles, whose vertices are called nodes. For each node $X$, we define its trio as the figure formed by three adjacent triangles with vertex $X$, such that their intersection is only $X$ and they are not congruent in pairs. a) Determine the maximum possible area of a trio. b) Show that there exists a node whose trios can cover the entire hexagon, and a node whose trios cannot cover the entire hexagon. c) Determine the total number of triangles associated with the hexagon.

[b]p1.[/b] Zion, RJ, Cam, and Tre decide to start learning languages. The four most popular languages that Duke offers are Spanish, French, Latin, and Korean. If each friend wants to learn exactly three of these four languages, how many ways can they pick courses such that they all attend at least one course together? [b]p2. [/b] Suppose we wrote the integers between $0001$ and $2019$ on a blackboard as such: $$000100020003 · · · 20182019.$$ How many $0$’s did we write? [b]p3.[/b] Duke’s basketball team has made $x$ three-pointers, $y$ two-pointers, and $z$ one-point free throws, where $x, y, z$ are whole numbers. Given that $3|x$, $5|y$, and $7|z$, find the greatest number of points that Duke’s basketball team could not have scored. [b]p4.[/b] Find the minimum value of $x^2 + 2xy + 3y^2 + 4x + 8y + 12$, given that $x$ and $y$ are real numbers. Note: calculus is not required to solve this problem. [b]p5.[/b] Circles $C_1, C_2$ have radii $1, 2$ and are centered at $O_1, O_2$, respectively. They intersect at points $ A$ and $ B$, and convex quadrilateral $O_1AO_2B$ is cyclic. Find the length of $AB$. Express your answer as $x/\sqrt{y}$ , where $x, y$ are integers and $y$ is square-free. [b]p6.[/b] An infinite geometric sequence $\{a_n\}$ has sum $\sum_{n=0}^{\infty} a_n = 3$. Compute the maximum possible value of the sum $\sum_{n=0}^{\infty} a_{3n} $. [b]p7.[/b] Let there be a sequence of numbers $x_1, x_2, x_3,...$ such that for all $i$, $$x_i = \frac{49}{7^{\frac{i}{1010}} + 49}.$$ Find the largest value of $n$ such that $$\left\lfloor \sum_{i=1}{n} x_i \right\rfloor \le 2019.$$ [b]p8.[/b] Let $X$ be a $9$-digit integer that includes all the digits $1$ through $9$ exactly once, such that any $2$-digit number formed from adjacent digits of $X$ is divisible by $7$ or $13$. Find all possible values of $X$. [b]p9.[/b] Two $2025$-digit numbers, $428\underbrace{\hbox{99... 99}}_{\hbox{2019 \,\, 9's}}571$ and $571\underbrace{\hbox{99... 99}}_{\hbox{2019 \,\, 9's}}428$ , form the legs of a right triangle. Find the sum of the digits in the hypotenuse. [b]p10.[/b] Suppose that the side lengths of $\vartriangle ABC$ are positive integers and the perimeter of the triangle is $35$. Let $G$ the centroid and $I$ be the incenter of the triangle. Given that $\angle GIC = 90^o$ , what is the length of $AB$? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

How many spanning trees does the following graph (with $6$ vertices and $9$ edges) have? (A spanning tree is a subset of edges that spans all of the vertices of the original graph, but does not contain any cycles.) [img]https://cdn.artofproblemsolving.com/attachments/0/4/0e53e0fbb141b66a7b1c08696be2c5dfe68067.png[/img]

There are $n$ students standing in line positions $1$ to $n$. While the teacher looks away, some students change their positions. When the teacher looks back, they are standing in line again. If a student who was initially in position $i$ is now in position $j$, we say the student moved for $|i-j|$ steps. Determine the maximal sum of steps of all students that they can achieve.

On the control board of a nuclear station, there are $n$ electric switches ($n > 0$), all in one row. Each switch has two possible positions: up and down. The switches are connected to each other in such a way that, whenever a switch moves down from its upper position, its right neighbour (if it exists) automatically changes position. At the beginning, all switches are down. The operator of the board first changes the position of the leftmost switch once, then the position of the second leftmost switch twice etc., until eventually he changes the position of the rightmost switch n times. How many switches are up after all these operations?

Decompose a $6\times 6$ square into unit squares and consider the $49$ vertices of these unit squares. We call a square good if its vertices are among the $49$ points and if its sides and diagonals do not lie on the gridlines of the $6\times 6$ square. [list=a] [*]Find the total number of good squares. [*]Prove that there exist two good disjoint squares such that the smallest distance between their vertices is $1/\sqrt{5}.$ [/list]

There is a city with $n$ metro stations, each located at a vertex of a regular n-polygon. Metro Line 1 is a line which only connects two non-neighboring stations $A$ and $B$. Metro Line 2 is a cyclic line which passes through all the stations in a shape of regular n-polygon. For each line metro can run in any direction, and $A$ and $B$ are the stations which one can transfer into other line. The line between two neighboring stations is called 'metro interval'. For each station there is one stationmaster, and there are at least one female stationmaster and one male stationmaster. If $n$ is odd, prove that for any integer $k$ $(0<k<n)$ there is a path that starts from a station with a male stationmaster and ends at a station with a female stationmaster, passing through $k$ metro intervals.

[b]p1.[/b] To every face of a given cube a new cube of the same size is glued. The resulting solid has how many faces? [b]p2.[/b] A father and his son returned from a fishing trip. To make their catches equal the father gave to his son some of his fish. If, instead, the son had given his father the same number of fish, then father would have had twice as many fish as his son. What percent more is the father's catch more than his son's? [b]p3.[/b] A radio transmitter has $4$ buttons. Each button controls its own switch: if the switch is OFF the button turns it ON and vice versa. The initial state of switches in unknown. The transmitter sends a signal if at least $3$ switches are ON. What is the minimal number of times you have to push the button to guarantee the signal is sent? [b]p4.[/b] $19$ matches are placed on a table to show the incorrect equation: $XXX + XIV = XV$. Move exactly one match to change this into a correct equation. [b]p5.[/b] Cut the grid shown into two parts of equal area by cutting along the lines of the grid. [img]https://cdn.artofproblemsolving.com/attachments/c/1/7f2f284acf3709c2f6b1bea08835d2fb409c44.png[/img] [b]p6.[/b] A family of funny dwarfs consists of a dad, a mom, and a child. Their names are: $A$, $R$, and $C$ (not in order). During lunch, $C$ made the statements: “$R$ and $A$ have different genders” and “$R$ and $A$ are my parents”, and $A$ made the statements “I am $C$'s dad” and “I am $R$'s daughter.” In fact, each dwarf told truth once and told a lie once. What is the name of the dad, what is the name of the child, and is the child a son or a daughter? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Ivan has a $m \times n$ board, and he color some squares black, so that no three black squares form a L-triomino up to rotations and reflections. What is the maximal number of black squares that Ivan can color? [i]Proposed by Ivan Chan Kai Chin[/i]

Given $n$ students in the plane such that the $\frac{n(n-1)}{2}$ distances are pairwise distinct. Each student gives a candy each to the $k$ students closest to him. Given that each student receives the same amount of candies, determine all possible values of $n$ in terms of $k$. [i]Proposed by Wong Jer Ren[/i]

For each finite set $ U$ of nonzero vectors in the plane we define $ l(U)$ to be the length of the vector that is the sum of all vectors in $ U.$ Given a finite set $ V$ of nonzero vectors in the plane, a subset $ B$ of $ V$ is said to be maximal if $ l(B)$ is greater than or equal to $ l(A)$ for each nonempty subset $ A$ of $ V.$ (a) Construct sets of 4 and 5 vectors that have 8 and 10 maximal subsets respectively. (b) Show that, for any set $ V$ consisting of $ n \geq 1$ vectors the number of maximal subsets is less than or equal to $ 2n.$