Let \( n \) be a positive integer. A graph on \( 2n - 1 \) vertices is given such that the size of the largest clique in the graph is \( n \). Prove that there exists a vertex that is present in every clique of size \( n\)

$m$ and $n$ are two nonnegative integers. In the Philosopher's Chess, The chessboard is an infinite grid of identical regular hexagons and a new piece named the Donkey moves on it as follows: Starting from one of the hexagons, the Donkey moves $m$ cells in one of the $6$ directions, then it turns $60$ degrees clockwise and after that moves $n$ cells in this new direction until it reaches it's final cell. At most how many cells are in the Philosopher's chessboard such that one cannot go from anyone of them to the other with a finite number of movements of the Donkey? [i]Proposed by Shayan Dashmiz[/i]

Alice and Bob play a game where they start from a complete graph with $n$ vertices and take turns removing a single edge from the graph, with Alice taking the first turn. The first player to disconnect the graph loses. Compute the sum of all $n$ between $2$ and $100$ inclusive such that Alice has a winning strategy. (A complete graph is one where there is an edge between every pair of vertices.)

Island Hopping Holidays offer short holidays to $64$ islands, labeled Island $i, 1 \le i \le 64$. A guest chooses any Island $a$ for the first night of the holiday, moves to Island $b$ for the second night, and finally moves to Island $c$ for the third night. Due to the limited number of boats, we must have $b \in T_a$ and $c \in T_b$, where the sets $T_i$ are chosen so that (a) each $T_i$ is non-empty, and $i \notin T_i$, (b) $\sum^{64}_{i=1} |T_i| = 128$, where $|T_i|$ is the number of elements of $T_i$. Exhibit a choice of sets $T_i$ giving at least $63\cdot 64 + 6 = 4038$ possible holidays. Note that c = a is allowed, and holiday choices $(a, b, c)$ and $(a',b',c')$ are considered distinct if $a \ne a'$ or $b \ne b'$ or $c \ne c'$.

Define a $ k$-[i]clique[/i] to be a set of $ k$ people such that every pair of them are acquainted with each other. At a certain party, every pair of 3-cliques has at least one person in common, and there are no 5-cliques. Prove that there are two or fewer people at the party whose departure leaves no 3-clique remaining.

Jeff has a deck of $12$ cards: $4$ $L$s, $4$ $M$s, and $4$ $T$s. Armaan randomly draws three cards without replacement. The probability that he takes $3$ $L$s can be written as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m +n$.

(Game) In an Indian reservatory there are $15$ totem poles arranged according to the left figure. Silent Stream and Red Fire used to play the following game: In turns they stretch ropes between two-two poles in such a way that every stretched rope is parallel to a side of the big triangle and no rope can go along a pole that is already touched by another rope. Furthermore, if instead of a rope one can stretch out a straight line extension of the rope, then one should stretch out this extension. The one who cannot stretch out more rope according to the rules loses. [i]Win two games in a row against the organizers! You can decide that you want to start or to be the second player. The figure on the right depicts the first three steps of a game. First Silent Stream stretches the blue rope, then Red Fire stretches the red one, finally Silent Stream stretches the blue one.[/i] [img]https://cdn.artofproblemsolving.com/attachments/f/8/3b8b9e38a8a477da288566ecb26036bfc7e615.png[/img]

Andrew, Benji, and Carlson want to split a pile of $5$ indistinguishable left shoes and $7$ indistinguishable right shoes. Andrew is practical and wants the same number of left and right shoes. Benji is greedy and wants the most shoes out of the three of them. Carlson is a trickster and wants Benji to have a different number of left and right shoes. How many ways are there to split up the shoes in a way that suits everyone’s desires?

A table $2 \times 2024$ is filled with positive integers. Specifically, the first row is filled with numbers from the set $\{1, \ldots, 2023\}$. It turned out that for any two columns the difference of numbers from the first row is divisible by the difference of numbers from the second row, while all numbers in the second row are pairwise different. Is it true for sure that the numbers in the first row are equal? Ivan Kukharchuk

Gary plays the following game with a fair $n$-sided die whose faces are labeled with the positive integers between $1$ and $n$, inclusive: if $n = 1$, he stops; otherwise he rolls the die, and starts over with a $k$-sided die, where $k$ is the number his $n$-sided die lands on. (In particular, if he gets $k = 1$, he will stop rolling the die.) If he starts out with a $6$-sided die, what is the expected number of rolls he makes?

Given an $1$ x $n$ table ($n\geq 2$), two players alternate the moves in which they write the signs + and - in the cells of the table. The first player always writes +, while the second always writes -. It is not allowed for two equal signs to appear in the adjacent cells. The player who can’t make a move loses the game. Which of the players has a winning strategy?

Given a $2015$-tuple $(a_1,a_2,\ldots,a_{2015})$ in each step we choose two indices $1\le k,l\le 2015$ with $a_k$ even and transform the $2015$-tuple into $(a_1,\ldots,\dfrac{a_k}{2},\ldots,a_l+\dfrac{a_k}{2},\ldots,a_{2015})$. Prove that starting from $(1,2,\ldots,2015)$ in finite number of steps one can reach any permutation of $(1,2,\ldots,2015)$.

The cell of a $(2m +1) \times (2n +1)$ board are painted in two colors - white and black. The unit cell of a row (column) is called [i]dominant[/i] on the row (the column) if more than half of the cells that row (column) have the same color as this cell. Prove that at least $m + n-1$ cells on the board are dominant in both their row and column.

Consider a 3 × 4 rectangular table where each cell can be colored using one of three available colors. Determine the number of different ways the table can be colored such that no two cells sharing a common side have the same color. It is not necessary to use all three colors in each coloring. [i]Proposed by Prudencio Guerrero Fernández, Cuba[/i]

Every time Josh and Ron tap their screens, one of three emojis appears, each with equal probability: barbecue, bacon, or burger. Josh taps his screen until he gets a sequence of barbecue, bacon, and burger consecutively (in that specific order.) Ron taps his screen until he gets a sequence of three bacons in a row. Let $M$ and $N$ be the expected number of times Josh and Ron tap their screens, respectively. What is $|M-N|$?

Does there exist a row of Pascal’s Triangle containing four distinct values $a,b,c$ and $d$ such that $b = 2a$ and $d = 2c$? Recall that Pascal’s triangle is the pattern of numbers that begins as follows [img]https://cdn.artofproblemsolving.com/attachments/2/1/050e56f0f1f1b2a9c78481f03acd65de50c45b.png[/img] where the elements of each row are the sums of pairs of adjacent elements of the prior row. For example, $10 =4+6$. Also note that the last row displayed above contains the four elements $a = 5,b = 10,d = 10,c = 5$, satisfying $b = 2a$ and $d = 2c$, but these four values are NOT distinct.

[b]p1.[/b] In this problem, a half deck of cards consists of $26$ cards, each labeled with an integer from $1$ to $13$. There are two cards labeled $1$, two labeled $2$, two labeled $3$, etc. A certain math class has $13$ students. Each day, the teacher thoroughly shuffles a half deck of cards and deals out two cards to each student. Each student then adds the two numbers on the cards received, and the resulting $13$ sums are multiplied together to form a product $P$. If $P$ is an even number, the class must do math homework that evening. Show that the class always must do math homework. [b]p2.[/b] Twenty-six people attended a math party: Archimedes, Bernoulli, Cauchy, ..., Yau, and Zeno. During the party, Archimedes shook hands with one person, Bernoulli shook hands with two people, Cauchy shook hands with three people, and similarly up through Yau, who shook hands with $25$ people. How many people did Zeno shake hands with? Justify that your answer is correct and that it is the only correct answer. [b]p3.[/b] Prove that there are no integers $m, n \ge 1$ such that $$\sqrt{m+\sqrt{m+\sqrt{m+...+\sqrt{m}}}}=n$$ where there are $2006$ square root signs. [b]p4.[/b] Let $c$ be a circle inscribed in a triangle ABC. Let $\ell$ be the line tangent to $c$ and parallel to $AC$ (with $\ell \ne AC$). Let $P$ and $Q$ be the intersections of $\ell$ with $AB$ and $BC$, respectively. As $ABC$ runs through all triangles of perimeter $1$, what is the longest that the line segment $PQ$ can be? Justify your answer. [b]p5.[/b] Each positive integer is assigned one of three colors. Show that there exist distinct positive integers $x, y$ such that $x$ and $y$ have the same color and $|x -y|$ is a perfect square. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Some time ago there was a war across the world. In the plane $n$ lines are moving, with the regions contained by the lines being the territories of the countries at war. Each line moves parallel to itself with constant speed (each with its own speed), and no line can reverse its direction. Some of the original countries disappeared (a country disappears iff its area is converted to zero) and within the course of the time, other countries appeared. After some time, the presidents of the existing countries made a treaty to end the war, created the United Nations, and all borders ceased movement. The UN then counted the total numbers of sovereign states that were destroyed and the existing ones, obtaining a total of $k$. Prove that $k\leq \frac{n^3+5n}{6}+1$. Is is possible to have equality?

Given a finite set $S \subset \mathbb{R}^3$, define $f(S)$ to be the mininum integer $k$ such that there exist $k$ planes that divide $\mathbb{R}^3$ into a set of regions, where no region contains more than one point in $S$. Suppose that \[M(n) = \max\{f(S) : |S| = n\} \text{ and } m(n) = \min\{f(S) : |S| = n\}.\] Evaluate $M(200) \cdot m(200)$.

In a party of eight people, each pair of people either know each other or do not know each other. Each person knows exactly three of the others. Determine whether the following two conditions can be satisfied simultaneously: [list] – for any three people, at least two do not know each other; – for any four people there are at least two who know each other. [/list]

All points in the plane are colored in $n$ colors. In each line, there are point of no more than two colors. What is the maximum number of colors?

A class consists of $7$ boys and $13$ girls. During the first three months of the school year, each boy has communicated with each girl at least once. Prove that there exist two boys and two girls such that both boys communicated with both girls first time in the same month.

In a chess festival that is held in a school with $2017$ students, each pair of students played at most one match versus each other. In the end, it is seen that for any pair of students which have played a match versus each other, at least one of them has played at most $22$ matches. What is the maximum possible number of matches in this event?

Prove that there exists a positive constant $c$ such that the following statement is true: Consider an integer $n > 1$, and a set $\mathcal S$ of $n$ points in the plane such that the distance between any two different points in $\mathcal S$ is at least 1. It follows that there is a line $\ell$ separating $\mathcal S$ such that the distance from any point of $\mathcal S$ to $\ell$ is at least $cn^{-1/3}$. (A line $\ell$ separates a set of points S if some segment joining two points in $\mathcal S$ crosses $\ell$.) [i]Note. Weaker results with $cn^{-1/3}$ replaced by $cn^{-\alpha}$ may be awarded points depending on the value of the constant $\alpha > 1/3$.[/i] [i]Proposed by Ting-Feng Lin and Hung-Hsun Hans Yu, Taiwan[/i]

For all integers $x,y$, a non-negative integer $f(x,y)$ is written on the point $(x,y)$ on the coordinate plane. Initially, $f(0,0) = 4$ and the value written on all remaining points is $0$. For integers $n, m$ that satisfies $f(n,m) \ge 2$, define '[color=#9a00ff]Seehang[/color]' as the act of reducing $f(n,m)$ by $1$, selecting 3 of $f(n,m+1), f(n,m-1), f(n+1,m), f(n-1,m)$ and increasing them by 1. Prove that after a finite number of '[color=#0f0][color=#9a00ff]Seehang[/color][/color]'s, it cannot be $f(n,m)\le 1$ for all integers $n,m$.