Let $n \geq 3$ be an integer, and consider a set of $n$ points in three-dimensional space such that: [list=i] [*] every two distinct points are connected by a string which is either red, green, blue, or yellow; [*] for every three distinct points, if the three strings between them are not all of the same colour, then they are of three different colours; [*] not all the strings have the same colour. [/list] Find the maximum possible value of $n$.

Justin throws a standard six-sided die three times in a row and notes the number of dots on the top face after each roll. How many different sequences of outcomes could he get?

[b]p1.[/b] In the picture below, the two parallel cuts divide the square into three pieces of equal area. The distance between the two parallel cuts is $d$. The square has length $s$. Find and prove a formula that expresses $s$ as a function of $d$. [img]https://cdn.artofproblemsolving.com/attachments/c/b/666074d28de50cdbf338a2c667f88feba6b20c.png[/img] [b]p2.[/b] Let $S$ be a subset of $\{1, 2, 3, . . . 10, 11\}$. We say that $S$ is lucky if no two elements of $S$ differ by $4$ or $7$. (a) Give an example of a lucky set with five elements. (b) Is it possible to find a lucky set with six elements? Explain why or why not.[/quote] [b]p3.[/b] Find polynomials $p(x)$ and $q(x)$ with real coefficients such that (a) $p(x) - q(x) = x^3 + x^2 - x - 1$ for all real $x$, (b) $p(x) > 0$ for all real $x$, (c) $q(x) > 0$ for all real $x$. [b]p4.[/b] A permutation on $\{1, 2, 3, …, n\}$ is a rearrangement of the symbols. For example $32154$ is a permutation on $\{1, 2, 3, 4, 5\}$. Given a permutation $a_1a_2a_3…a_n$, an inversion is a pair of $a_i$ and $a_j$ such that $a_i > a_j$ but $i < j$. For example, $32154$ has $4$ inversions. Suppose you are only allowed to exchange adjacent symbols. For any permutation, show that the minimum number of exchanges required to put all the symbols in their natural positions (that is, $123 …n$) is the number of inversions. [b]p5.[/b] We say a number $N$ is a nontrivial sum of consecutive positive integers if it can be written as the sum of $2$ or more consecutive positive integers. What is the set of numbers from $1000$ to $2000$ that are NOT nontrivial sums of consecutive positive integers? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

A polyhedron has $7n$ faces. Show that there exist $n + 1$ of the polyhedron's faces that all have the same number of edges.

Mario has a deck of seven pairs of matching number cards and two pairs of matching Jokers, for a total of $18$ cards. He shuffles the deck, then draws the cards from the top one by one until he holds a pair of matching Jokers. The expected number of complete pairs that Mario holds at the end (including the Jokers) is $\frac{m}{n}$ , where $m$, $n$ are positive integers and gcd $(m,n) = 1$. Find $100m + n$.

Chip and Dale play the following game. Chip starts by splitting $222$ nuts between two piles, so Dale can see it. In response, Dale chooses some number $N$ from $1$ to $222$. Then Chip moves nuts from the piles he prepared to a new (third) pile until there will be exactly $N$ nuts in any one or two piles. When Chip accomplishes his task, Dale gets an exact amount of nuts that Chip moved. What is the maximal number of nuts that Dale can get for sure, no matter how Chip acts? (Naturally, Dale wants to get as many nuts as possible, while Chip wants to lose as little as possible).

A natural number is written on the blackboard. Whenever number $ x$ is written, one can write any of the numbers $ 2x \plus{} 1$ and $ \frac {x}{x \plus{} 2}$. At some moment the number $ 2008$ appears on the blackboard. Show that it was there from the very beginning.

Each vertex $v$ and each edge $e$ of a graph $G$ are assigned numbers $f(v)\in\{1,2\}$ and $f(e)\in\{1,2,3\}$, respectively. Let $S(v)$ be the sum of numbers assigned to the edges incident to $v$ plus the number $f(v)$. We say that an assignment $f$ is [i]cool [/i]if $S(u) \ne S(v)$ for every pair $(u,v)$ of adjacent (i.e. connected by an edge) vertices in $G$. Prove that for every graph there exists a cool assignment.

Let $k$ be a positive integer and $S$ be a set of sets which have $k$ elements. For every $A,B \in S$ and $A\neq B$ we have $A \Delta B \in S$. Find all values of $k$ when $|S|=1023$ and $|S|=2023$. Note:$A \Delta B = (A \setminus B) \cup (B \setminus A)$

In a room, there are $2005$ boxes, each of them containing one or several sorts of fruits, and of course an integer amount of each fruit. [b]a)[/b] Show that we can find $669$ boxes, which altogether contain at least a third of all apples and at least a third of all bananas. [b]b)[/b] Can we always find $669$ boxes, which altogether contain at least a third of all apples, at least a third of all bananas and at least a third of all pears?

Let \( X \) be a finite set of real numbers, \( d \) be a real number, and \(\lambda_1, \lambda_2, \cdots, \lambda_{2025}\) be 2025 non-zero real numbers. Define \[A = \left\{ (x_1, x_2, \cdots, x_{2025}) : x_1, x_2, \cdots, x_{2025} \in X \text{ and } \sum_{i=1}^{2025} \lambda_i x_i = d \right\},\] \[B = \left\{ (x_1, x_2, \cdots, x_{2024}) : x_1, x_2, \cdots, x_{2024} \in X \text{ and } \sum_{i=1}^{2024} (-1)^i x_i = 0 \right\},\] \[C = \left\{ (x_1, x_2, \cdots, x_{2026}) : x_1, x_2, \cdots, x_{2026} \in X \text{ and } \sum_{i=1}^{2026} (-1)^i x_i = 0 \right\}.\] Show that \( |A|^2 \leq |B| \cdot |C| \).

A game in a casino uses the diagram shown. At the start a ball appears at $S$. Each time the player presses a button, the ball moves to one of the adjacent letters with equal probability. The game ends when one of the following two things happens: (i) The ball returns to $S$, the player loses. (ii) The ball reaches $G$, the player wins. Find the probability that the player wins and the expected duration of a game.

Given a positive integer $n$, does there exist a planar polygon and a point in its plane such that every line through that point meets the boundary of the polygon at exactly $2n$ points?

Leo and Paul are at the Berkeley BART station and are racing to San Francisco. Leo is planning to take the line that takes him directly to SF, and because he has terrible BART luck, his train will arrive in some integer number of minutes, with probability $\frac i{210}$ for $1\le i\le20$ at any given minute. Paul will take a second line, whose trains always arrive before Leo’s train, with uniform probability. However, Paul must also make a transfer to a 3rd line, whose trains arrive with uniform probability between $0$ and $10$ minutes after Paul reaches the transfer station. What is the probability that Leo gets to SF before Paul does?

Bessie the cow is hungry and wants to eat some oranges, which she has an infinite supply of. Bessie starts with a fullness level of $0$, and each orange that she eats increases her fullness level by $85$. She can also eat lemons, and each time she eats a lemon, her fullness level is halved, rounding down. What is the smallest number of lemons that Bessie should have in order to be able to attain every possible nonnegative integer fullness level?

[b]p1.[/b] We say that six positive integers form a magic triangle if they are arranged in a triangular array as in the figure below in such a way that each number in the top two rows is equal to the sum of its two neighbors in the row directly below it. The triangle shown is magic because $4 = 1 + 3$, $5 = 3 + 2$, and $9 = 4 + 5$. $$9$$ $$4\,\,\,\,5$$ $$1\,\,\,\,3\,\,\,\,2$$ (a) Find a magic triangle such that the numbers at the three corners are $10$, $20$, and $2010$, with $2010$ at the top. (b) Find a magic triangle such that the numbers at the three corners are $20$, $201$, and $2010$, with $2010$ at the top, or prove that no such triangle exists. [b]p2.[/b] (a) The equalities $\frac12+\frac13+\frac16= 1$ and $\frac12+\frac13+\frac17+\frac{1}{42}= 1$ express $1$ as a sum of the reciprocals of three (respectively four) distinct positive integers. Find five positive integers $a < b < c <d < e$ such that $$\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{d}+\frac{1}{e}= 1.$$ (b) Prove that for any integer $m \ge 3$, there exist $m$ positive integers $d_1 < d_2 <... < d_m$ such that $$\frac{1}{d_1}+\frac{1}{d_2}+ ... +\frac{1}{d_m}= 1.$$ [b]p3.[/b] Suppose that $P(x) = a_nx^n +... + a_1x + a_0$ is a polynomial of degree n with real coefficients. Say that the real number $b$ is a balance point of $P$ if for every pair of real numbers $a$ and $c$ such that $b$ is the average of $a$ and $c$, we have that $P(b)$ is the average of $P(a)$ and $P(c)$. Assume that $P$ has two distinct balance points. Prove that $n$ is at most $1$, i.e., that $P$ is a linear function. [b]p4.[/b] A roller coaster at an amusement park has a train consisting of $30$ cars, each seating two people next to each other. $60$ math students want to take as many rides as they can, but are told that there are two rules that cannot be broken. First, all $60$ students must ride each time, and second, no two students are ever allowed to sit next to each other more than once. What is the maximal number of roller coaster rides that these students can take? Justify your answer. [b]p5.[/b] Let $ABCD$ be a convex quadrilateral such that the lengths of all four sides and the two diagonals of $ABCD$ are rational numbers. If the two diagonals $AC$ and $BD$ intersect at a point $M$, prove that the length of $AM$ is also a rational number. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $S$ be a $k$-element set. [i](a)[/i] Find the number of mappings $f : S \to S$ such that \[\text{(i) } f(x) \neq x \text{ for } x \in S, \quad \text{(ii) } f(f(x)) = x \text{ for }x \in S.\] [i](b)[/i] The same with the condition $\text{(i)}$ left out.

Let $f_0$ be the function from $\mathbb{Z}^2$ to $\{0,1\}$ such that $f_0(0,0)=1$ and $f_0(x,y)=0$ otherwise. For each positive integer $m$, let $f_m(x,y)$ be the remainder when \[ f_{m-1}(x,y) + \sum_{j=-1}^{1} \sum_{k=-1}^{1} f_{m-1}(x+j,y+k) \] is divided by $2$. Finally, for each nonnegative integer $n$, let $a_n$ denote the number of pairs $(x,y)$ such that $f_n(x,y) = 1$. Find a closed form for $a_n$. [i]Proposed by Bobby Shen[/i]

There are $N$ piles each consisting of a single nut. Two players in turns play the following game. At each move, a player combines two piles that contain coprime numbers of nuts into a new pile. A player who can not make a move, loses. For every $N > 2$ determine which of the players, the first or the second, has a winning strategy.

The 40 unit squares of the 9 9-table (see below) are labeled. The horizontal or vertical row of 9 unit squares is good if it has more labeled unit squares than unlabeled ones. How many good (horizontal and vertical) rows totally could have the table?

Two players $A$ and $B$ have one stone each on a $100 \times 100$ chessboard. They move their stones one after the other, and a move means moving one's stone to a neighbouring field (horizontally or vertically, not diagonally). At the beginning of the game, the stone of $A$ lies in the lower left corner, and the one of $B$ in the lower right corner. Player $A$ starts. Prove: Player $A$ is, independently from that what $B$ does, able to reach, after finitely many steps, the field $B$'s stone is lying on at that moment.

Hesam chose $10$ distinct positive integers and he gave all pairwise $\gcd$'s and pairwise ${\text lcm}$'s (a total of $90$ numbers) to Masoud. Can Masoud always find the first $10$ numbers, just by knowing these $90$ numbers? [i]Proposed by Morteza Saghafian [/i]

Let $n \geq 2$ be a positive integer. There are $2n$ cities $M_1, M_2, \ldots, M_{2n}$ in the country of Mathlandia. Currently there roads only between $M_1$ and $M_2, M_3, \ldots, M_n$ and the king wants to build more roads so that it is possible to reach any city from every other city. The cost to build a road between $M_i$ and $M_j$ is $k_{i, j}>0$. Let $$K=\sum_{j=n+1}^{2n} k_{1,j}+\sum_{2 \leq i<j \leq 2n} k_{i, j}.$$Prove that the king can fulfill his plan at cost no more than $\frac{2K}{3n-1}$.

Lucy starts by writing $s$ integer-valued $2022$-tuples on a blackboard. After doing that, she can take any two (not necessarily distinct) tuples $\mathbf{v}=(v_1,\ldots,v_{2022})$ and $\mathbf{w}=(w_1,\ldots,w_{2022})$ that she has already written, and apply one of the following operations to obtain a new tuple: \begin{align*} \mathbf{v}+\mathbf{w}&=(v_1+w_1,\ldots,v_{2022}+w_{2022}) \\ \mathbf{v} \lor \mathbf{w}&=(\max(v_1,w_1),\ldots,\max(v_{2022},w_{2022})) \end{align*} and then write this tuple on the blackboard. It turns out that, in this way, Lucy can write any integer-valued $2022$-tuple on the blackboard after finitely many steps. What is the smallest possible number $s$ of tuples that she initially wrote?

The number 7 is written on a board. Alice and Bob in turn (Alice begins) write an additional digit in the number on the board: it is allowed to write the digit at the beginning (provided the digit is nonzero), between any two digits or at the end. If after someone’s turn the number on the board is a perfect square then this person wins. Is it possible for a player to guarantee the win? [i]Alexandr Gribalko[/i]