A crazy physicist discovered a new kind of particle wich he called an imon, after some of them mysteriously appeared in his lab. Some pairs of imons in the lab can be entangled, and each imon can participate in many entanglement relations. The physicist has found a way to perform the following two kinds of operations with these particles, one operation at a time. (i) If some imon is entangled with an odd number of other imons in the lab, then the physicist can destroy it. (ii) At any moment, he may double the whole family of imons in the lab by creating a copy $I'$ of each imon $I$. During this procedure, the two copies $I'$ and $J'$ become entangled if and only if the original imons $I$ and $J$ are entangled, and each copy $I'$ becomes entangled with its original imon $I$; no other entanglements occur or disappear at this moment. Prove that the physicist may apply a sequence of such operations resulting in a family of imons, no two of which are entangled.

A tourist going to visit the [i]Complant[/i], found that: a) in this country $1024$ cities, numbered by integers from $0$ to $1023$ , b) two cities with numbers $m$ and $n$ are connected by a straight line if and only if the binary entries of numbers $m$ and $n$ they differ exactly in one digit, c) during the stay of a tourist in that country $8$ roads will be closed for scheduled repairs. Prove that a tourist can make a closed route along the existing roads of [i]Complant[/i], passing through each of its cities exactly once.

Given a sequence of integers $A_1,A_2,\cdots A_{99}$ such that for every sub-sequence that contains $m$ consecutive elements, there exist not more than $max\{ \frac{m}{3} ,1\}$ odd integers. Let $S=\{ (i,j) \ | i<j \}$ such that $A_i$ is even and $A_j$ is odd. Find $max\{ |S|\}$.

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.

Given a poistive integer $ m, $ determine the smallest integer $ n\ge 2 $ such that for any coloring of the $ n^2 $ unit squares of a $ n\times n $ square with $ m $ colors, there are, at least, two unit squares $ (i,j),(k,l) $ that share the same color, where $ 1\le i,j,k,l\le n,i\neq j,k\neq l. $ [i]American Mathematical Monthly[/i]

A quota of diplomas at the All-Russian Olympiad should be strictly less than $45\%$. More than $20$ students took part in the olympiad. After the olympiad the Authorities declared the results low because the quota of diplomas was significantly less than $45\%$. The Jury responded that the quota was already maximum possible on this olympiad or any other olympiad with smaller number of participants. Then the Authorities ordered to increase the number of participants for the next olympiad so that the quota of diplomas became at least two times closer to $45\%$. Prove that the number of participants should be at least doubled.

A $\pm 1$-[i]sequence[/i] is a sequence of $2022$ numbers $a_1, \ldots, a_{2022},$ each equal to either $+1$ or $-1$. Determine the largest $C$ so that, for any $\pm 1$-sequence, there exists an integer $k$ and indices $1 \le t_1 < \ldots < t_k \le 2022$ so that $t_{i+1} - t_i \le 2$ for all $i$, and $$\left| \sum_{i = 1}^{k} a_{t_i} \right| \ge C.$$

A total of $n$ people compete in a mathematical match which contains $15$ problems where $n>12$. For each problem, $1$ point is given for a right answer and $0$ is given for a wrong answer. Analysing each possible situation, we find that if the sum of points every group of $12$ people get is no less than $36$, then there are at least $3$ people that got the right answer of a certain problem, among the $n$ people. Find the least possible $n$.

Consider a natural number $ n $ for which it exist a natural number $ k $ and $ k $ distinct primes so that $ n=p_1\cdot p_2\cdots p_k. $ [b]a)[/b] Find the number of functions $ f:\{ 1, 2,\ldots , n\}\longrightarrow\{ 1,2,\ldots ,n\} $ that have the property that $ f(1)\cdot f(2)\cdots f\left( n \right) $ divides $ n. $ [b]b)[/b] If $ n=6, $ find the number of functions $ f:\{ 1, 2,3,4,5,6\}\longrightarrow\{ 1,2,3,4,5,6\} $ that have the property that $ f(1)\cdot f(2)\cdot f(3)\cdot f(4)\cdot f(5)\cdot f(6) $ divides $ 36. $

[b]p1.[/b] Let $p > 5$ be a prime. It is known that the average of all of the prime numbers that are at least $5$ and at most $p$ is $12$. Find $p$. [b]p2.[/b] The numbers $1, 2,..., n$ are written down in random order. What is the probability that $n-1$ and $n$ are written next to each other? (Give your answer in term of $n$.) [b]p3.[/b] The Duke Blue Devils are playing a basketball game at home against the UNC Tar Heels. The Tar Heels score $N$ points and the Blue Devils score $M$ points, where $1 < M,N < 100$. The first digit of $N$ is $a$ and the second digit of $N$ is $b$. It is known that $N = a+b^2$. The first digit of $M$ is $b$ and the second digit of $M$ is $a$. By how many points do the Blue Devils win? [b]p4.[/b] Let $P(x)$ be a polynomial with integer coefficients. It is known that $P(x)$ gives a remainder of $1$ upon polynomial division by $x + 1$ and a remainder of $2$ upon polynomial division by $x + 2$. Find the remainder when $P(x)$ is divided by $(x + 1)(x + 2)$. [b]p5.[/b] Dracula starts at the point $(0,9)$ in the plane. Dracula has to pick up buckets of blood from three rivers, in the following order: the Red River, which is the line $y = 10$; the Maroon River, which is the line $y = 0$; and the Slightly Crimson River, which is the line $x = 10$. After visiting all three rivers, Dracula must then bring the buckets of blood to a castle located at $(8,5)$. What is the shortest distance that Dracula can walk to accomplish this goal? [b]p6.[/b] Thirteen hungry zombies are sitting at a circular table at a restaurant. They have five identical plates of zombie food. Each plate is either in front of a zombie or between two zombies. If a plate is in front of a zombie, that zombie and both of its neighbors can reach the plate. If a plate is between two zombies, only those two zombies may reach it. In how many ways can we arrange the plates of food around the circle so that each zombie can reach exactly one plate of food? (All zombies are distinct.) [b]p7.[/b] Let $R_I$ , $R_{II}$ ,$R_{III}$ ,$R_{IV}$ be areas of the elliptical region $$\frac{(x - 10)^2}{10}+ \frac{(y-31)^2}{31} \le 2009$$ that lie in the first, second, third, and fourth quadrants, respectively. Find $R_I -R_{II} +R_{III} -R_{IV}$ . [b]p8.[/b] Let $r_1, r_2, r_3$ be the three (not necessarily distinct) solutions to the equation $x^3+4x^2-ax+1 = 0$. If $a$ can be any real number, find the minimum possible value of $$\left(r_1 +\frac{1}{r_1} \right)^2+ \left(r_2 +\frac{1}{r_2} \right)^2+ \left(r_3 +\frac{1}{r_3} \right)^2$$ [b]p9.[/b] Let $n$ be a positive integer. There exist positive integers $1 = a_1 < a_2 <... < a_n = 2009$ such that the average of any $n - 1$ of elements of $\{a_1, a_2,..., a_n\}$ is a positive integer. Find the maximum possible value of $n$. [b]p10.[/b] Let $A(0) = (2, 7, 8)$ be an ordered triple. For each $n$, construct $A(n)$ from $A(n - 1)$ by replacing the $k$th position in $A(n - 1)$ by the average (arithmetic mean) of all entries in $A(n - 1)$, where $k \equiv n$ (mod $3$) and $1 \le k \le 3$. For example, $A(1) = \left( \frac{17}{3} , 7, 8 \right)$ and $A(2) = \left( \frac{17}{3} , \frac{62}{9}, 8\right)$. It is known that all entries converge to the same number $N$. Find the value of $N$. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

A family of finite sets $\left\{ A_{1},A_{2},.......,A_{m}\right\} $is called [i]equipartitionable [/i] if there is a function $\varphi:\cup_{i=1}^{m}$$\rightarrow\left\{ -1,1\right\} $ such that $\sum_{x\in A_{i}}\varphi\left(x\right)=0$ for every $i=1,.....,m.$ Let $f\left(n\right)$ denote the smallest possible number of $n$-element sets which form a non-equipartitionable family. Prove that a) $f(4k +2) = 3$ for each nonnegative integer $k$, b) $f\left(2n\right)\leq1+m d\left(n\right)$, where $m d\left(n\right)$ denotes the least positive non-divisor of $n.$

Let $n$ and $k$ be arbitrary non-negative integers. Suppose we have drawn $2kn+1$ (different) diagonals of a convex $n$-gon. Show that there exists a broken line formed by $2k+1$ of these diagonals that passes through no point more than once. Prove also that this is not necessarily true when we draw only $kn$ diagonals.

In the classroom of at least four students the following holds: no matter which four of them take seats around a round table, there is always someone who either knows both of his neighbours, or does not know either of his neighbours. Prove that it is possible to divide the students into two groups such that in one of them, all students know one another, and in the other, none of the students know each other. (Note: if student A knows student B, then student B knows student A as well.)

There is a chessboard with $m$ columns and $n$ rows. In each blanks, an integer is given. If a rectangle $R$ (in this chessboard) has an integer $h$ satisfying the following two conditions, we call $R$ as a 'shelf'. (i) All integers contained in $R$ are bigger than $h$. (ii) All integers in blanks, which are not contained in $R$ but meet with $R$ at a vertex or a side, are not bigger than $h$. Assume that all integers are given to make shelves as much as possible. Find the number of shelves.

Let $n$ be a positive integer. How many integer solutions $(i, j, k, l) , \ 1 \leq i, j, k, l \leq n$, does the following system of inequalities have: \[1 \leq -j + k + l \leq n\]\[1 \leq i - k + l \leq n\]\[1 \leq i - j + l \leq n\]\[1 \leq i + j - k \leq n \ ?\]

The cells of an $8\times 8$ chessboard are all coloured in white. A move consists in inverting the colours of a rectangle $1 \times 3$ horizontal or vertical (the white cells become black and conversely). Is it possible to colour all the cells of the chessboard in black in a finite number of moves ?

[u]Round 5[/u] [b]p13.[/b] Points $A, B, C$, and $D$ lie in a plane with $AB = 6$, $BC = 5$, and $CD = 5$, and $AB$ is perpendicular to $BC$. Point E lies on line $AD$ such that $D \ne E$, $AE = 3$ and $CE = 5$. Find $DE$. [b]p14.[/b] How many ordered pairs of integers $(x,y)$ are solutions to $x^2y = 36 + y$? [b]p15.[/b] Chicken nuggets come in boxes of two sizes, $a$ nuggets per box and $b$ nuggets per box. We know that $899$ nuggets is the largest number of nuggets we cannot obtain with some combination of $a$-sized boxes and $b$-sized boxes. How many different pairs $(a, b)$ are there with $a < b$? [u]Round 6[/u] [b]p16.[/b] You are playing a game with coins with your friends Alice and Bob. When all three of you flip your respective coins, the majority side wins. For example, if Alice, Bob, and you flip Heads, Tails, Heads in that order, then you win. If Alice, Bob, and you flip Heads, Heads, Tails in that order, then you lose. Notice that more than one person will “win.” Alice and Bob design their coins as follows: a value $p$ is chosen randomly and uniformly between $0$ and $1$. Alice then makes a biased coin that lands on heads with probability $p$, and Bob makes a biased coin that lands on heads with probability $1 -p$. You design your own biased coin to maximize your chance of winning without knowing $p$. What is the probability that you win? [b]p17.[/b] There are $N$ distinct students, numbered from $1$ to $N$. Each student has exactly one hat: $y$ students have yellow hats, $b$ have blue hats, and $r$ have red hats, where $y + b + r = N$ and $y, b, r > 0$. The students stand in a line such that all the $r$ people with red hats stand in front of all the $b$ people with blue hats. Anyone wearing red is standing in front of everyone wearing blue. The $y$ people with yellow hats can stand anywhere in the line. The number of ways for the students to stand in a line is $2016$. What is $100y + 10b + r$? [b]p18.[/b] Let P be a point in rectangle $ABCD$ such that $\angle APC = 135^o$ and $\angle BPD = 150^o$. Suppose furthermore that the distance from P to $AC$ is $18$. Find the distance from $P$ to $BD$. [u]Round 7 [/u] [b]p19.[/b] Let triangle $ABC$ be an isosceles triangle with $|AB| = |AC|$. Let $D$ and $E$ lie on $AB$ and $AC$, respectively. Suppose $|AD| = |BC| = |EC|$ and triangle $ADE$ is isosceles. Find the sum of all possible values of $\angle BAC$ in radians. Write your answer in the form $2 arcsin \left( \frac{a}{b}\right) + \frac{c}{d} \pi$, where $\frac{a}{b}$ and $\frac{c}{d}$ are in lowest terms, $-1 \le \frac{a}{b} \le 1$, and $-1 \le \frac{c}{d} \le 1$. [b]p20.[/b] Kevin is playing a game in which he aims to maximize his score. In the $n^{th}$ round, for $n \ge 1$, a real number between $0$ and $\frac{1}{3^n}$ is randomly generated. At each round, Kevin can either choose to have the randomly generated number from that round as his score and end the game, or he can choose to pass on the number and continue to the next round. Once Kevin passes on a number, he CANNOT claim that number as his score. Kevin may continue playing for as many rounds as he wishes. If Kevin plays optimally, the expected value of his score is $a + b\sqrt{c}$ where $a, b$, and $c$ are integers and $c$ is positive and not divisible by any positive perfect square other than $1$. What is $100a + 10b + c$? [b]p21.[/b] Lisa the ladybug (a dimensionless ladybug) lives on the coordinate plane. She begins at the origin and walks along the grid, at each step moving either right or up one unit. The path she takes ends up at $(2016, 2017)$. Define the “area” of a path as the area below the path and above the $x$-axis. The sum of areas over all paths that Lisa can take can be represented as as $a \cdot {{4033} \choose {2016}}$ . What is the remainder when $a$ is divided by $1000$? PS. You should use hide for answers. Rounds 1-4 have been posted [url=https://artofproblemsolving.com/community/c4h2782871p24446475]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Natural numbers $1,2,...,500$ are written on a blackboard. Two players $A$ and $B$ consecutively make moves, $A$ starts. Each move a player chooses two numbers $n$ and $2n$ and erases them from the blackboard. If a player cannot perform a valid move, he loses. Which player can guarantee a win?

Given a positive integer $n \geq 3$, let $C_{n}$ be the collection of all $n$-tuples $a=(a_{1},a_{2},...,a_{n})$ of nonnegative reals $a_i$, $i=1,...,n$, such that $a_{1}+a_{2}+...+a_{n}=1$. For $k \in \left \{ 1,...,n-1 \right \}$ and $a \in C_{n}$, consider the sum set $\sigma_{k}(a) = \left \{a_{1}+...+a_{k},a_{2}+...+a_{k+1},...,a_{n-k+1}+...+a_{n} \right \}$. Show the following. (a) There exist $m_k=\max\{\min\sigma_k(a):a\in\mathcal{C}_n\}$ and $M_k=\min\{\max\sigma_k(a):a\in\mathcal{C}_n\}$. (b) It holds that $\displaystyle{1\leq\sum_{k=1}^{n-1}(\frac{1}{M_k}-\frac{1}{m_k})\leq n-2}$. Moreover, on the left side, equality is attained only for finitely many values of $n$, whereas on the right side, equality holds for infinitely values of $n$.

Prove: there exist integer $x_1,x_2,\cdots x_{10},y_1,y_2,\cdots y_{10}$ satisfying the following conditions: $(1)$ $|x_i|,|y_i|\le 10^{10} $ for all $1\le i \le 10$ $(2)$ Define the set \[S = \left\{ \left( \sum_{i=1}^{10} a_i x_i, \sum_{i=1}^{10} a_i y_i \right) : a_1, a_2, \cdots, a_{10} \in \{0, 1\} \right\},\] then \(|S| = 1024\)，and any rectangular strip of width 1 covers at most two points of S.

In a square of side length $60$, $121$ distinct points are given. Show that among them there exists three points which are vertices of a triangle with an area not exceeding $30$.

There is a bottle with a flat and circular bottom, closed and partially filled of wine, so that its level does not exceed the cylindrical part. Discuss in which cases the capacity of the bottle can be calculated without opening it, having only one double graduated decimeter; and if possible, describe how it would be calculated. (Problem of the Italian [i]Gara Mathematica[/i]).

A total of $a \cdot b \cdot c$ cubical boxes are joined together in a $a \times b \times c$ rectangular stack, where $a, b, c \ge 2$. A bee is found inside one of the boxes. It can fly from one box to another through a hole in the wall, but not through edges or corners. Also, it cannot fly outside the stack. For which triples $(a, b, c)$ is it possible for the bee to fly through all of the boxes exactly once, and end up in the same box where it started?

Two runners start running along a circular track of unit length from the same starting point and int he same sense, with constant speeds $v_1$ and $v_2$ respectively, where $v_1$ and $v_2$ are two distinct relatively prime natural numbers. They continue running till they simultneously reach the starting point. Prove that (a) at any given time $t$, at least one of the runners is at a distance not more than $\frac{[\frac{v_1 + v_2}{2}]}{v_1 + v_2}$ units from the starting point. (b) there is a time $t$ such that both the runners are at least $\frac{[\frac{v_1 + v_2}{2}]}{v_1 + v_2}$ units away from the starting point. (All disstances are measured along the track). $[x]$ is the greatest integer function.

Let $\mathcal{P}$ be a partition of $\{1,2,\dots ,2024\}$ into sets of two elements, such that for any $\{a,b\}\in\mathcal{P}$, either $|a-b|=1$ or $|a-b|=506$. Suppose that $\{1518,1519\}\in\mathcal{P}$. Determine the pair of $505$ in the partition.