For any finite sets $X$ and $Y$ of positive integers, denote by $f_X(k)$ the $k^{\text{th}}$ smallest positive integer not in $X$, and let $$X*Y=X\cup \{ f_X(y):y\in Y\}.$$Let $A$ be a set of $a>0$ positive integers and let $B$ be a set of $b>0$ positive integers. Prove that if $A*B=B*A$, then $$\underbrace{A*(A*\cdots (A*(A*A))\cdots )}_{\text{ A appears $b$ times}}=\underbrace{B*(B*\cdots (B*(B*B))\cdots )}_{\text{ B appears $a$ times}}.$$ [i]Proposed by Alex Zhai, United States[/i]

We put four numbers $1,2, 3,4$ around a circle in order. One starts at the number $1$ and every step, he moves to an adjacent number on either side. How many ways he can move such that sum of the numbers he visits in his path (including the starting number) is equal to $21$?

A cycling group that has $4n$ members will have several cycling events, such that: a) Two cycling events are done every week; once on Saturday and once on Sunday. b) Exactly $2n$ members participate in any cycling event. c) No member may participate in both cycling events of a week. d) After all cycling events are completed, the number of events where each pair of members meet is the same for all pairs of members. Prove that after all cycling events are completed, the number of events where each group of three members meet is the same value $t$ for all groups of three members, and that for $n \ge 2$, $t$ is divisible by $n-1$.

For each positive integer $k,$ let $t(k)$ be the largest odd divisor of $k.$ Determine all positive integers $a$ for which there exists a positive integer $n,$ such that all the differences \[t(n+a)-t(n); t(n+a+1)-t(n+1), \ldots, t(n+2a-1)-t(n+a-1)\] are divisible by 4. [i]Proposed by Gerhard Wöginger, Austria[/i]

Given a permutation of $1,2,3,\dots,n$, with consecutive elements $a,b,c$ (in that order), we may perform either of the [i]moves[/i]: [list] [*] If $a$ is the median of $a$, $b$, and $c$, we may replace $a,b,c$ with $b,c,a$ (in that order) [*] If $c$ is the median of $a$, $b$, and $c$, we may replace $a,b,c$ with $c,a,b$ (in that order) [/list] What is the least number of sets in a partition of all $n!$ permutations, such that any two permutations in the same set are obtainable from each other by a sequence of moves? [i]Proposed by Milan Haiman[/i]

Some numbers from $1$ to $100$ are painted red so that the following two conditions are met: $\bullet$ The number $1 $ is painted red. $\bullet$ If the numbers other than $a$ and $b$ are painted red then no number between $a$ and $b$ divides the number $ab$. What is the maximum number of numbers that can be painted red?

Let $\Bbb{N}$ be the set of positive integers. For each subset $\mathcal{X}$ of $\Bbb{N}$ we define the set $\Delta(\mathcal{X})$ as the set of all numbers $| m - n |,$ where $m$ and $n$ are elements of $\mathcal{X}$, ie: $$\Delta (\mathcal{X}) = \{ |m-n| \ | \ m, n \in \mathcal{X} \}$$ Let $\mathcal A$ and $\mathcal B$ be two infinite, disjoint sets whose union is $\Bbb{N.}$ a) Prove that the set $\Delta (\mathcal A) \cap \Delta (\mathcal B)$ has infinitely many elements. b) Prove that there exists an infinite subset $\mathcal C$ of $\Bbb{N}$ such that $\Delta (\mathcal C)$ is a subset of $\Delta (\mathcal A) \cap \Delta (\mathcal B).$

On a circle with radius $1$ the points $A_1, A_2,..., A_n$ lie such that every arc $A_iA_{i+i}$ has length $\frac{2\pi}{n}= a$. Given that there exists a set $V$ consisting of $ k$ of these points ($k < n$), which has the property that each of the arc lengths $a$, $2a$$,...$, $(n- 1)a$ can be obtained in exactly one way be taken as the length of an arc traversed in a positive sense, beginning and ending in a point of $V$. Express $n$ in terms of $k$ and give the set $V$ for the case $n = 7$.

There are lamps in every field of $n\times n$ table. At start all the lamps are off. A move consists of chosing $m$ consecutive fields in a row or a column and changing the status of that $m$ lamps. Prove that you can reach a state in which all the lamps are on only if $m$ divides $n.$

A board of size $2015 \times 2015$ is covered with sub-boards of size $2 \times 2$, each of which is painted like chessboard. Each sub-board covers exactly $4$ squares of the board and each square of the board is covered with at least one square of a sub-board (the painted of the sub-boards can be of any shape). Prove that there is a way to cover the board in such a way that there are exactly $2015$ black squares visible. What is the maximum number of visible black squares?

We are given n objects of identical appearance, but different mass, and a balance which can be used to compare any two objects (but only one object can be placed in each pan at a time). How many times must we use the balance to find the heaviest object and the lightest object?

$100$ red points divide a blue circle into $100$ arcs such that their lengths are all positive integers from $1$ to $100$ in an arbitrary order. Prove that there exist two perpendicular chords with red endpoints.

[b]p1.[/b] Sometimes one finds in an old park a tetrahedral pile of cannon balls, that is, a pile each layer of which is a tightly packed triangular layer of balls. A. How many cannon balls are in a tetrahedral pile of cannon balls of $N$ layers? B. How high is a tetrahedral pile of cannon balls of $N$ layers? (Assume each cannon ball is a sphere of radius $R$.) [b]p2.[/b] A prime is an integer greater than $1$ whose only positive integer divisors are itself and $1$. A. Find a triple of primes $(p, q, r)$ such that $p = q + 2$ and $q = r + 2$ . B. Prove that there is only one triple $(p, q, r)$ of primes such that $p = q + 2$ and $q = r + 2$ . [b]p3.[/b] The function $g$ is defined recursively on the positive integers by $g(1) =1$, and for $n>1$ , $g(n)= 1+g(n-g(n-1))$ . A. Find $g(1)$ , $g(2)$ , $g(3)$ and $g(4)$ . B. Describe the pattern formed by the entire sequence $g(1) , g(2 ), g(3), ...$ C. Prove your answer to Part B. [b]p4.[/b] Let $x$ , $y$ and $z$ be real numbers such that $x + y + z = 1$ and $xyz = 3$ . A. Prove that none of $x$ , $y$ , nor $z$ can equal $1$. B. Determine all values of $x$ that can occur in a simultaneous solution to these two equations (where $x , y , z$ are real numbers). [b]p5.[/b] A round robin tournament was played among thirteen teams. Each team played every other team exactly once. At the conclusion of the tournament, it happened that each team had won six games and lost six games. A. How many games were played in this tournament? B. Define a [i]circular triangle[/i] in a round robin tournament to be a set of three different teams in which none of the three teams beat both of the other two teams. How many circular triangles are there in this tournament? C. Prove your answer to Part B. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Given an integer $n\ge 2$, compute $\sum_{\sigma} \textrm{sgn}(\sigma) n^{\ell(\sigma)}$, where all $n$-element permutations are considered, and where $\ell(\sigma)$ is the number of disjoint cycles in the standard decomposition of $\sigma$.

Show that if at least five persons are sitting at a round table, then it is possible to rearrange them so that everyone has two new neighbors.

There are $\frac{k(k+1)}{2}+1$ points on the planes, some are connected by disjoint segments ( also point can not lies on segment, that connects two other points). It is true, that plane is divided to some parallelograms and one infinite region. What maximum number of segments can be drawn ? [i] A.Kupavski, A. Polyanski[/i]

A mother and her son are playing. At first, the son divides a ${300}\mathrm{\;g}$ wheel of cheese into 4 slices. Then the mother divides ${280}\mathrm{\;g}$ of butter between two plates. At last, the son puts the cheese slices on those plates. The son wins if on each plate the amount of cheese is not less than the amount of butter (otherwise the mother wins). Who of them can win irrespective of the opponent's actions? Alexandr Shapovalov

Find the number of non-isosceles triangles (up to congruence) with integral side lengths, in which the sum of the two shorter sides is $19$.

Among the $n$ inhabitants of an island, every two are either friends or enemies. Some day, the chief of the island orders that each inhabitant (including himself) makes and wears a necklace consisting of marbles, in such a way that the necklaces of two friends have at least one marble of the same type and that the necklaces of two enemies differ at all marbles. (A necklace may also be marbleless). Show that the chief’s order can be achieved by using $\left\lfloor\frac{n^2}4\right\rfloor$ different types of stones, but not necessarily by using fewer types.

There are only two letters in the Mumu tribe alphabet: M and $U$. The word in the Mumu language is any sequence of letters $M$ and $U$, in which next to each letter $M$ there is a letter $U$ (for example, $UUU$ and $UMMUM$ are words and $MMU$ is not). Let $f(m,u)$ denote the number of words in the Mumu language which have $m$ times the letter $M$ and $u$ times the letter $U$. Prove that $f (m, u) - f (2u - m + 1, u) = f (m, u - 1) - f (2u - m + 1, u - 1)$ for any $u \ge 2,3 \le m \le 2u$.

There are $N{}$ mess-loving clerks in the office. Each of them has some rubbish on the desk. The mess-loving clerks leave the office for lunch one at a time (after return of the preceding one). At that moment all those remaining put half of rubbish from their desks on the desk of the one who left. Can it so happen that after all of them have had lunch the amount of rubbish at the desk of each one will be the same as before lunch if a) $N = 2{}$ and b) $N = 10$? [i]Alexey Zaslavsky[/i]

We call the sum of any $k$ of $n$ given numbers (with distinct indices) a $k$-sum. Given $n$, find all $k$ such that, whenever more than half of $k$-sums of numbers $a_{1},a_{2},...,a_{n}$ are positive, the sum $a_{1}+a_{2}+...+a_{n}$ is positive as well.

Given is a convex polygon $ P$ with $ n$ vertices. Triangle whose vertices lie on vertices of $ P$ is called [i]good [/i] if all its sides are unit length. Prove that there are at most $ \frac {2n}{3}$ [i]good[/i] triangles. [i]Author: Vyacheslav Yasinskiy, Ukraine[/i]

[b] Problem 6.[/b] Let $m\geq 5$ and $n$ are given natural numbers, and $M$ is regular $2n+1$-gon. Find the number of the convex $m$-gons with vertices among the vertices of $M$, who have at least one acute angle. [i]Alexandar Ivanov[/i]

Alice and Bob play a game together as a team on a $100 \times 100$ board with all unit squares initially white. Alice sets up the game by coloring exactly $k$ of the unit squares red at the beginning. After that, a legal move for Bob is to choose a row or column with at least $10$ red squares and color all of the remaining squares in it red. What is the smallest $k$ such that Alice can set up a game in such a way that Bob can color the entire board red after finitely many moves? Proposed by [i]Nikola Velov, Macedonia[/i]