Two boys are given a bag of potatoes, each bag containing $150$ tubers. They take turns transferring the potatoes, where in each turn they transfer a non-zero tubers from their bag to the other boy's bag. Their moves must satisfy the following condition: In each move, a boy must move more tubers than he had in his bag before any of his previous moves (if there were such moves). So, with his first move, a boy can move any non-zero quantity, and with his fifth move, a boy can move $200$ tubers, if before his first, second, third and fourth move, the numbers of tubers in his bag was less than $200$. What is the maximal total number of moves the two boys can do? [i]Proposed by E. Molchanov[/i]

There are $n \ge 2$ houses on the northern side of a street. Going from the west to the east, the houses are numbered from 1 to $n$. The number of each house is shown on a plate. One day the inhabitants of the street make fun of the postman by shuffling their number plates in the following way: for each pair of neighbouring houses, the currnet number plates are swapped exactly once during the day. How many different sequences of number plates are possible at the end of the day?

In the spherical shaped planet $X$ there are $2n$ gas stations. Every station is paired with one other station , and every two paired stations are diametrically opposite points on the planet. Each station has a given amount of gas. It is known that : if a car with empty (large enough) tank starting from any station it is always to reach the paired station with the initial station (it can get extra gas during the journey). Find all naturals $n$ such that for any placement of $2n$ stations for wich holds the above condotions, holds: there always a gas station wich the car can start with empty tank and go to all other stations on the planet.(Consider that the car consumes a constant amount of gas per unit length.)

We have $10$ boxes of different sizes, each one big enough to contain all the smaller boxes when put side by side. We may nest the boxes however we want (and how deeply we want), as long as we put smaller boxes in larger ones. At the end, all boxes should be directly or indirectly nested in the largest box. How many ways can we nest the boxes?

[u]Round 1[/u] [b]p1.[/b] Is it possible to draw some number of diagonals in a convex hexagon so that every diagonal crosses EXACTLY three others in the interior of the hexagon? (Diagonals that touch at one of the corners of the hexagon DO NOT count as crossing.) [b]p2.[/b] A $ 3\times 3$ square grid is filled with positive numbers so that (a) the product of the numbers in every row is $1$, (b) the product of the numbers in every column is $1$, (c) the product of the numbers in any of the four $2\times 2$ squares is $2$. What is the middle number in the grid? Find all possible answers and show that there are no others. [b]p3.[/b] Each letter in $HAGRID$'s name represents a distinct digit between $0$ and $9$. Show that $$HAGRID \times H \times A\times G\times R\times I\times D$$ is divisible by $3$. (For example, if $H=1$, $A=2$, $G=3$, $R = 4$, $I = 5$, $D = 64$, then $HAGRID \times H \times A\times G\times R\times I\times D= 123456\times 1\times2\times3\times4\times5\times 6$). [b]p4.[/b] You walk into a room and find five boxes sitting on a table. Each box contains some number of coins, and you can see how many coins are in each box. In the corner of the room, there is a large pile of coins. You can take two coins at a time from the pile and place them in different boxes. If you can add coins to boxes in this way as many times as you like, can you guarantee that each box on the table will eventually contain the same number of coins? [b]p5.[/b] Alex, Bob and Chad are playing a table tennis tournament. During each game, two boys are playing each other and one is resting. In the next game the boy who lost a game goes to rest, and the boy who was resting plays the winner. By the end of tournament, Alex played a total of $10$ games, Bob played $15$ games, and Chad played $17$ games. Who lost the second game? [u]Round 2[/u] [b]p6.[/b] After going for a swim in his vault of gold coins, Scrooge McDuck decides he wants to try to arrange some of his gold coins on a table so that every coin he places on the table touches exactly three others. Can he possibly do this? You need to justify your answer. (Assume the gold coins are circular, and that they all have the same size. Coins must be laid at on the table, and no two of them can overlap.) [b]p7.[/b] You have a deck of $50$ cards, each of which is labeled with a number between $1$ and $25$. In the deck, there are exactly two cards with each label. The cards are shuffled and dealt to $25$ students who are sitting at a round table, and each student receives two cards. The students will now play a game. On every move of the game, each student takes the card with the smaller number out of his or her hand and passes it to the person on his/her right. Each student makes this move at the same time so that everyone always has exactly two cards. The game continues until some student has a pair of cards with the same number. Show that this game will eventually end. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

For real numbers $a_1, a_2, \ldots, a_{200}$, we consider the value $S = a_1a_2 + a_2a_3 + \ldots + a_{199}a_{200} + a_{200}a_1$. In one operation, you can change the sign of any number (that is, change $a_i$ to $-a_i$), and then calculate the value of $S$ for the new numbers again. What is the smallest number of operations needed to always be able to make $S$ nonnegative? [i]Proposed by Oleksii Masalitin[/i]

Each integer should be colored by one of three colors, including red. Each number which can be represent as a sum of two numbers of different colors should be red. Each color should be used. Is this coloring possible?

Suppose two polygons may be glued together at an edge if and only if corresponding edges of the same length are made to coincide. A $3\times 4$ rectangle is cut into $n$ pieces by making straight line cuts. What is the minimum value of $n$ so that it’s possible to cut the pieces in such a way that they may be glued together two at a time into a polygon with perimeter at least $2021$?

Let $n$ be a positive integer. Denote by $a(n)$ the ways of expression $n=x_1+x_2+\dots$ where $x_1\leqslant x_2 \leqslant\dots$ are positive integers and $x_i+1$ is a power of $2$ for each $i$. Denote by $b(n)$ the ways of expression $n=y_1+y_2+\dots$ where $y_i$ is a positive integer and $2y_i\leqslant y_{i+1}$ for each $i$. Prove that $a(n)=b(n)$.

On a table, there are $2025$ empty boxes numbered $1,2,\dots ,2025$, and $2025$ balls with weights $1,2,\dots ,2025$. Starting with Vadim, Vadim and Marian take turns selecting a ball from the table and placing it into an empty box. After all $2025$ turns, there is exactly one ball in each box. Denote the weight of the ball in box $i$ by $w_i$. Marian wins if $$\sum_{i=1}^{2025}i\cdot w_i\equiv 0 \pmod{23}.$$ If both players play optimally, can Marian guarantee a win? [i]Proposed by Pavel Ciurea[/i]

Let $m$ and $n$ be positive integers. Find all pairs of non-negative integers $a$ and $b$ that always satisfy the following condition: Given any configuration of $m$ white dots and $n$ black dots on a circle, there always exist a line cutting the circle into two arcs, one of which consists of exactly $a$ white dots and $b$ black dots. [i]Proposed by Tan Min Heng[/i]

A graph $G$ with $n$ vertices is called [i]cool[/i] if we can label each vertex with a different positive integer not greater than $\frac{n^2}{4}$ and find a set of non-negative integers $D$ so that there is an edge between two vertices iff the difference between their labels is in $D$. Show that if $n$ is sufficiently large we can always find a graph with $n$ vertices which is not cool.

A real number is assigned to each vertex of a triangular prism so that the number on any vertex is the arithmetic mean of the numbers on the three adjacent vertices. Prove that all six numbers are equal.

In the land of Microbablia the alphabet has only two letters, ‘A’ and ‘B’. Not surprisingly, the inhabitants are obsessed with the band ABBA. Words in the local dialect with a high ABBA-factor are considered particularly lucky. To compute the ABBA-factor of a word you just count the number of occurrences of ABBA within the word (not necessarily consecutively). So for instance AABA has ABBA-factor $0$, ABBA has ABBA-factor $1$, AABBBA has ABBA-factor $6$, and ABBABBA has ABBA factor $8$. What is the greatest possible ABBA-factor for a $100$ letter word?

There are $1 000 000$ soldiers in a line. The sergeant splits the line into $100$ segments (the length of different segments may be different) and permutes the segments (not changing the order of soldiers in each segment) forming a new line. The sergeant repeats this procedure several times (splits the new line in segments of the same lengths and permutes them in exactly the same way as the first time). Every soldier originally from the first segment recorded the number of performed procedures that took him to return to the first segment for the first time. Prove that at most $100$ of these numbers are different.

Petya has $10, 000$ balls, among them there are no two balls of equal weight. He also has a device, which works as follows: if he puts exactly $10$ balls on it, it will report the sum of the weights of some two of them (but he doesn't know which ones). Prove that Petya can use the device a few times so that after a while he will be able to choose one of the balls and accurately tell its weight.

Adithya and Bill are playing a game on a connected graph with $n > 2$ vertices, two of which are labeled $A$ and $B$, so that $A$ and $B$ are distinct and non-adjacent and known to both players. Adithya starts on vertex $A$ and Bill starts on $B$. Each turn, both players move simultaneously: Bill moves to an adjacent vertex, while Adithya may either move to an adjacent vertex or stay at his current vertex. Adithya loses if he is on the same vertex as Bill, and wins if he reaches $B$ alone. Adithya cannot see where Bill is, but Bill can see where Adithya is. Given that Adithya has a winning strategy, what is the maximum possible number of edges the graph may have? (Your answer may be in terms of $n$.) [i]Proposed by Steven Liu[/i]

Let be two natural numbers $ n $ and $ a. $ [b]a)[/b] Prove that there exists an $ n\text{-tuplet} $ of natural numbers $ \left( a_1,a_2,\ldots ,a_n\right) $ that satisfy the following equality. $$ 1+\frac{1}{a} =\prod_{i=1}^n \left( 1+\frac{1}{a_i} \right) $$ [b]b)[/b] Show that there exist only finitely such $ n\text{-tuplets} . $

A positive integer $m(\ge 2$) is given. From circle $C_1$ with a radius 1, construct $C_2, C_3, C_4, ... $ through following acts: In the $i$th act, select a circle $P_i$ inside $C_i$ with a area $\frac{1}{m}$ of $C_i$. If such circle dosen't exist, the act ends. If not, let $C_{i+1}$ a difference of sets $C_i -P_i$. Prove that this act ends within a finite number of times.

Given is a positive integer $n \geq 2$ and three pairwise disjoint sets $A, B, C$, each of $n$ distinct real numbers. Denote by $a$ the number of triples $(x, y, z) \in A \times B \times C$ satisfying $x<y<z$ and let $b$ denote the number of triples $(x, y, z) \in A \times B \times C$ such that $x>y>z$. Prove that $n$ divides $a-b$.

The $\underline{\text{path number}}$ of a graph is the minimum number of paths we need to partition the vertices of a graph. Given a connected graph with the independence number $k > 1$, what is the maximum possible value for the path number in this graph? Find the answer in terms of $k$. The independence number of a graph $\textbf{G}$ is the maximum possible number $k$, such that there exist $k$ pairwise non-adjacent vertices in $\textbf{G}$.

There are $n$ points on a flat piece of paper, any two of them at a distance of at least $2$ from each other. An inattentive pupil spills ink on a part of the paper such that the total area of the damaged part equals $\frac 32$. Prove that there exist two vectors of equal length less than $1$ and with their sum having a given direction, such that after a translation by either of these two vectors no points of the given set remain in the damaged area.

Let positive integers $M$, $m$, $n$ be such that $1\leq m \leq n$, $1\leq M \leq \dfrac {m(m+1)} {2}$, and let $A \subseteq \{1,2,\ldots,n\}$ with $|A|=m$. Prove there exists a subset $B\subseteq A$ with $$0 \leq \sum_{b\in B} b - M \leq n-m.$$ ([i]Dan Schwarz[/i])

The Bank of Zürich issues coins with an $H$ on one side and a $T$ on the other side. Alice has $n$ of these coins arranged in a line from left to right. She repeatedly performs the following operation: if some coin is showing its $H$ side, Alice chooses a group of consecutive coins (this group must contain at least one coin) and flips all of them; otherwise, all coins show $T$ and Alice stops. For instance, if $n = 3$, Alice may perform the following operations: $THT \to HTH \to HHH \to TTH \to TTT$. She might also choose to perform the operation $THT \to TTT$. For each initial configuration $C$, let $m(C)$ be the minimal number of operations that Alice must perform. For example, $m(THT) = 1$ and $m(TTT) = 0$. For every integer $n \geq 1$, determine the largest value of $m(C)$ over all $2^n$ possible initial configurations $C$. [i]Massimiliano Foschi, Italy[/i]

In Occidentalia there are $20$ different companies, each looking to hire $15$ new employees. A group of $300$ applicants interview each of the companies. Each company qualifies each applicant as suitable or not suitable to work in it, in such a way that each of them finds exactly $p$ suitable applicants, with $p > 15$. and each applicant is found suitable by at least one company. What is the smallest of $p $f or which it is always possible to assign $15$ applicants to each company, given that each company is assigned only applicants that it considers appropriate, and that each of the $300$ applicants is assigned to a company?