Initially, Aslı distributes $1000$ balls to $30$ boxes as she wishes. After that, Aslı and Zehra make alternated moves which consists of taking a ball in any wanted box starting with Aslı. One who takes the last ball from any box takes that box to herself. What is the maximum number of boxes can Aslı guarantee to take herself regardless of Zehra's moves?

$10$ cities are connected by one-way air routes in a way so that each city can be reached from any other by several connected flights. Let $n$ be the smallest number of flights needed for a tourist to visit every city and return to the starting city. Clearly $n$ depends on the flight schedule. Find the largest $n$ and the corresponding flight schedule.

A sequence $P_1, \dots, P_n$ of points in the plane (not necessarily diferent) is [i]carioca[/i] if there exists a permutation $a_1, \dots, a_n$ of the numbers $1, \dots, n$ for which the segments $$P_{a_1}P_{a_2}, P_{a_2}P_{a_3}, \dots, P_{a_n}P_{a_1}$$ are all of the same length. Determine the greatest number $k$ such that for any sequence of $k$ points in the plane, $2023-k$ points can be added so that the sequence of $2023$ points is [i]carioca[/i].

A circle is circumscribed around an isosceles triangle whose two base angles are equal to $x^{\circ}$. Two points are chosen independently and randomly on the circle, and a chord is drawn between them. The probability that the chord intersects the triangle is $\frac{14}{25}.$ Find the sum of the largest and smallest possible value of $x$.

Let $n$ be a natural number. A tiling of a $2n \times 2n$ board is a placing of $2n^2$ dominos (of size $2 \times 1$ or $1 \times 2$) such that each of them covers exactly two squares of the board and they cover all the board.Consider now two [i]sepearate tilings[/i] of a $2n \times 2n$ board: one with red dominos and the other with blue dominos. We say two squares are red neighbours if they are covered by the same red domino in the red tiling; similarly define blue neighbours. Suppose we can assign a non-zero integer to each of the squares such that the number on any square equals the difference between the numbers on it's red and blue neighbours i.e the number on it's red neigbhbour minus the number on its blue neighbour. Show that $n$ is divisible by $3$ [i] Proposed by Tejaswi Navilarekallu [/i]

In a country there are $n>100$ cities and initially no roads. The government randomly determined the cost of building a two-way road between any two cities, using all amounts from $1$ to $\frac{n(n-1)}{2}$ thalers once (all options are equally likely). The mayor of each city chooses the cheapest of the $n-1$ roads emanating from that city and it is built (this may be the mutual desired of the mayors of both cities being connected, or only one of the two). After the construction of these roads, the cities are divided into $M$ connected components (between cities of the same connected component, you can get along the constructed roads, possibly via other cities, but this is not possible for cities of different components). Find the expected value of the random variable $M$. [i]Proposed by F. Petrov[/i]

Determine the number of distinct right-angled triangles such that its three sides are of integral lengths, and its area is $999$ times of its perimeter. (Congruent triangles are considered identical.)

On the table are $300$ coins. Petya, Vasya and Tolya play the next game. They go in turn in the following order: Petya, Vasya, Tolya, Petya. Vasya, Tolya, etc. In one move, Petya can take $1, 2, 3$, or $4$ coins from the table, Vasya, $1$ or $2$ coins, and Tolya, too, $1$ or $2$ coins. Can Vasya and Tolya agree so that, as if Petya were playing, one of them two will take the last coin off the table?

Baron Munchausen claims that he got a map of a country that consists of five cities. Each two cities are connected by a direct road. Each road intersects no more than one another road (and no more than once). On the map, the roads are colored in yellow or red, and while circling any city (along its border) one can notice that the colors of crossed roads alternate. Can Baron's claim be true?

Let $ n$ be an integer greater than $ 3.$ Points $ V_{1},V_{2},...,V_{n},$ with no three collinear, lie on a plane. Some of the segments $ V_{i}V_{j},$ with $ 1 \le i < j \le n,$ are constructed. Points $ V_{i}$ and $ V_{j}$ are [i]neighbors[/i] if $ V_{i}V_{j}$ is constructed. Initially, chess pieces $ C_{1},C_{2},...,C_{n}$ are placed at points $ V_{1},V_{2},...,V_{n}$ (not necessarily in that order) with exactly one piece at each point. In a move, one can choose some of the $ n$ chess pieces, and simultaneously relocate each of the chosen piece from its current position to one of its neighboring positions such that after the move, exactly one chess piece is at each point and no two chess pieces have exchanged their positions. A set of constructed segments is called [i]harmonic[/i] if for any initial positions of the chess pieces, each chess piece $ C_{i}(1 \le i \le n)$ is at the point $ V_{i}$ after a finite number of moves. Determine the minimum number of segments in a harmonic set.

[u]Round 1[/u] [b]1.1.[/b] A person asks for help every $3$ seconds. Over a time period of $5$ minutes, how many times will they ask for help? [b]1.2.[/b] In a big bag, there are $14$ red marbles, $15$ blue marbles, and$ 16$ white marbles. If Anuj takes a marble out of the bag each time without replacement, how many marbles does Anuj need to remove to be sure that he will have at least $3$ red marbles? [b]1.3.[/b] If Josh has $5$ distinct candies, how many ways can he pick $3$ of them to eat? [u]Round 2[/u] [b]2.1.[/b] Annie has a circular pizza. She makes $4$ straight cuts. What is the minimum number of slices of pizza that she can make? [b]2.2.[/b] What is the sum of the first $4$ prime numbers that can be written as the sum of two perfect squares? [b]2.3.[/b] Consider a regular octagon $ABCDEFGH$ inscribed in a circle of area $64\pi$. If the length of arc $ABC$ is $n\pi$, what is $n$? [u]Round 3[/u] [b]3.1.[/b] Let $ABCDEF$ be an equiangular hexagon with consecutive sides of length $6, 5, 3, 8$, and $3$. Find the length of the sixth side. [b]3.2.[/b] Jack writes all of the integers from $ 1$ to $ n$ on a blackboard except the even primes. He selects one of the numbers and erases all of its digits except the leftmost one. He adds up the new list of numbers and finds that the sum is $2020$. What was the number he chose? [b]3.3.[/b] Our original competition date was scheduled for April $11$, $2020$ which is a Saturday. The numbers $4116$ and $2020$ have the same remainder when divided by $x$. If $x$ is a prime number, find the sum of all possible $x$. [u]Round 4[/u] [b]4.1.[/b] The polynomials $5p^2 + 13pq + cq^2$ and $5p^2 + 13pq - cq^2$ where $c$ is a positive integer can both be factored into linear binomials with integer coefficients. Find $c$. [b]4.2.[/b] In a Cartesian coordinate plane, how many ways are there to get from $(0, 0)$ to $(2, 3)$ in $7$ moves, if each move consists of a moving one unit either up, down, left, or right? [b]4.3.[/b] Bob the Builder is building houses. On Monday he finds an empty field. Each day starting on Monday, he finishes building a house at noon. On the $n$th day, there is a $\frac{n}{8}$ chance that a storm will appear at $3:14$ PM and destroy all the houses on the field. At any given moment, Bob feels sad if and only if there is exactly $1$ house left on the field that is not destroyed. The probability that he will not be sad on Friday at $6$ PM can be expressed as $p/q$ in simplest form. Find $p + q$. PS. You should use hide for answers. Rounds 5-8 have been posted [url=https://artofproblemsolving.com/community/c3h2784570p24468605]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Given a polyhedron $P$. Mikita claims that he can write one integer on each face of $P$ such that not all the written numbers are zeros, and for each vertex $V$ of $P$ the sum of numbers on faces containing $V$ is equals to 0. Matvei claims that he can write one integer in each vertex of $P$ such that not all the written numbers are zeros, and for each face $F$ of $P$ the sum of numbers in vertices belonging to $F$ is equals to 0. Show that if the number of edges of polyhedron $P$ is odd, then at least one of the boys is right.

Three fair six-sided dice are thrown. Determine the probability that the sum of the numbers on the three top faces is $6$.

There is graph $ G_0$ on vertices $ A_1, A_2, \ldots, A_n$. Graph $ G_{n \plus{} 1}$ on vertices $ A_1, A_2, \ldots, A_n$ is constructed by the rule: $ A_i$ and $ A_j$ are joined only if in graph $ G_n$ there is a vertices $ A_k\neq A_i, A_j$ such that $ A_k$ is joined with both $ A_i$ and $ A_j$. Prove that the sequence $ \{G_n\}_{n\in\mathbb{N}}$ is periodic after some term with period $ T \le 2^n$.

An interstellar hotel has $100$ rooms with capacities $101,102,\ldots, 200$ people. These rooms are occupied by $n$ people in total. Now a VIP guest is about to arrive and the owner wants to provide him with a personal room. On that purpose, the owner wants to choose two rooms $A$ and $B$ and move all guests from $A$ to $B$ without exceeding its capacity. Determine the largest $n$ for which the owner can be sure that he can achieve his goal no matter what the initial distribution of the guests is.

Let positive integers $m,n$ satisfy $n=2^m-1$. $P_n =\{1,2,\cdots ,n\}$ is a set that contains $n$ points on an axis. A grasshopper on the axis can leap from one point to another adjacent point. Find the maximal value of $m$ satisfying following conditions: (a) $x, y$ are two arbitrary points in $P_n$; (b) starting at point $x$, the grasshopper leaps $2012$ times and finishes at point $y$; (the grasshopper is allowed to travel $x$ and $y$ more than once) (c) there are even number ways for the grasshopper to do (b).

The pupil is training in the square equation solution. Having the recurrent equation solved, he stops, if it doesn't have two roots, or solves the next equation, with the free coefficient equal to the greatest root, the coefficient at $x$ equal to the least root, and the coefficient at $x^2$ equal to $1$. Prove that the process cannot be infinite. What maximal number of the equations he will have to solve?

A $9 \times 9$ square is divided into $81$ unit cells. Some of the cells are coloured. The distance between the centres of any two coloured cells is more than $2$. (a) Give an example of colouring with $17$ coloured cells. (b) Prove that the numbers of coloured cells cannot exceed $17$. (S. Fomin, Leningrad)

Find the number of binary sequences $S$ of length $2015$ such that for any two segments $I_1, I_2$ of $S$ of the same length, we have • The sum of digits of $I_1$ differs from the sum of digits of $I_2$ by at most $1$, • If $I_1$ begins on the left end of S then the sum of digits of $I_1$ is not greater than the sum of digits of $I_2$, • If $I_2$ ends on the right end of S then the sum of digits of $I_2$ is not less than the sum of digits of $I_1$. Lê Anh Vinh

Let $G$ be a tournoment such that it's edges are colored either red or blue. Prove that there exists a vertex of $G$ like $v$ with the property that, for every other vertex $u$ there is a mono-color directed path from $v$ to $u$.

In a certain Zoom meeting, there are $4$ students. How many ways are there to split them into any number of distinguishable breakout rooms, each with at least $ 1$ student?

Unit cubes are made into beads by drilling a hole through them along a diagonal. The beads are put on a string in such a way that they can move freely in space under the restriction that the vertices of two neighboring cubes are touching. Let $ A$ be the beginning vertex and $ B$ be the end vertex. Let there be $ p \times q \times r$ cubes on the string $ (p, q, r \geq 1).$ [i](a)[/i] Determine for which values of $ p, q,$ and $ r$ it is possible to build a block with dimensions $ p, q,$ and $ r.$ Give reasons for your answers. [i](b)[/i] The same question as (a) with the extra condition that $ A \equal{} B.$

Cucumber river in the Flower city has parallel banks with the distance between them $1$ metre. It has some islands with the total perimeter $8$ metres. Mr. Know-All claims that it is possible to cross the river in a boat from the arbitrary point, and the trajectory will not exceed $3$ metres. Is he right?

There are 2019 cards in a box. Each card has a number written on one of its sides and a letter on the other side. Amy and Ben play the following game: in the beginning Amy takes all the cards, places them on a line and then she flips as many cards as she wishes. Each time Ben touches a card he has to flip it and its neighboring cards. Ben is allowed to have as many as 2019 touches. Ben wins if all the cards are on the numbers' side, otherwise Amy wins. Determine who has a winning strategy.

Tim the Beaver can make three different types of geometrical figures: squares, regular hexagons, and regular octagons. Tim makes a random sequence $F_0$, $F_1$, $F_2$, $F_3$, $...$ of figures as follows: $\bullet$ $F_0$ is a square. $\bullet$ For every positive integer $i$, $F_i$ is randomly chosen to be one of the $2$ figures distinct from $F_{i-1}$ (each chosen with equal probability $\frac12$ ). $\bullet$ Tim takes $4$ seconds to make squares, $6$ to make hexagons, and $8$ to make octagons. He makes one figure after another, with no breaks in between. Suppose that exactly $17$ seconds after he starts making $F_0$, Tim is making a figure with $n$ sides. What is the expected value of $n$?