One hundred and one beetles are crawling in the plane. Some of the beetles are friends. Every one hundred beetles can position themselves so that two of them are friends if and only if they are at unit distance from each other. Is it always true that all one hundred and one beetles can do the same?

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]).

Given two sets $A, B$ of positive real numbers such that: $|A| = |B| =n$; $A \neq B$ and $S(A)=S(B)$, where $|X|$ is the number of elements and $S(X)$ is the sum of all elements in set $X$. Prove that we can fill in each unit square of a $n\times n$ square with positive numbers and some zeros such that: a) the set of the sum of all numbers in each row equals $A$; b) the set of the sum of all numbers in each column equals $A$. c) there are at least $(n-1)^{2}+k$ zero numbers in the $n\times n$ array with $k=|A \cap B|$.

Let's define [i]almost mean[/i] of numbers $a_1, a_2, \ldots, a_n$ as $\frac{a_1 + a_2 + \ldots + a_n}{n+1}$. Oleksiy has positive real numbers $b_1, b_2, \ldots, b_{2023}$, not necessarily distinct. For each pair $(i, j)$ with $1 \leq i, j \leq 2023$, Oleksiy wrote on a board [i]almost mean[/i] of numbers $b_i, b_{i+1}, \ldots, b_j$. Prove that there are at least $45$ distinct numbers on the board. [i]Proposed by Anton Trygub[/i]

Let $ n,k$ be given positive integers satisfying $ k\le 2n \minus{} 1$. On a table tennis tournament $ 2n$ players take part, they play a total of $ k$ rounds match, each round is divided into $ n$ groups, each group two players match. The two players in different rounds can match on many occasions. Find the greatest positive integer $ m \equal{} f(n,k)$ such that no matter how the tournament processes, we always find $ m$ players each of pair of which didn't match each other.

Two distinct squares of the $8\times8$ chessboard $C$ are said to be adjacent if they have a vertex or side in common. Also, $g$ is called a $C$-gap if for every numbering of the squares of $C$ with all the integers $1, 2, \ldots, 64$ there exist twoadjacent squares whose numbers differ by at least $g$. Determine the largest $C$-gap $g$.

Suppose we have a simple polygon (that is it does not intersect itself, but not necessarily convex). Show that this polygon has a diameter which is completely inside the polygon and the two arcs it creates on the polygon perimeter (the two arcs have 2 vertices in common) both have at least one third of the vertices of the polygon.

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 $u$.

A country has two capitals and several towns. Some of them are connected by roads. Some of the roads are toll roads where a fee is charged for driving along them. It is known that any route from the south capital to the north capital contains at least ten toll roads. Prove that all toll roads can be distributed among ten companies so that anybody driving from the south capital to the north capital must pay each of these companies. [i](5 points)[/i]

The set of quadruples $(a,b,c,d)$ where each of $a,b,c,d$ is either $0$ or $1$ is [i]called vertices of the four dimensional unit cube[/i] or [i]4-cube[/i] for short. Two vertices are called [i]adjacent[/i], if their respective quadruples differ by one variable only. Each two adjacent vertices are connected by an edge. A robot is moving through the edges of the 4-cube starting from $(0,0,0,0)$ and each turn consists of passing an edge and moving to adjacent vertex. In how many ways can the robot go back to $(0,0,0,0)$ after $4042$ turns? Note that it is [u]NOT[/u] forbidden for the robot to pass through $(0,0,0,0)$ before the $4042$-nd turn.

In some club, each member is on two commissions. Furthermore, it is known that two any commissions always have exactly one member in common. Knowing there are five commissions. How many members does the club have?

Let $ABC$ be an equilateral triangle and $\mathcal{E}$ the set of all points contained in the three segments $AB$, $BC$, and $CA$ (including $A$, $B$, and $C$). Determine whether, for every partition of $\mathcal{E}$ into two disjoint subsets, at least one of the two subsets contains the vertices of a right-angled triangle.

[u]Round 5[/u] [b]p13.[/b] A unit square is rotated $30^o$ counterclockwise about one of its vertices. Determine the area of the intersection of the original square with the rotated one. [b]p14.[/b] Suppose points $A$ and $B$ lie on a circle of radius $4$ with center $O$, such that $\angle AOB = 90^o$. The perpendicular bisectors of segments $OA$ and $OB$ divide the interior of the circle into four regions. Find the area of the smallest region. [b]p15.[/b] Let $ABCD$ be a quadrilateral such that $AB = 4$, $BC = 6$, $CD = 5$, $DA = 3$, and $\angle DAB = 90^o$. There is a point $I$ inside the quadrilateral that is equidistant from all the sides. Find $AI$. [u]Round 6[/u] [i]The answer to each of the three questions in this round depends on the answer to one of the other questions. There is only one set of correct answers to these problems; however, each question will be scored independently, regardless of whether the answers to the other questions are correct. [/i] [b]p16.[/b] Let $C$ be the answer to problem $18$. Compute $$\left( 1 - \frac{1}{2^2} \right) \left( 1 - \frac{1}{3^2} \right) ... \left( 1 - \frac{1}{C^2} \right).$$ [b]p17.[/b] Let $A$ be the answer to problem $16$. Let $PQRS$ be a square, and let point $M$ lie on segment $PQ$ such that $MQ = 7PM$ and point $N$ lie on segment $PS$ such that $NS = 7PN$. Segments $MS$ and $NQ$ meet at point $X$. Given that the area of quadrilateral $PMXN$ is $A - \frac12$, find the side length of the square. [b]p18.[/b] Let $B$ be the answer to problem $17$ and let $N = 6B$. Find the number of ordered triples $(a, b, c)$ of integers between $0$ and $N - 1$, inclusive, such that $a + b + c$ is divisible by $N$. [u]Round 7[/u] [b]p19.[/b] Let $k$ be the units digit of $\underbrace{7^{7^{7^{7^{7^{7^{7}}}}}}}_{Seven \,\,7s}$ . What is the largest prime factor of the number consisting of $k$ $7$’s written in a row? [b]p20.[/b] Suppose that $E = 7^7$ , $M = 7$, and $C = 7·7·7$. The characters $E, M, C, C$ are arranged randomly in the following blanks. $$... \times ... \times ... \times ... $$ Then one of the multiplication signs is chosen at random and changed to an equals sign. What is the probability that the resulting equation is true? [b]p21[/b]. During a recent math contest, Sophy Moore made the mistake of thinking that $133$ is a prime number. Fresh Mann replied, “To test whether a number is divisible by $3$, we just need to check whether the sum of the digits is divisible by $3$. By the same reasoning, to test whether a number is divisible by $7$, we just need to check that the sum of the digits is a multiple of $7$, so $133$ is clearly divisible by $7$.” Although his general principle is false, $133$ is indeed divisible by $7$. How many three-digit numbers are divisible by $7$ and have the sum of their digits divisible by $7$? [u]Round 8[/u] [b]p22.[/b] A [i]look-and-say[/i] sequence is defined as follows: starting from an initial term $a_1$, each subsequent term $a_k$ is found by reading the digits of $a_{k-1}$ from left to right and specifying the number of times each digit appears consecutively. For example, $4$ would be succeeded by $14$ (“One four.”), and $31337$ would be followed by $13112317$ (“One three, one one, two three, one seven.”) If $a_1$ is a random two-digit positive integer, find the probability that $a_4$ is at least six digits long. [b]p23.[/b] In triangle $ABC$, $\angle C = 90^o$. Point $P$ lies on segment $BC$ and is not $B$ or $C$. Point $I$ lies on segment $AP$, and $\angle BIP = \angle PBI = \angle CAB$. If $\frac{AP}{BC} = k$, express $\frac{IP}{CP}$ in terms of $k$. [b]p24.[/b] A subset of $\{1, 2, 3, ... , 30\}$ is called [i]delicious [/i] if it does not contain an element that is $3$ times another element. A subset is called super delicious if it is delicious and no delicious set has more elements than it has. Determine the number of super delicious subsets. PS. You sholud use hide for answers. First rounds have been posted [url=https://artofproblemsolving.com/community/c4h2784267p24464980]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

[u]Round 1[/u] [b]p1.[/b] An iscoceles triangle has one angle equal to $100$ degrees, what is the degree measure of one of the two remaining angles. [b]p2.[/b] Tanmay picks four cards from a standard deck of $52$ cards at random. What is the probability he gets exactly one Ace, exactly exactly one King, exactly one Queen, exactly one Jack and exactly one Ten? [b]p3.[/b] What is the sum of all the factors of $2014$? [u]Round 2[/u] [b]p4.[/b] Which number under $1000$ has the greatest number of factors? [b]p5.[/b] How many $10$ digit primes have all distinct digits? [b]p6.[/b] In a far o universe called Manhattan, the distance between two points on the plane $P = (x_1, y_1)$ and $Q = (x_2, y_2)$ is defined as $d(P,Q) = |x_1-x_2|+|y_1-y_2|$. Let $S$ be the region of points that are a distance of $\le 7$ away from the origin $(0, 0)$. What is the area of $S$? [u]Round 3[/u] [b]p7.[/b] How many factors does $13! + 14! + 15!$ have? [b]p8.[/b] How many zeroes does $45!$ have consecutively at the very end in its representation in base $45$? [b]p9.[/b] A sequence of circles $\omega_0$, $\omega_1$, $\omega_2$, ... is drawn such that: $\bullet$ $\omega_0$ has a radius of $1$. $\bullet$ $\omega_{i+1}$ has twice the radius of $\omega_i$. $\bullet$ $\omega_i$ is internally tangent to $\omega_{i+1}$. Let $A$ be a point on $\omega_0$ and $B$ be a point on $\omega_{10}$. What is the maximum possible value of $AB$? [u]Round 4[/u] [b]p10.[/b] A $3-4-5$ triangle is constructed. Then a similar triangle is constructed with the shortest side of the first triangle being the new hypotenuse for the second triangle. This happens an infinite amount of times. What is the maximum area of the resulting figure? [b]p11.[/b] If an unfair coin is flipped $4$ times and has a $3/64$ chance of coming heads exactly thrice, what is the probability the coin comes tails on a single flip. [b]p12.[/b] Find all triples of positive integers $(a, b, c)$ that satisfy $2a = 1+bc$, $2b = 1+ac$, and $2c = 1 + ab$. [u]Round 5[/u] [b]p13.[/b] $6$ numbered points on a plane are placed so that they can create a regular hexagon $P_1P_2P_3P_4P_5P_6$ if connected. If a triangle is drawn to include a certain amount of points in it, how many triangles are there that hold a different set of points? (note: the triangle with $P_1$ and $P_2$ is not the same as the one with $P_3$ and $P_4$). [b]p14.[/b] Let $S$ be the set of all numbers of the form $n(2n + 1)(3n + 2)(4n + 3)(5n + 4)$ for $n \ge 1$. What is the largest number that divides every member of $S$? [b]p15. [/b]Jordan tosses a fair coin until he gets heads at least twice. What is the expected number of flips of the coin that he will make? PS. You should use hide for answers. Rounds 6-10 have been posted [url=https://artofproblemsolving.com/community/c3h3156859p28695035]here[/url].. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $ F_n$ be a set of all possible connected figures, that consist of $ n$ unit cells. For each element $ f_n$ of this set, let $ S(f_n)$ be the area of that minimal rectangle that covers $ f_n$ and each side of the rectangle is parallel to the corresponding side of the cell. Find $ max(S(f_n))$,where $ f_n\in F_n$? Remark: Two cells are called connected if they have a common edge.

What is the least possible number of the checkers being required a) for the $8\times 8$ chess-board, b) for the $n\times n$ chess-board, to provide the property: [i]Every line (of the chess-board fields) parallel to the side or diagonal is occupied by at least one checker[/i] ?

In the plan $6$ points were located such that the distance between two damages of them is greater than or equal to $1$. Prove that it is possible to choose two of those points such that their distance is greater than or equal to $2 \cos{18}$ Observation: It might help you to know that $\cos{18} = 0.95105\ldots$ and $\cos{24} = 0.91354\ldots$

The plane is partitioned into congruent equilateral triangles such that any two of them which are not disjoint have either a common vertex or a common side. Is there a circle containing exactly 1988 points in its interior?

Given two closed broken lines in the plane with odd numbers of edges. All the lines, containing those edges are different, and not a triple of them intersects in one point. Prove that it is possible to chose one edge from each line such, that the chosen edges will be the opposite sides of a convex quadrangle.

For a permutation $P = (p_1, p_2, ... , p_n)$ of $(1, 2, ... , n)$ define $X(P)$ as the number of $j$ such that $p_i < p_j$ for every $i < j$. What is the expected value of $X(P)$ if each permutation is equally likely?

There is a WORD game with the following rules. There are finite number of relations $U_{i}\longrightarrow V_{i}$($U_{i},V_{i}$ are words). There is are two words $A,B$. We start from $A$, and we want to reach to $B$. At each step we can change one subword $U_{i}$ to $V_{i}$. Prove that there does not exist an algorithm that picks up $A,B$ and $U_{i}$'s,$V_{i}$'s and decides whether we can reach from $A$ to $B$ or not.

Let $ G(V,E)$ be a connected graph, and let $ d_G(x,y)$ denote the length of the shortest path joining $ x$ and $ y$ in $ G$. Let $ r_G(x)\equal{} \max \{ d_G(x,y) : \; y \in V \ \}$ for $ x \in V$, and let $ r(G)\equal{} \min \{ r_G(x) : \;x \in V\ \}$. Show that if $ r(G) \geq 2$, then $ G$ contains a path of length $ 2r(G)\minus{}2$ as an induced subgraph. [i]V. T. Sos[/i]

An $n\times n$ chessboard is given, where $n$ is an even positive integer. On every line, the unit squares are to be permuted, subject to the condition that the resulting table has to be symmetric with respect to its main diagonal (the diagonal from the top-left corner to the bottom-right corner). We say that a board is [i]alternative[/i] if it has at least one pair of complementary lines (two lines are complementary if the unit squares on them which lie on the same column have distinct colours). Otherwise, we call the board [i]nonalternative[/i]. For what values of $n$ do we always get from the $n\times n$ chessboard an alternative board?\\ \\ [i](Alexandru Petrescu and Andra Elena Mircea)[/i]

Let $k$ be a positive integer. The organising commitee of a tennis tournament is to schedule the matches for $2k$ players so that every two players play once, each day exactly one match is played, and each player arrives to the tournament site the day of his first match, and departs the day of his last match. For every day a player is present on the tournament, the committee has to pay $1$ coin to the hotel. The organisers want to design the schedule so as to minimise the total cost of all players' stays. Determine this minimum cost.

Albert, Bernard, and Cheryl are playing marbles. At the beginning, each of them brings 5 red marbles, 7 green marbles and 13 blue marbles and in the middle of the table, there is a box of infinitely many red, blue and green marbles. In each turn, each player may choose 2 marbles of different color and replace them with 2 marbles of the third color. After a finite number of steps, this conversation happens. Albert : " I have only red marbles" Bernard : "I have only blue marbles" Cheryl: "I have only green marbles" Which of the three are lying?