Let $S$ be a nonempty set of positive integers. We say that a positive integer $n$ is [i]clean[/i] if it has a unique representation as a sum of an odd number of distinct elements from $S$. Prove that there exist infinitely many positive integers that are not clean.

Henry invited $2N$ guests to his birthday party. He has $N$ white hats and $N$ black hats. He wants to place hats on his guests and split his guests into one or several dancing circles so that in each circle there would be at least two people and the colors of hats of any two neighbours would be different. Prove that Henry can do this in exactly $(2N)!$ different ways. (All the hats with the same color are identical, all the guests are obviously distinct, $(2N)! = 1 \cdot 2 \cdot . . . \cdot (2N)$.) Gleb Pogudin

Some positive integers, sum of which is $23$, are written in sequential form. Neither one of the terms nor the sum of some consecutive terms in the sequence is equal to $3$. [b]a) [/b]Is it possible that the sequence contains exactly $11$ terms? [b]b)[/b]Is it possible that the sequence contains exactly $12$ terms?

Let $F$ be a finite non-empty set of integers and let $n$ be a positive integer. Suppose that $\bullet$ Any $x \in F$ may be written as $x=y+z$ for some $y$, $z \in F$; $\bullet$ If $1 \leq k \leq n$ and $x_1$, ..., $x_k \in F$, then $x_1+\cdots+x_k \neq 0$. Show that $F$ has at least $2n+2$ elements.

$n$ people (with names $1,2,\dots,n$) are around a table. Some of them are friends. At each step 2 friend can change their place. Find a necessary and sufficient condition for friendship relation between them that with these steps we can always reach to all of posiible permutations.

A $3000\times 3000$ square is tiled by dominoes (i. e. $1\times 2$ rectangles) in an arbitrary way. Show that one can color the dominoes in three colors such that the number of the dominoes of each color is the same, and each dominoe $d$ has at most two neighbours of the same color as $d$. (Two dominoes are said to be [i]neighbours[/i] if a cell of one domino has a common edge with a cell of the other one.)

A rectangle $ABCD$ is given whose sides have lengths $3$ and $2n$, where $n$ is a natural number. Denote by $U(n)$ the number of ways in which one can cut the rectangle into rectangles of side lengths $1$ and $2$. $(a)$ Prove that \[U(n + 1)+U(n -1) = 4U(n);\] $(b)$ Prove that \[U(n) =\frac{1}{2\sqrt{3}}[(\sqrt{3} + 1)(2 +\sqrt{3})^n + (\sqrt{3} - 1)(2 -\sqrt{3})^n].\]

Consider a regular 2n-gon $ P$,$A_1,A_2,\cdots ,A_{2n}$ in the plane ,where $n$ is a positive integer . We say that a point $S$ on one of the sides of $P$ can be seen from a point $E$ that is external to $P$ , if the line segment $SE$ contains no other points that lie on the sides of $P$ except $S$ .We color the sides of $P$ in 3 different colors (ignore the vertices of $P$,we consider them colorless), such that every side is colored in exactly one color, and each color is used at least once . Moreover ,from every point in the plane external to $P$ , points of most 2 different colors on $P$ can be seen .Find the number of distinct such colorings of $P$ (two colorings are considered distinct if at least one of sides is colored differently). [i]Proposed by Viktor Simjanoski, Macedonia[/i] JBMO 2017, Q4

Consider the part of a lattice given by the corners $(0, 0), (n, 0), (n, 2)$ and $(0, 2)$. From a lattice point $(a, b)$ one can move to $(a + 1, b)$ or to $(a + 1, b + 1)$ or to $(a, b - 1$), provided that the second point is also contained in the part of the lattice. How many ways are there to move from $(0, 0)$ to $(n, 2)$ considering these rules?

Luke chose a set of three different dates $a,b,c$ in the month of May, where in any year, if one makes a calendar with a sheet of grid paper the centers of the cells with dates $a,b,c$ would form an isosceles right triangle or a straight line. How many sets can be chosen? [img]https://cdn.artofproblemsolving.com/attachments/7/3/dbf90fdc81fc0f0d14c32020b69df53b67b397.png[/img]

Let $ S \equal{} \{x_1, x_2, \ldots, x_{k \plus{} l}\}$ be a $ (k \plus{} l)$-element set of real numbers contained in the interval $ [0, 1]$; $ k$ and $ l$ are positive integers. A $ k$-element subset $ A\subset S$ is called [i]nice[/i] if \[ \left |\frac {1}{k}\sum_{x_i\in A} x_i \minus{} \frac {1}{l}\sum_{x_j\in S\setminus A} x_j\right |\le \frac {k \plus{} l}{2kl}\] Prove that the number of nice subsets is at least $ \dfrac{2}{k \plus{} l}\dbinom{k \plus{} l}{k}$. [i]Proposed by Andrey Badzyan, Russia[/i]

What is the least number of rooks that can be placed on a standard $8 \times 8$ chessboard so that all the white squares are attacked? (A rook also attacks the square it is on, in addition to every other square in the same row or column.)

A strip consists of $n$ squares which are numerated in their order by integers $1,2,3,..., n$. In the beginning, one square is empty while each remaining square contains one piece. Whenever a square contains a piece and its some neighbouring square contains another piece while the square immediately following the neighbouring square is empty, one may raise the first piece over the second one to the empty square, removing the second piece from the strip. Find all possibilites which square can be initially empty, if it is possible to reach a state where the strip contains only one piece and a) $n = 2008$, b) $n = 2009$.

[b]Q[/b] On a \(10 \times 10\) chess board whose colors of square are green and blue in an arbitrary way and we are simultaneously allowed to switch all the colors of all squares in any \((2 \times 2)\) and \((5\times 5)\) region. Can we transform any coloring of the board into one where all squares are blue ? Give a proper explanation of your answer. Note. that if a unit square is part of both the $2\times 2$ and $5\times 5$ region,then its color switched is twice(i.e switching is additive) [i]Proposed by Aritra12[/i]

A music streaming service proposes songs classified in $10$ musical genres, so that each song belong to one and only one gender. The songs are played one after the other: the first $17$ are chosen by the user, but starting from the eighteenth the service automatically determines which song to play. Elisabetta has noticed that, if one makes the classification of which genres they appear several times during the last $17$ songs played, the new song always belongs to the genre at the top of the ranking or, in case of same merit, at one of the first genres. Prove that, however, the first $17$ tracks are chosen, from a certain point onwards the songs proposed are all of the same kind.

An ant in the $xy$-plane is at the origin facing in the positive $x$-direction. The ant then begins a progression of moves, on the $n^{th}$ of which it first walks $\frac{1}{5^n}$ units in the direction it is facing and then turns $60^o$ degrees to the left. After a very large number of moves, the ant’s movements begins to converge to a certain point; what is the $y$-value of this point?

[b]p1.[/b] There exist real numbers $B$, $M$, and $T$ such that $B + M + T = 23$ and $B - M - T = 20$. Compute $M + T$. [b]p2.[/b] Kaity has a rectangular garden that measures $10$ yards by $12$ yards. Austin’s triangular garden has side lengths $6$ yards, $8$ yards, and $10$ yards. Compute the ratio of the area of Kaity’s garden to the area of Austin’s garden. [b]p3.[/b] Nikhil’s mom and brother both have ages under $100$ years that are perfect squares. His mom is $33$ years older than his brother. Compute the sum of their ages. [b]p4.[/b] Madison wants to arrange $3$ identical blue books and $2$ identical pink books on a shelf so that each book is next to at least one book of the other color. In how many ways can Madison arrange the books? [b]p5.[/b] Two friends, Anna and Bruno, are biking together at the same initial speed from school to the mall, which is $6$ miles away. Suddenly, $1$ mile in, Anna realizes that she forgot her calculator at school. If she bikes $4$ miles per hour faster than her initial speed, she could head back to school and still reach the mall at the same time as Bruno, assuming Bruno continues biking towards the mall at their initial speed. In miles per hour, what is Anna and Bruno’s initial speed, before Anna has changed her speed? (Assume that the rate at which Anna and Bruno bike is constant.) [b]p6.[/b] Let a number be “almost-perfect” if the sum of its digits is $28$. Compute the sum of the third smallest and third largest almost-perfect $4$-digit positive integers. [b]p7.[/b] Regular hexagon $ABCDEF$ is contained in rectangle $PQRS$ such that line $\overline{AB}$ lies on line $\overline{PQ}$, point $C$ lies on line $\overline{QR}$, line $\overline{DE}$ lies on line $\overline{RS}$, and point $F$ lies on line $\overline{SP}$. Given that $PQ = 4$, compute the perimeter of $AQCDSF$. [img]https://cdn.artofproblemsolving.com/attachments/6/7/5db3d5806eaefa00d7fc90fb786a41c0466a90.png[/img] [b]p8.[/b] Compute the number of ordered pairs $(m, n)$, where $m$ and $n$ are relatively prime positive integers and $mn = 2520$. (Note that positive integers $x$ and $y$ are relatively prime if they share no common divisors other than $1$. For example, this means that $1$ is relatively prime to every positive integer.) [b]p9.[/b] A geometric sequence with more than two terms has first term $x$, last term $2023$, and common ratio $y$, where $x$ and $y$ are both positive integers greater than $1$. An arithmetic sequence with a finite number of terms has first term $x$ and common difference $y$. Also, of all arithmetic sequences with first term $x$, common difference $y$, and no terms exceeding $2023$, this sequence is the longest. What is the last term of the arithmetic sequence? [b]p10.[/b] Andrew is playing a game where he must choose three slips, uniformly at random and without replacement, from a jar that has nine slips labeled $1$ through $9$. He wins if the sum of the three chosen numbers is divisible by $3$ and one of the numbers is $1$. What is the probability Andrew wins? [b]p11.[/b] Circle $O$ is inscribed in square $ABCD$. Let $E$ be the point where $O$ meets line segment $\overline{AB}$. Line segments $\overline{EC}$ and $\overline{ED}$ intersect $O$ at points $P$ and $Q$, respectively. Compute the ratio of the area of triangle $\vartriangle EPQ$ to the area of triangle $\vartriangle ECD$. [b]p12.[/b] Define a recursive sequence by $a_1 = \frac12$ and $a_2 = 1$, and $$a_n =\frac{1 + a_{n-1}}{a_{n-2}}$$ for n ≥ 3. The product $a_1a_2a_3 ... a_{2023}$ can be expressed in the form $a^b \cdot c^d \cdot e^f$ , where $a$, $b$, $c$, $d$, $e$, and $f$ are positive (not necessarily distinct) integers, and a, c, and e are prime. Compute $a + b + c + d + e + f$. [b]p13.[/b] An increasing sequence of $3$-digit positive integers satisfies the following properties: $\bullet$ Each number is a multiple of $2$, $3$, or $5$. $\bullet$ Adjacent numbers differ by only one digit and are relatively prime. (Note that positive integers x and y are relatively prime if they share no common divisors other than $1$.) What is the maximum possible length of the sequence? [b]p14.[/b] Circles $O_A$ and $O_B$ with centers $A$ and $B$, respectively, have radii $3$ and $8$, respectively, and are internally tangent to each other at point $P$. Point $C$ is on circle $O_A$ such that line $\overline{BC}$ is tangent to circle $OA$. Extend line $\overline{PC}$ to intersect circle $O_B$ at point $D \ne P$. Compute $CD$. [b]p15.[/b] Compute the product of all real solutions $x$ to the equation $x^2 + 20x - 23 = 2 \sqrt{x^2 + 20x + 1}$. [b]p16.[/b] Compute the number of divisors of $729, 000, 000$ that are perfect powers. (A perfect power is an integer that can be written in the form $a^b$, where $a$ and $b$ are positive integers and $b > 1$.) [b]p17.[/b] The arithmetic mean of two positive integers $x$ and $y$, each less than $100$, is $4$ more than their geometric mean. Given $x > y$, compute the sum of all possible values for $x + y$. (Note that the geometric mean of $x$ and $y$ is defined to be $\sqrt{xy}$.) [b]p18.[/b] Ankit and Richard are playing a game. Ankit repeatedly writes the digits $2$, $0$, $2$, $3$, in that order, from left to right on a board until Richard tells him to stop. Richard wins if the resulting number, interpreted as a base-$10$ integer, is divisible by as many positive integers less than or equal to $12$ as possible. For example, if Richard stops Ankit after $7$ digits have been written, the number would be $2023202$, which is divisible by $1$ and $2$. Richard wants to win the game as early as possible. Assuming Ankit must write at least one digit, after how many digits should Richard stop Ankit? [b]p19.[/b] Eight chairs are set around a circular table. Among these chairs, two are red, two are blue, two are green, and two are yellow. Chairs that are the same color are identical. If rotations and reflections of arrangements of chairs are considered distinct, how many arrangements of chairs satisfy the property that each pair of adjacent chairs are different colors? [b]p20.[/b] Four congruent spheres are placed inside a right-circular cone such that they are all tangent to the base and the lateral face of the cone, and each sphere is tangent to exactly two other spheres. If the radius of the cone is $1$ and the height of the cone is $2\sqrt2$, what is the radius of one of the spheres? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Alice and Bob play a game on the infinite side of a checkered strip, in which the cells are numbered with consecutive integers from left to right (..., $-2$, $-1$, $0$, $1$, $2$, ...). Alice in her turn puts one cross in any free cell, and Bob in his turn puts zeros in any 2020 free cells. Alice will win if he manages to get such 4 cells marked with crosses, the corresponding cell numbers will form an arithmetic progression. Bob's goal in this game is to prevent Alice from winning. They take turns and Alice moves first. Will Alice be able to win no matter how Bob plays?

Through a point in the interior of a sphere we put three pairwise perpendicular planes. Those planes dissect the surface of the sphere in eight curvilinear triangles. Alternately the triangles are coloured black and wide to make the sphere surface look like a checkerboard. Prove that exactly half of the sphere's surface is coloured black.

$100$ integers are arranged in a circle. Each number is greater than the sum of the two subsequent numbers (in a clockwise order). Determine the maximal possible number of positive numbers in such circle. [i](S.Berlov)[/i]

Let $f:X\rightarrow X$, where $X=\{1,2,\ldots ,100\}$, be a function satisfying: 1) $f(x)\neq x$ for all $x=1,2,\ldots,100$; 2) for any subset $A$ of $X$ such that $|A|=40$, we have $A\cap f(A)\neq\emptyset$. Find the minimum $k$ such that for any such function $f$, there exist a subset $B$ of $X$, where $|B|=k$, such that $B\cup f(B)=X$.

We define a beehive of order $n$ as follows: a beehive of order 1 is one hexagon To construct a beehive of order $n$, take a beehive of order $n-1$ and draw a layer of hexagons in the exterior of these hexagons. See diagram for examples of $n=2,3$ Initially some hexagons are infected by a virus. If a hexagon has been infected, it will always be infected. Otherwise, it will be infected if at least 5 out of the 6 neighbours are infected. Let $f(n)$ be the minimum number of infected hexagons in the beginning so that after a finite time, all hexagons become infected. Find $f(n)$.

Let $ S \equal{}\{1,2,3, \ldots, 2n\}$ ($ n \in \mathbb{Z}^\plus{}$). Ddetermine the number of subsets $ T$ of $ S$ such that there are no 2 element in $ T$ $ a,b$ such that $ |a\minus{}b|\equal{}\{1,n\}$

The function $F$ is a one-to-one transformation of the plane into itself that maps rectangles into rectangles (rectangles are closed; continuity is not assumed). Prove that $F$ maps squares into squares.

Let $n$ be a positive integer. Determine the smallest positive integer $k$ with the following property: it is possible to mark $k$ cells on a $2n \times 2n$ board so that there exists a unique partition of the board into $1 \times 2$ and $2 \times 1$ dominoes, none of which contain two marked cells.