Let $m$ and $n$ be fixed positive integers. Tsvety and Freyja play a game on an infinite grid of unit square cells. Tsvety has secretly written a real number inside of each cell so that the sum of the numbers within every rectangle of size either $m$ by $n$ or $n$ by $m$ is zero. Freyja wants to learn all of these numbers. One by one, Freyja asks Tsvety about some cell in the grid, and Tsvety truthfully reveals what number is written in it. Freyja wins if, at any point, Freyja can simultaneously deduce the number written in every cell of the entire infinite grid (If this never occurs, Freyja has lost the game and Tsvety wins). In terms of $m$ and $n$, find the smallest number of questions that Freyja must ask to win, or show that no finite number of questions suffice. [i]Nikolai Beluhov[/i]

It is given $n$ a natural number and a circle with circumference $n$. On the circle, in clockwise direction, numbers $0,1,2,\dots n-1$ are written, in this order and in the same distance to each other. Every number is colored red or blue, and there exists a non-zero number of numbers of each color. It is known that there exists a set $S\subsetneq \{0,1,2,\dots n-1\}, |S|\geq 2$, for wich it holds: if $(x,y), x<y$ is a circle sector whose endpoints are of distinct colors, whose distance $y-x$ is in $S$, then $y$ is in $S$. Prove that there is a divisor $d$ of $n$ different from $1$ and $n$ for wich holds: if $(x,y),x<y$ are different points of distinct colors, such that their distance is divisible by $d$, then both $x,y$ are divisible by $d$.

$A_1,A_2,\cdots,A_8$ are fixed points on a circle. Determine the smallest positive integer $n$ such that among any $n$ triangles with these eight points as vertices, two of them will have a common side.

How many ways are there to line up $19$ girls (all of different heights) in a row so that no girl has a shorter girl both in front of and behind her?

There are $36$ gangsters bands.And there are war between some bands. Every gangster can belongs to several bands and every 2 gangsters belongs to different set of bands. Gangster can not be in feuding bands. Also for every gangster is true, that every band, where this gangster is not in, is in war with some band, where this gangster is in. What is maximum number of gangsters in city?

Does there exist an infinite set $ M$ of straight lines on the coordinate plane such that (i) no two lines are parallel, and (ii) for any integer point there is a line from $ M$ containing it?

Thirty-four countries participated in a jury session of the IMO, each represented by the leader and the deputy leader of the team. Before the meeting, some participants exchanged handshakes, but no team leader shook hands with his deputy. After the meeting, the leader of the Illyrian team asked every other participant the number of people they had shaken hands with, and all the answers she got were different. How many people did the deputy leader of the Illyrian team greet ?

In the game of [i]Ring Mafia[/i], there are $2019$ counters arranged in a circle. $673$ of these counters are mafia, and the remaining $1346$ counters are town. Two players, Tony and Madeline, take turns with Tony going first. Tony does not know which counters are mafia but Madeline does. On Tony’s turn, he selects any subset of the counters (possibly the empty set) and removes all counters in that set. On Madeline’s turn, she selects a town counter which is adjacent to a mafia counter and removes it. Whenever counters are removed, the remaining counters are brought closer together without changing their order so that they still form a circle. The game ends when either all mafia counters have been removed, or all town counters have been removed. Is there a strategy for Tony that guarantees, no matter where the mafia counters are placed and what Madeline does, that at least one town counter remains at the end of the game? [i]Proposed by Andrew Gu[/i]

Frankie the Frog starts his morning at the origin in $\mathbb{R}^2$. He decides to go on a leisurely stroll, consisting of $3^1+3^{10}+3^{11}+3^{100}+3^{111}+3^{1000}$ moves, starting with the first move. On the $n$th move, he hops a distance of $$\max\{k\in\mathbb{Z}:3^k|n\}+1,$$ then turns $90^{\circ}$ counterclockwise. What is the square of the distance from his final position to the origin?

[u]Set 7[/u] [b]G19.[/b] How many ordered triples $(x, y, z)$ with $1 \le x, y, z \le 50$ are there such that both $x + y + z$ and $xy + yz + zx$ are divisible by$ 6$? [b]G20.[/b] Triangle $ABC$ has orthocenter $H$ and circumcenter $O$. If $D$ is the foot of the perpendicular from $A$ to $BC$, then $AH = 8$ and $HD = 3$. If $\angle AOH = 90^o$, find $BC^2$. [b]G21.[/b] Nate flips a fair coin until he gets two heads in a row, immediately followed by a tails. The probability that he flips the coin exactly $12$ times is $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m + n$. [u]Set 8[/u] [b]G22.[/b] Let $f$ be a function defined by $f(1) = 1$ and $$f(n) = \frac{1}{p}f\left(\frac{n}{p}\right)f(p) + 2p - 2,$$ where $p$ is the least prime dividing $n$, for all integers $n \ge 2$. Find $f(2022)$. [b]G23.[/b] Jessica has $15$ balls numbered $1$ through $15$. With her left hand, she scoops up $2$ of the balls. With her right hand, she scoops up $2$ of the remaining balls. The probability that the sum of the balls in her left hand is equal to the sum of the balls in her right hand can be expressed as $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m + n$. [b]G24.[/b] Let $ABCD$ be a cyclic quadrilateral such that its diagonal $BD = 17$ is the diameter of its circumcircle. Given $AB = 8$, $BC = CD$, and that a line $\ell$ through A intersects the incircle of $ABD$ at two points $P$ and $Q$, the maximum area of $CP Q$ can be expressed as a fraction $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Find $m + n$. [u]Set 9[/u] [i]This set consists of three estimation problems, with scoring schemes described.[/i] [b]G25.[/b] Estimate $N$, the total number of participants (in person and online) at MOAA this year. An estimate of $e$ gets a total of max $ \left( 0, \lfloor 150 \left( 1- \frac{|N-e|}{N}\right) \rfloor -120 \right)$ points. [b]G26.[/b] If $A$ is the the total number of in person participants at MOAA this year, and $B$ is the total number of online participants at MOAA this year, estimate $N$, the product $AB$. An estimate of $e$ gets a total of max $(0, 30 - \lceil \log10(8|N - e| + 1)\rceil )$ points. [b]G27.[/b] Estimate $N$, the total number of letters in all the teams that signed up for MOAA this year, both in person and online. An estimate of e gets a total of max $(0, 30 - \lceil 7 log5(|N - E|)\rceil )$ points. PS. You should use hide for answers. Sets 1-3 have been posted [url=https://artofproblemsolving.com/community/c3h3131303p28367061]here [/url] and 4-6 [url=https://artofproblemsolving.com/community/c3h3131305p28367080]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Determine the number of $3-$digit numbers in base $10$ having at least one $5$ and at most one $3$.

Let $F$ be a polygon the boundary of which is a broken line with vertices in the knots (units) of a given in advance regular square network. If $k$ is the count of knots of the network situated over the boundary of $F$, and $\ell$ is the count of the knots of the network lying inside $F$, prove that if the surface of every square from the network is $1$, then the surface $S$ of $F$ is calculated with the formulae: $$S=\frac k2+\ell-1$$ [i]V. Chukanov[/i]

On the plane we mark $n$ ($n > 2$) straight lines passing through one point $O$ in such a way that for any two of them there is a marked straight line that bisects one of the pairs of vertical angles, formed by these straight lines. Prove that the drawn straight lines divide full angle into equal parts.

[b]p1.[/b] Compute the greatest integer less than or equal to $$\frac{10 + 12 + 14 + 16 + 18 + 20}{21}$$ [b]p2.[/b] Let$ A = 1$.$B = 2$, $C = 3$, $...$, $Z = 26$. Find $A + B +M + C$. [b]p3.[/b] In Mr. M's farm, there are $10$ cows, $8$ chickens, and $4$ spiders. How many legs are there (including Mr. M's legs)? [b]p4.[/b] The area of an equilateral triangle with perimeter $18$ inches can be expressed in the form $a\sqrt{b}{c}$ , where $a$ and $c$ are relatively prime and $b$ is not divisible by the square of any prime. Find $a + b + c$. [b]p5.[/b] Let $f$ be a linear function so $f(x) = ax + b$ for some $a$ and $b$. If $f(1) = 2017$ and $f(2) = 2018$, what is $f(2019)$? [b]p6.[/b] How many integers $m$ satisfy $4 < m^2 \le 216$? [b]p7.[/b] Allen and Michael Phelps compete at the Olympics for swimming. Allen swims $\frac98$ the distance Phelps swims, but Allen swims in $\frac59$ of Phelps's time. If Phelps swims at a rate of $3$ kilometers per hour, what is Allen's rate of swimming? The answer can be expressed as $m/n$ for relatively prime positive integers $m, n$. Find $m + n$. [b]p8.[/b] Let $X$ be the number of distinct arrangements of the letters in "POONAM," $Y$ be the number of distinct arrangements of the letters in "ALLEN" and $Z$ be the number of distinct arrangements of the letters in "NITHIN." Evaluate $\frac{X+Z}{Y}$ : [b]p9.[/b] Two overlapping circles, both of radius $9$ cm, have centers that are $9$ cm apart. The combined area of the two circles can be expressed as $\frac{a\pi+b\sqrt{c}+d}{e}$ where $c$ is not divisible by the square of any prime and the fraction is simplified. Find $a + b + c + d + e$. [b]p10.[/b] In the Boxborough-Acton Regional High School (BARHS), $99$ people take Korean, $55$ people take Maori, and $27$ people take Pig Latin. $4$ people take both Korean and Maori, $6$ people take both Korean and Pig Latin, and $5$ people take both Maori and Pig Latin. $1$ especially ambitious person takes all three languages, and and $100$ people do not take a language. If BARHS does not oer any other languages, how many students attend BARHS? [b]p11.[/b] Let $H$ be a regular hexagon of side length $2$. Let $M$ be the circumcircle of $H$ and $N$ be the inscribed circle of $H$. Let $m, n$ be the area of $M$ and $N$ respectively. The quantity $m - n$ is in the form $\pi a$, where $a$ is an integer. Find $a$. [b]p12.[/b] How many ordered quadruples of positive integers $(p, q, r, s)$ are there such that $p + q + r + s \le 12$? [b]p13.[/b] Let $K = 2^{\left(1+ \frac{1}{3^2} \right)\left(1+ \frac{1}{3^4} \right)\left(1+ \frac{1}{3^8}\right)\left(1+ \frac{1}{3^{16}} \right)...}$. What is $K^8$? [b]p14.[/b] Neetin, Neeton, Neethan, Neethine, and Neekhil are playing basketball. Neetin starts out with the ball. How many ways can they pass 5 times so that Neethan ends up with the ball? [b]p15.[/b] In an octahedron with side lengths $3$, inscribe a sphere. Then inscribe a second sphere tangent to the first sphere and to $4$ faces of the octahedron. The radius of the second sphere can be expressed in the form $\frac{\sqrt{a}-\sqrt{b}}{c}$ , where the square of any prime factor of $c$ does not evenly divide into $b$. Compute $a + b + c$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Every integer is painted white or black, so that if $m$ is white then $m + 20$ is also white, and if $k$ is black then $k + 35$ is also black. For which $n$ can exactly $n$ of the numbers $1, 2, ..., 50$ be white?

Is it possible to fill a $3 \times 3$ grid with each of the numbers $1,2,\ldots,9$ once each such that the sum of any two numbers sharing a side is prime?

Fix positive integers $n$ and $k$ such that $2 \le k \le n$ and a set $M$ consisting of $n$ fruits. A [i]permutation[/i] is a sequence $x=(x_1,x_2,\ldots,x_n)$ such that $\{x_1,\ldots,x_n\}=M$. Ivan [i]prefers[/i] some (at least one) of these permutations. He realized that for every preferred permutation $x$, there exist $k$ indices $i_1 < i_2 < \ldots < i_k$ with the following property: for every $1 \le j < k$, if he swaps $x_{i_j}$ and $x_{i_{j+1}}$, he obtains another preferred permutation. \\ Prove that he prefers at least $k!$ permutations.

There're $n$ students whose names are different from each other. Everyone has $n-1$ envelopes initially with the others' name and address written on them respectively. Everyone also has at least one greeting card with her name signed on it. Everyday precisely a student encloses a greeting card (which can be the one received before) with an envelope (the name on the card and the name on envelope cannot be the same) and post it to the appointed student by a same day delivery. Prove that when no one can post the greeting cards in this way any more: (i) Everyone still has at least one card; (ii) If there exist $k$ students $p_1, p_2, \cdots, p_k$ so that $p_i$ never post a card to $p_{i+1}$, where $i = 1,2, \cdots, k$ and $p_{k+1} = p_1$, then these $k$ students have prepared the same number of greeting cards initially.

Let $f(n)$ be the smallest number of 1s needed to represent the positive integer $n$ using only 1s, $+$ signs, $\times$ signs and brackets $(,)$. For example, you could represent 80 with 13 1s as follows: $(1+1+1+1+1)(1+1+1+1)(1+1+1+1)$. Show that $3 \log(n) \leq \log(3)f(n) \leq 5 \log(n)$ for $n > 1$.

The graph $G$ with 2014 vertices doesn’t contain any 3-cliques. If the set of the degrees of the vertices of $G$ is $\{1,2,...,k\}$, find the greatest possible value of $k$.

Let $P_1, P_2, \dots, P_n$ be points on the plane. There is an edge between distinct points $P_x, P_y$ if and only if $x \mid y$. Find the largest $n$, such that the graph can be drawn with no crossing edges.

A $(0_x, 1_y, 2_z)$-string is an infinite ternary string such that: [list] [*] If there is a $0$ in position $i$ then there is a $1$ in position $i + x$, [*] if there is a $1$ in position $j$ then there is a $2$ in position $j + y$, [*] if there is a $2$ in position $k$ then there is a $0$ in position $k + z$. [/list] For how many ordered triples of positive integers $(x, y, z)$ with $x, y, z \leq 100$ does there exist $(0_x, 1_y, 2_z)$-string?

In a village, there are $n$ houses with $n>2$ and all of them are not collinear. We want to generate a water resource in the village. For doing this, point $A$ is [i]better[/i] than point $B$ if the sum of the distances from point $A$ to the houses is less than the sum of the distances from point $B$ to the houses. We call a point [i]ideal[/i] if there doesn’t exist any [i]better[/i] point than it. Prove that there exist at most $1$ [i]ideal[/i] point to generate the resource.

On a $(4n + 2)\times (4n + 2)$ square grid, a turtle can move between squares sharing a side.The turtle begins in a corner square of the grid and enters each square exactly once, ending in the square where she started. In terms of $n$, what is the largest positive integer $k$ such that there must be a row or column that the turtle has entered at least $k$ distinct times?

There are $30$ children, $a_1,a_2,...,a_{30}$ seated clockwise in a circle on the floor. The teacher walks behind the children in the clockwise direction with a box of $1000$ candies. She drops a candy behind the first child $a_1$. She then skips one child and drops a candy behind the third child, $a_3$. Now she skips two children and drops a candy behind the next child, $a_6$. She continues this way, at each stage skipping one child more than at the preceding stage before dropping a candy behind the next child. How many children will never receive a candy? Justify your answer.