Some regular polygons are inscribed in a circle. Fedir turned some of them, so all polygons have a common vertice. Prove that the number of vertices did not increase.

Ten localities are served by two international airlines such that there exists a direct service (without stops) between any two of these localities and all airline schedules offer round-trip service between the cities they serve. Prove that at least one of the airlines can offer two disjoint round trips each containing an odd number of landings.

[u]Round 1[/u] [b]1.1.[/b] There are $99$ dogs sitting in a long line. Starting with the third dog in the line, if every third dog barks three times, and all the other dogs each bark once, how many barks are there in total? [b]1.2.[/b] Indigo notices that when she uses her lucky pencil, her test scores are always $66 \frac23 \%$ higher than when she uses normal pencils. What percent lower is her test score when using a normal pencil than her test score when using her lucky pencil? [b]1.3.[/b] Bill has a farm with deer, sheep, and apple trees. He mostly enjoys looking after his apple trees, but somehow, the deer and sheep always want to eat the trees' leaves, so Bill decides to build a fence around his trees. The $60$ trees are arranged in a $5\times 12$ rectangular array with $5$ feet between each pair of adjacent trees. If the rectangular fence is constructed $6$ feet away from the array of trees, what is the area the fence encompasses in feet squared? (Ignore the width of the trees.) [u]Round 2[/u] [b]2.1.[/b] If $x + 3y = 2$, then what is the value of the expression $9^x * 729^y$? [b]2.2.[/b] Lazy Sheep loves sleeping in, but unfortunately, he has school two days a week. If Lazy Sheep wakes up each day before school's starting time with probability $1/8$ independent of previous days, then the probability that Lazy Sheep wakes up late on at least one school day over a given week is $p/q$ for relatively prime positive integers $p, q$. Find $p + q$. [b]2.3.[/b] An integer $n$ leaves remainder $1$ when divided by $4$. Find the sum of the possible remainders $n$ leaves when divided by $20$. [u]Round 3[/u] [b]3.1. [/b]Jake has a circular knob with three settings that can freely rotate. Each minute, he rotates the knob $120^o$ clockwise or counterclockwise at random. The probability that the knob is back in its original state after $4$ minutes is $p/q$ for relatively prime positive integers $p, q$. Find $p + q$. [b]3.2.[/b] Given that $3$ not necessarily distinct primes $p, q, r$ satisfy $p+6q +2r = 60$, find the sum of all possible values of $p + q + r$. [b]3.3.[/b] Dexter's favorite number is the positive integer $x$, If $15x$ has an even number of proper divisors, what is the smallest possible value of $x$? (Note: A proper divisor of a positive integer is a divisor other than itself.) [u]Round 4[/u] [b]4.1.[/b] Three circles of radius $1$ are each tangent to the other two circles. A fourth circle is externally tangent to all three circles. The radius of the fourth circle can be expressed as $\frac{a\sqrt{b}-\sqrt{c}}{d}$ for positive integers $a, b, c, d$ where $b$ is not divisible by the square of any prime and $a$ and $d$ are relatively prime. Find $a + b + c + d$. [b]4.2. [/b]Evaluate $$\frac{\sqrt{15}}{3} \cdot \frac{\sqrt{35}}{5} \cdot \frac{\sqrt{63}}{7}... \cdot \frac{\sqrt{5475}}{73}$$ [b]4.3.[/b] For any positive integer $n$, let $f(n)$ denote the number of digits in its base $10$ representation, and let $g(n)$ denote the number of digits in its base $4$ representation. For how many $n$ is $g(n)$ an integer multiple of $f(n)$? PS. You should use hide for answers. Rounds 5-8 have been posted [url=https://artofproblemsolving.com/community/c3h2784571p24468619]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Sets $A, B$, and $C$ satisfy $|A| = 92$, $|B| = 35$, $|C| = 63$, $|A\cap B| = 16$, $|A\cap C| = 51$, $|B\cap C| = 19$. Compute the number of possible values of$ |A \cap B \cap C|$.

From a bag containing 5 pairs of socks, each pair a different color, a random sample of 4 single socks is drawn. Any complete pairs in the sample are discarded and replaced by a new pair draw from the bag. The process continues until the bag is empty or there are 4 socks of different colors held outside the bag. What is the probability of the latter alternative?

[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].

[b]p1.[/b] $4$ balls are distributed uniformly at random among $6$ bins. What is the expected number of empty bins? [b]p2.[/b] Compute ${150 \choose 20 }$ (mod $221$). [b]p3.[/b] On the right triangle $ABC$, with right angle at$ B$, the altitude $BD$ is drawn. $E$ is drawn on $BC$ such that AE bisects angle $BAC$ and F is drawn on $AC$ such that $BF$ bisects angle $CBD$. Let the intersection of $AE$ and $BF$ be $G$. Given that $AB = 15$,$ BC = 20$, $AC = 25$, find $\frac{BG}{GF}$ . [b]p4.[/b] What is the largest integer $n$ so that $\frac{n^2-2012}{n+7}$ is also an integer? [b]p5.[/b] What is the side length of the largest equilateral triangle that can be inscribed in a regular pentagon with side length $1$? [b]p6.[/b] Inside a LilacBall, you can find one of $7$ different notes, each equally likely. Delcatty must collect all $7$ notes in order to restore harmony and save Kanto from eternal darkness. What is the expected number of LilacBalls she must open in order to do so? PS. You had better use hide for answers.

Jim has two fair $6$-sided dice, one whose faces are labelled from $1$ to $6$, and the second whose faces are labelled from $3$ to $8$. Twice, he randomly picks one of the dice (each die equally likely) and rolls it. Given the sum of the resulting two rolls is $9$, if $\frac{m}{n}$ is the probability he rolled the same die twice where $m, n$ are relatively prime positive integers, then what is $m + n$?

Stoyan and Nikolai have two $100\times 100$ chess boards. Both of them number each cell with the numbers $1$ to $10000$ in some way. Is it possible that for every two numbers $a$ and $b$, which share a common side in Nikolai's board, these two numbers are at a knight's move distance in Stoyan's board (that is, a knight can move from one of the cells to the other one with a move)? [i]Nikolai Beluhov[/i]

A wizard kidnaps $31$ members from party $A$, $28$ members from party $B$, $23$ members from party $C$, and $19$ members from party $D$, keeping them isolated in individual rooms in his castle, where he forces them to work. Every day, after work, the kidnapped people can walk in the park and talk with each other. However, when three members of three different parties start talking with each other, the wizard reconverts them to the fourth party (there are no conversations with $4$ or more people involved). a) Find out whether it is possible that, after some time, all of the kidnapped people belong to the same party. If the answer is yes, determine to which party they will belong. b) Find all quartets of positive integers that add up to $101$ that if they were to be considered the number of members from the four parties, it is possible that, after some time, all of the kidnapped people belong to the same party, under the same rules imposed by the wizard.

The numbers from $1$ to $500$ are written on the board. Two players $A$ and $B$ erase alternately one number at a time, and $A$ deletes the first number. If the sum of the last two number on the board is divisible by $3$, $B$ wins, otherwise $A$ wins. Which player can lay out a strategy that ensures this player's victory?

A number of $n$ lamps ($n\ge 3$) are put at $n$ vertices of a regular $n$-gon. Initially, all the lamps are off. In each step. Lisa will choose three lamps that are located at three vertices of an isosceles triangle and change their states (from off to on and vice versa). Her aim is to turn on all the lamps. At least how many steps are required to do so?

Let $M=\{1,2,\dots,49\}$ be the set of the first $49$ positive integers. Determine the maximum integer $k$ such that the set $M$ has a subset of $k$ elements such that there is no $6$ consecutive integers in such subset. For this value of $k$, find the number of subsets of $M$ with $k$ elements with the given property.

Let $S = \lbrace 1, 2, \ldots, 15 \rbrace$. Let $A_1, A_2, \ldots, A_n$ be $n$ subsets of $S$ which satisfy the following conditions: [b]I.[/b] $|A_i| = 7, i = 1, 2, \ldots, n$; [b]II.[/b] $|A_i \cap A_j| \leq 3, 1 \leq i < j \leq n$ [b]III.[/b] For any 3-element subset $M$ of $S$, there exists $A_k$ such that $M \subset A_k$. Find the smallest possible value of $n$.

A coin is tossed 9 times. Hence $2^{9}$ different outcomes are possible. In how many cases 2 consecutive heads does not appear? [list=1] [*] 34 [*] 55 [*] 89 [*] None of these [/list]

On a blackboard a positive integer $n_0$ is written. Two players, $A$ and $B$ are playing a game, which respects the following rules: $-$ acting alternatively per turn, each player deletes the number written on the blackboard $n_k$ and writes instead one number denoted with $n_{k+1}$ from the set $\left\{n_k-1, \dsp \left\lfloor\frac {n_k}3\right\rfloor\right\}$; $-$ player $A$ starts first deleting $n_0$ and replacing it with $n_1\in\left\{n_0-1, \dsp \left\lfloor\frac {n_0}3\right\rfloor\right\}$; $-$ the game ends when the number on the table is 0 - and the player who wrote it is the winner. Find which player has a winning strategy in each of the following cases: a) $n_0=120$; b) $n_0=\dsp \frac {3^{2002}-1}2$; c) $n_0=\dsp \frac{3^{2002}+1}2$.

[i](5 pts)[/i] Let $n$ be a positive integer, $K = \{1, 2, \dots, n\}$, and $\sigma : K \rightarrow K$ be a function with the property that $\sigma(i) = \sigma(j)$ if and only if $i = j$ (in other words, $\sigma$ is a \textit{bijection}). Show that there is a positive integer $m$ such that \[ \underbrace{\sigma(\sigma( \dots \sigma(i) \dots ))}_\textrm{$m$ times} = i \] for all $i \in K$.

Each pair of vertices of a regular $67$-gon is joined by a line segment. Suppose $n$ of these segments are selected, and each of them is painted one of ten available colors. Find the minimum possible value of $n$ for which, regardless of which $n$ segments were selected and how they were painted, there will always be a vertex of the polygon that belongs to seven segments of the same color.

Reimu has a wooden cube. In each step, she creates a new polyhedron from the previous one by cutting off a pyramid from each vertex of the polyhedron along a plane through the trisection point on each adjacent edge that is closer to the vertex. For example, the polyhedron after the first step has six octagonal faces and eight equilateral triangular faces. How many faces are on the polyhedron after the fifth step?

Let $n$ ($\ge 4$) be an even integer. We label $n$ pairwise distinct real numbers arbitrarily on the $n$ vertices of a regular $n$-gon, and label the $n$ sides clockwise as $e_1, e_2, \ldots, e_n$. A side is called [i]positive[/i] if the numbers on both endpoints are increasing in clockwise direction. An unordered pair of distinct sides $\left\{ e_i,e_j \right\}$ is called [i]alternating[/i] if it satisfies both conditions: (i) $2 \mid (i+j)$; and (ii) if one rearranges the four numbers on the vertices of these two sides $e_i$ and $e_j$ in increasing order $a < b < c < d$, then $a$ and $c$ are the numbers on the two endpoints of one of sides $e_i$ or $e_j$. Prove that the number of alternating pairs of sides and the number of positive sides are of different parity.

Consider an infinite grid $\mathcal G$ of unit square cells. A [i]chessboard polygon[/i] is a simple polygon (i.e. not self-intersecting) whose sides lie along the gridlines of $\mathcal G$. Nikolai chooses a chessboard polygon $F$ and challenges you to paint some cells of $\mathcal G$ green, such that any chessboard polygon congruent to $F$ has at least $1$ green cell but at most $2020$ green cells. Can Nikolai choose $F$ to make your job impossible? [i]Nikolai Beluhov[/i]

$10$ persons sit around a circular table and on the table there are $22$ vases. Two persons can see each other if and only if there are no vases aligned with them. Prove that there are at least two people who can see each other.

In each square of a $3 \times 3$ board a real number is written. The element of the $i$ -th row and the $j$ -th column is equal to abso;uteof the difference of the sum of the elements of column $j$ and the sum of the elements of row $i$. Prove that every element of the board is equal to the sum or difference of two other elements on the board.

Keith has $10$ coins labeled $1$ through $10$, where the $i$th coin has weight $2^i$. The coins are all fair, so the probability of flipping heads on any of the coins is $\tfrac{1}{2}$. After flipping all of the coins, Keith takes all of the coins which land heads and measures their total weight, $W$. If the probability that $137\le W\le 1061$ is $\tfrac{m}{n}$ for coprime positive integers $m,n$, determine $m+n$.

Given $n$ points in a circle, Arnaldo write 0 or 1 in all the points. Bernado can do a operation, he can chosse some point and change its number and the numbers of the points on the right and left side of it. Arnaldo wins if Bernado can´t change all the numbers in the circle to 0, and Bernado wins if he can a) Show that Bernado can win if $n=101$ b) Show that Arnaldo wins if $n=102$