On a circle there are $n$ points such that every point has a distinct number. Determine the number of ways of choosing $k$ points such that for any point there are at least 3 points between this point and the nearest point. (clockwise) ($n,k\geq 2$)

Let a $9$-digit number be balanced if it has all numerals $1$ to $9$. Let $S$ be the sequence of the numerals which is constructed by writing all balanced numbers in increasing order consecutively. Find the least possible value of $k$ such that any two subsequences of $S$ which has consecutive $k$ numerals are different from each other.

The vertices of a regular $2012$-gon are labelled with the numbers $1$ through $2012$ in some order. Call a vertex a peak if its label is larger than the label of its two neighbours, and a valley if its label is smaller than the label of its two neighbours. Show that the total number of peaks is equal to the total number of valleys.

Let $k\geq 2$ be an integer. We want to determine the weight of $n$ balls. One try consists of choosing two balls, and we are given the sum of the weights of the two chosen balls. We know that at most $k$ of the answers can be wrong. Let $f_k(n)$ denote the smallest number for which it is true that we can always find the weights of the balls with $f_k(n)$ tries (the tries don't have to be decided in advance). Prove that there exist numbers $a_k$ and $b_k$ for which $|f_k(n)-a_kn|\leq b_k$ holds. [i]Proposed by Surányi László, Budapest and Bálint Virág, Toronto[/i]

[b]p1.[/b] What is $2 \cdot 24 + 20 \cdot 24 + 202 \cdot 4 + 2024$? [b]p2.[/b] Jerry has $300$ legos. Tie can either make cars, which require $17$ legos, or bikes, which require $13$ legos. Assuming he uses all of his legos, how many ordered pairs $(a, b)$ are there such that he makes $a$ cars and $b$ bikes? [b]p3.[/b] Patrick has $7$ unique textbooks: $2$ Geometry books, $3$ Precalculus books and $2$ Algebra II books. How many ways can he arrange his books on a bookshelf such that all the books of the same subjects are adjacent to each other? [b]p4.[/b] After a hurricane, a $32$ meter tall flagpole at the Act on-Boxborough Regional High School snapped and fell over. Given that the snapped part remains in contact with the original pole, and the top of the polo falls $24$ meters away from the bottom of the pole, at which height did the polo snap? (Assume the flagpole is perpendicular to the ground.) [b]p5.[/b] Jimmy is selling lemonade. Iio has $200$ cups of lemonade, and he will sell them all by the end of the day. Being the ethically dubious individual he is, Jimmy intends to dilute a few of the cups of lemonade with water to conserve resources. Jimmy sells each cup for $\$4$. It costs him $\$ 1$ to make a diluted cup of lemonade, and it costs him $\$2.75$ to make a cup of normal lemonade. What is the minimum number of diluted cups Jimmy must sell to make a profit of over $\$400$? [b]p6.[/b] Jeffrey has a bag filled with five fair dice: one with $4$ sides, one with $6$ sides, one with $8$ sides, one with $12$ sides, and one with $20$ sides. The dice are numbered from $1$ to the number of sides on the die. Now, Marco will randomly pick a die from .Jeffrey's bag and roll it. The probability that Marco rolls a $7$ can be expressed as $\frac{a}{b}$ where $a$ and $b$ are relatively prime positive integers. Find $a+b$. [b]p7.[/b] What is the remainder when the sum of the first $2024$ odd numbers is divided by $6072$? [b]p8.[/b] A rhombus $ABCD$ with $\angle A = 60^o$ and $AB = 600$ cm is drawn on a piece of paper. Three ants start moving from point $A$ to the three other points on the rhombus. One ant walks from $A$ to $B$ at a leisurely speed of $10$ cm/s. The second ant runs from $A$ to $C$ at a slightly quicker pace of $6\sqrt3$ cm/s, arriving to $C$ $x$ seconds after the first ant. The third ant travels from $A$ to $B$ to $D$ at a constant speed, arriving at $D$ $x$ seconds after the second ant. The speed of the last ant can be written as $\frac{m}{n}$ cm/s, where $m$ and $n$ are relatively prime positive integers. Find $mn$. [b]p9.[/b] This year, the Apple family has harvested so many apples that they cannot sell them all! Applejack decides to make $40$ glasses of apple cider to give to her friends. If Twilight and Fluttershy each want $1$ or $2$ glasses; Pinkie Pic wants cither $2$, $14$, or $15$ glasses; Rarity wants an amount of glasses that is a power of three; and Rainbow Dash wants any odd number of glasses, then how many ways can Applejack give her apple cider to her friends? Note: $1$ is considered to be a power of $3$. [b]p10.[/b] Let $g_x$ be a geometric sequence with first term $27$ and successive ratio $2n$ (so $g_{x+1}/g_x = 2n$). Then, define a function $f$ as $f(x) = \log_n(g_x)$, where $n$ is the base of the logarithm. It is known that the sum of the first seven terms of $f(x)$ is $42$. Find $g_2$, the second term of the geometric sequence. Note: The logarithm base $b$ of $x$, denoted $\log_b(x)$ is equal to the value $y$ such that $b^y = x$. In other words, if $\log_b(x) = y$, then $b^y = x$. [b]p11.[/b] Let $\varepsilon$ be an ellipse centered around the origin, such that its minor axis is perpendicular to the $x$-axis. The length of the ellipse's major and minor axes is $8$ and $6$, respectively. Then, let $ABCD$ be a rectangle centered around the origin, such that $AB$ is parallel to the $x$-axis. The lengths of $AB$ and $BC$ are $8$ and $3\sqrt2$, respectively. The area outside the ellipse but inside the rectangle can be expressed as $a\sqrt{b}-c-d\pi$, for positive integers $a$, $b$, $c$, $d$ where $b$ is not divisible by a perfect square of any prime. Find $a + b + c + d$. [img]https://cdn.artofproblemsolving.com/attachments/e/c/9d943966763ee7830d037ef98c21139cf6f529.png[/img] [b]p12.[/b] Let $N = 2^7 \cdot 3^7 \cdot 5^5$. Find the number of ways to express $N$ as the product of squares and cubes, all of which are integers greater than $1$. [b]p13.[/b] Jerry and Eric are playing a $10$-card game where Jerry is deemed the ’’landlord" and Eric is deemed the ' peasant'’. To deal the cards, the landlord keeps one card to himself. Then, the rest of the $9$ cards are dealt out, such that each card has a $1/2$ chance to go to each player. Once all $10$ cards are dealt out, the landlord compares the number of cards he owns with his peasant. The probability that the landlord wins is the fraction of cards he has. (For example, if Jerry has $5$ cards and Eric has $2$ cards, Jerry has a$ 5/7$ ths chance of winning.) The probability that Jerry wins the game can be written as $\frac{p}{q}$ where $p$ and $q$ are relatively prime. Find $p + q$. [b]p14.[/b] Define $P(x) = 20x^4 + 24x^3 + 10x^2 + 21x+ 7$ to have roots $a$, $b$, $c$, and $d$. If $Q(x)$ has roots $\frac{1}{a-2}$,$\frac{1}{b-2}$,$ \frac{1}{c-2}$, $\frac{1}{d-2}$ and integer coefficients with a greatest common divisor of $1$, then find $Q(2)$. [b]p15.[/b] Let $\vartriangle ABC$ be a triangle with side lengths $AB = 14$, $BC = 13$, and $AC = 15$. The incircle of $\vartriangle ABC$ is drawn with center $I$, tangent to $\overline{AB}$ at $X$. The line $\overleftrightarrow{IX}$ intersects the incircle again at $Y$ and intersects $\overline{AC}$ at $Z$. The area of $\vartriangle AYZ$ can be expressed as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

[b]p1.[/b] The edges of a tetrahedron are all tangent to a sphere. Prove that the sum of the lengths of any pair of opposite edges equals the sum of the lengths of any other pair of opposite edges. (Two edges of a tetrahedron are said to be opposite if they do not have a vertex in common.) [b]p2.[/b] Find the simplest formula possible for the product of the following $2n - 2$ factors: $$\left(1+\frac12 \right),\left(1-\frac12 \right), \left(1+\frac13 \right) , \left(1-\frac13 \right),...,\left(1+\frac{1}{n} \right), \left(1-\frac{1}{n} \right)$$. Prove that your formula is correct. [b]p3.[/b] Solve $$\frac{(x + 1)^2+1}{x + 1} + \frac{(x + 4)^2+4}{x + 4}=\frac{(x + 2)^2+2}{x + 2}+\frac{(x + 3)^2+3}{x + 3}$$ [b]p4.[/b] Triangle $ABC$ is inscribed in a circle, $BD$ is tangent to this circle and $CD$ is perpendicular to $BD$. $BH$ is the altitude from $B$ to $AC$. Prove that the line $DH$ is parallel to $AB$. [img]https://cdn.artofproblemsolving.com/attachments/e/9/4d0b136dca4a9b68104f00300951837adef84c.png[/img] [b]p5.[/b] Consider the picture below as a section of a city street map. There are several paths from $A$ to $B$, and if one always walks along the street, the shortest paths are $15$ blocks in length. Find the number of paths of this length between $A$ and $B$. [img]https://cdn.artofproblemsolving.com/attachments/8/d/60c426ea71db98775399cfa5ea80e94d2ea9d2.png[/img] [b]p6.[/b] A [u]finite [/u] [u]graph [/u] is a set of points, called [u]vertices[/u], together with a set of arcs, called [u]edges[/u]. Each edge connects two of the vertices (it is not necessary that every pair of vertices be connected by an edge). The [u]order [/u] of a vertex in a finite graph is the number of edges attached to that vertex. [u]Example[/u] The figure at the right is a finite graph with $4$ vertices and $7$ edges. [img]https://cdn.artofproblemsolving.com/attachments/5/9/84d479c5dbd0a6f61a66970e46ab15830d8fba.png[/img] One vertex has order $5$ and the other vertices order $3$. Define a finite graph to be [u]heterogeneous [/u] if no two vertices have the same order. Prove that no graph with two or more vertices is heterogeneous. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

$n$ positive numbers are given. Is it always possible to find a convex polygon with $n+3$ edges and a triangulation of it so that the length of the diameters used in the triangulation are the given $n$ numbers? [i]Proposed by Morteza Saghafian[/i]

Each $1 \times 1$ square of an $n \times n$ table contains a different number. The smallest number in each row is marked, and these marked numbers are in different columns. Then the smallest number in each column is marked, and these marked numbers are in different rows. Prove that the two sets of marked numbers are identical. (V Klepcyn)

In a football tournament with $n\geq 2$ teams each two played a match. For a won match the victor gets 2 points and for a draw each one gets 1 point. In the final results there weren’t two teams with equal amount of points. It turned out that because of a mistake each match that was written in the results as won was actually a draw and each one that was written as draw was actually won. In the new ranking there were also no two teams with the same amount of points. Find all n for which it is possible for the two rankings to be opposite of each other, that is the first team in the first ranking is actually the last one, the second team is pre-last and so on.

The integers from $1$ to $36$ are written on a “mathlotto” ticket. When you buy a “mathlotto” ticket, you choose $6$ of these $36$ numbers. Then $6$ of the integers from $1$ to $36$ are drawn, and a winning ticket is one which does not contain any of them. Prove that (a) if you buy $9$ tickets, you can choose your numbers so that regardless of which numbers are drawn, you are guaranteed to have at least one winning ticket; (b) if you buy only $8$ tickets, it is possible for you not to have any winning tickets, regardless of how you choose your numbers. (S Tokarev)

Let $G$ be an undirected simple graph. Let $f(G)$ be the number of ways to orient all of the edges of $G$ in one of the two possible directions so that the resulting directed graph has no directed cycles. Show that $f(G)$ is a multiple of $3$ if and only if $G$ has a cycle of odd length.

Let be given $n$ points, no three of which are on a line. All the segments with endpoints in these points are colored so that two segments with a common endpoint are of different colors. Determine the least number of colors for which this is possible

What sizes of squares with integer sides can be completely covered without overlap by identical tiles consisting of three squares with side $1$ joined together in one $L$ shape? [center][img]https://cdn.artofproblemsolving.com/attachments/3/f/9fe95b05527857f7e44dfd033e6fb01e5d25a2.png[/img][/center]

21 people take a test with 15 true or false questions. It is known that every 2 people have at least 1 correct answer in common. What is the minimum number of people that could have correctly answered the question which the most people were correct on?

Let $n\ge 3$ be an integer and let $A_{1}, A_{2},\dots, A_{n}$ be $n$ distinct subsets of $S=\{1, 2,\dots, n\}$. Show that there exists $x\in S$ such that the n subsets $A_{i}-\{x\}, i=1,2,\dots n$ are also disjoint. what i have is [hide="this"]we may assume that the union of the $A_{i}$s is $S$. [/hide]

In Geneva there are $16$ secret agents, each of whom is watching one or more other agents. It is known that if agent $A$ is watching agent $B$, then $B$ is not watching $A$. Moreover, any $10$ agents can be ordered so that the first is watching the second, the second is watching the third, etc, the last is watching the first. Show that any $11$ agents can also be so ordered.

Natural numbers $1,2,...,500$ are written on a blackboard. Two players $A$ and $B$ consecutively make moves, $A$ starts. Each move a player chooses two numbers $n$ and $2n$ and erases them from the blackboard. If a player cannot perform a valid move, he loses. Which player can guarantee a win?

You wish to color every vertex, edge, face, and the interior of a cube one color each such that no two adjacent objects are the same color. Faces are adjacent if they share an edge. Edges are adjacent if they share a vertex. The interior is adjacent to all of its faces, edges, and vertices. Each face is adjacent to all of its edges and vertices. Each edge is adjacent to both of its vertices. What is the minimum number of colors required to do this?

[b]p1.[/b] What is the smallest positive integer $x$ such that $\frac{1}{x} <\sqrt{12011} - \sqrt{12006}$? [b]p2. [/b] Two soccer players run a drill on a $100$ foot by $300$ foot rectangular soccer eld. The two players start on two different corners of the rectangle separated by $100$ feet, then run parallel along the long edges of the eld, passing a soccer ball back and forth between them. Assume that the ball travels at a constant speed of $50$ feet per second, both players run at a constant speed of $30$ feet per second, and the players lead each other perfectly and pass the ball as soon as they receive it, how far has the ball travelled by the time it reaches the other end of the eld? [b]p3.[/b] A trapezoid $ABCD$ has $AB$ and $CD$ both perpendicular to $AD$ and $BC =AB + AD$. If $AB = 26$, what is $\frac{CD^2}{AD+CD}$ ? [b]p4.[/b] A hydrophobic, hungry, and lazy mouse is at $(0, 0)$, a piece of cheese at $(26, 26)$, and a circular lake of radius $5\sqrt2$ is centered at $(13, 13)$. What is the length of the shortest path that the mouse can take to reach the cheese that also does not also pass through the lake? [b]p5.[/b] Let $a, b$, and $c$ be real numbers such that $a + b + c = 0$ and $a^2 + b^2 + c^2 = 3$. If $a^5 + b^5 + c^5\ne 0$, compute $\frac{(a^3+b^3+c^3)(a^4+b^4+c^4)}{a^5+b^5+c^5}$. [b]p6. [/b] Let $S$ be the number of points with integer coordinates that lie on the line segment with endpoints $\left( 2^{2^2}, 4^{4^4}\right)$ and $\left(4^{4^4}, 0\right)$. Compute $\log_2 (S - 1)$. [b]p7.[/b] For a positive integer $n$ let $f(n)$ be the sum of the digits of $n$. Calculate $$f(f(f(2^{2006})))$$ [b]p8.[/b] If $a_1, a_2, a_3, a_4$ are roots of $x^4 - 2006x^3 + 11x + 11 = 0$, find $|a^3_1 + a^3_2 + a^3_3 + a^3_4|$. [b]p9.[/b] A triangle $ABC$ has $M$ and $N$ on sides $BC$ and $AC$, respectively, such that $AM$ and $BN$ intersect at $P$ and the areas of triangles $ANP$, $APB$, and $PMB$ are $5$, $10$, and $8$ respectively. If $R$ and $S$ are the midpoints of $MC$ and $NC$, respectively, compute the area of triangle $CRS$. [b]p10.[/b] Jack's calculator has a strange button labelled ''PS.'' If Jack's calculator is displaying the positive integer $n$, pressing PS will cause the calculator to divide $n$ by the largest power of $2$ that evenly divides $n$, and then adding 1 to the result and displaying that number. If Jack randomly chooses an integer $k$ between $ 1$ and $1023$, inclusive, and enters it on his calculator, then presses the PS button twice, what is the probability that the number that is displayed is a power of $2$? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

On an $8 \times 8$ chessboard, a rook stands on the bottom left corner square. We want to move it to the upper right corner, subject to the following rules: we have to move the rook exactly $9$ times, such that the length of each move is either $3$ or $4$. (It is allowed to mix the two lengths throughout the "journey".) How many ways are there to do this? In each move, the rook moves horizontally or vertically.

A deck consists of $2^n$ cards. The deck is shuffled using the following operation: if the cards are initially in the order $a_1,a_2,a_3,a_4,...,a_{2^n-1},a_{2^n}$ then after shuffling the order becomes $a_{2^{n-1}+1},a_1,a_{2^{n-1}+2},a_2,...,a_{2^n},a_{2^{n-1}}$ . Find the smallest number of such operations after which the original order of the cards is restored. (R. Palm)

A cube 10x10x10 is constructed from 1000 white unit cubes. Polly and Velly play the following game: Velly chooses a certain amount of parallelepipeds 1x1x10, no two of which have a common vertex or an edge, and repaints them in black. Polly can choose an arbitrary number of unit cubes and ask Velly for their color. What’s the least amount of unit cubes she has to choose so that she can determine the color of each unit cube?

$ $ $ $ $ $ $ $There is a group of more than three airports. For any two airports $A, B$ belonging to this group, if there is an aircraft from $A$ to $ $ $B$, there is an aircraft from $B$ to $ $ $A$. For a list of different airports $A_0,A_1,...A_n$, define this list as a '[color=#00f]route[/color]' if there is an aircraft from $A_i$ to $A_{i+1}$ for each $i=0,1,...,n-1$. Also, define the beginning of this [color=#00f]route[/color] as $A_0$, the end as $A_n$, and the length as $n$. ($n\in \mathbb N$) $ $ $ $ $ $ $ $Now, let's say that for any three different pairs of airports $(A,B,C)$, there is always a [color=#00f]route[/color] $P$ that satisfies the following condition. $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ [b]Condition[/b]: $P$ begins with $A$ and ends with $B$, and does not include $C$. $ $ $ $ $ $When the length of the longest of the existing [color=#00f]route[/color]s is $M$ ($\ge 2$), prove that any two [color=#00f]route[/color]s of length $M$ contain at least two different airports simultaneously.

Let $S =\{1, 2,..., 6\}$. How many functions $f : S \to S$ are there such that for all $s \in S$, $$f^5(s) = f(f(f(f(f(s))))) = 1?$$

Peter and Basil play the following game on a horizontal table $1\times{2019}$. Initially Peter chooses $n$ positive integers and writes them on a board. After that Basil puts a coin in one of the cells. Then at each move, Peter announces a number s among the numbers written on the board, and Basil needs to shift the coin by $s$ cells, if it is possible: either to the left, or to the right, by his decision. In case it is not possible to shift the coin by $s$ cells neither to the left, nor to the right, the coin stays in the current cell. Find the least $n$ such that Peter can play so that the coin will visit all the cells, regardless of the way Basil plays.