There are $2008$ blue, $2009$ red and $2010$ yellow chips on a table. At each step, one chooses two chips of different colors, and recolor both of them using the third color. Can all the chips be of the same color after some steps? Prove your answer.

Let \(n\ge3\) be a fixed integer, and let \(\alpha\) be a fixed positive real number. There are \(n\) numbers written around a circle such that there is exactly one \(1\) and the rest are \(0\)'s. An [i]operation[/i] consists of picking a number \(a\) in the circle, subtracting some positive real \(x\le a\) from it, and adding \(\alpha x\) to each of its neighbors. Find all pairs \((n,\alpha)\) such that all the numbers in the circle can be made equal after a finite number of operations. [i]Proposed by Anthony Wang[/i]

[b]p1.[/b] Find a real number $x$ for which $x\lfloor x \rfloor = 1234.$ Note: $\lfloor x\rfloor$ is the largest integer less than or equal to $x$. [b]p2.[/b] Let $C_1$ be a circle of radius $1$, and $C_2$ be a circle that lies completely inside or on the boundary of $C_1$. Suppose$ P$ is a point that lies inside or on $C_2$. Suppose $O_1$, and $O_2$ are the centers of $C_1$, and $C_2$, respectively. What is the maximum possible area of $\vartriangle O_1O_2P$? Prove your answer. [b]p3.[/b] The numbers $1, 2, . . . , 99$ are written on a blackboard. We are allowed to erase any two distinct (but perhaps equal) numbers and replace them by their nonnegative difference. This operation is performed until a single number $k$ remains on the blackboard. What are all the possible values of $k$? Prove your answer. Note: As an example if we start from $1, 2, 3, 4$ on the board, we can proceed by erasing $1$ and $2$ and replacing them by $1$. At that point we are left with $1, 3, 4$. We may then erase $3$ and $4$ and replacethem by $1$. The last step would be to erase $1$, $1$ and end up with a single $0$ on the board. [b]p4.[/b] Let $a, b$ be two real numbers so that $a^3 - 6a^2 + 13a = 1$ and $b^3 - 6b^2 + 13b = 19$. Find $a + b$. Prove your answer. [b]p5.[/b] Let $m, n, k$ be three positive integers with $n \ge k$. Suppose $A =\prod_{1\le i\le j\le m} gcd(n + i, k + j) $ is the product of $gcd(n + i, k + j)$, where $i, j$ range over all integers satisfying $1\le i\le j\le m$. Prove that the following fraction is an integer $$\frac{A}{(k + 1) \dots(k + m)}{n \choose k}.$$ Note: $gcd(a, b)$ is the greatest common divisor of $a$ and $b$, and ${n \choose k}= \frac{n!}{k!(n - k)!}$ PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $n> 2$ be a positive integer. Given is a horizontal row of $n$ cells where each cell is painted blue or red. We say that a block is a sequence of consecutive boxes of the same color. Arepito the crab is initially standing at the leftmost cell. On each turn, he counts the number $m$ of cells belonging to the largest block containing the square he is on, and does one of the following: If the square he is on is blue and there are at least $m$ squares to the right of him, Arepito moves $m$ squares to the right; If the square he is in is red and there are at least $m$ squares to the left of him, Arepito moves $m$ cells to the left; In any other case, he stays on the same square and does not move any further. For each $n$, determine the smallest integer $k$ for which there is an initial coloring of the row with $k$ blue cells, for which Arepito will reach the rightmost cell.

Let $X$ be a set of $n + 1$ elements, $n\geq 2$. Ordered $n$-tuples $(a_1,\ldots,a_n)$ and $(b_1,\ldots,b_n)$ formed from distinct elements of $X$ are called[i] disjoint [/i]if there exist distinct indices $1\leq i \neq j\leq n$ such that $a_i = b_j$. Find the maximal number of pairwise disjoint $n$-tuples.

1. A strip for playing "hopscotch" consists of ten squares numbered consecutively $1,2, \ldots, 10$. Clarissa and Marissa start from the center of the first square, jump 9 times to the centers of the other squares so that they visit each square once, and end up at the tenth square. (Jumps forward and backward are allowed.) Each jump of Clarissa was for the same distance as the corresponding jump of Marissa. Does this mean that they both visited the squares in the same order? Alexey Tolpygo

A natural number \(N\) is given. A cube with side length \(2N + 1\) is made up of \((2N + 1)^3\) unit cubes, each of which is either black or white. It turns out that among any $8$ cubes that share a common vertex and form a \(2 \times 2 \times 2\) cube, there are at most $4$ black cubes. What is the maximum number of black cubes that could have been used?

Consider all words containing only letters $A$ and $B$. For any positive integer $n$, $p(n)$ denotes the number of all $n$-letter words without four consecutive $A$'s or three consecutive $B$'s. Find the value of the expression \[\frac{p(2004)-p(2002)-p(1999)}{p(2001)+p(2000)}.\]

Uri must paint some integers from $1$ to $2022$ (inclusive) in red, such that none of the differences between two red numbers is a prime number. Determine the maximum number of numbers Uri can paint red. [b]Note 1:[/b] The [i]difference [/i]between two distinct numbers is the subtraction of the larger minus the smaller. [b]Note 2:[/b] $1$ is not a prime number.

A teacher wants to divide the $2010$ questions she asked in the exams during the school year into three folders of $670$ questions and give each folder to a student who solved all $670$ questions in that folder. Determine the minimum number of students in the class that makes this possible for all possible situations in which there are at most two students who did not solve any given question.

If you roll four fair $6$-sided dice, what is the probability that at least three of them will show the same value?

How many [i]connected subsequences [/i](i.e, consisting of one element or consecutive elements) of the following sequence are there: $1,2,...,100$? [b]A.[/b] $1010$ [b]B.[/b] $2020$ [b]C.[/b] $3030$ [b]D.[/b] $4040$ [b]E.[/b] $5050$

Twenty-five of the numbers $1, 2, \cdots , 50$ are chosen. Twenty-five of the numbers$ 51, 52, \cdots, 100$ are also chosen. No two chosen numbers differ by $0$ or $50$. Find the sum of all $50$ chosen numbers.

There is a football championship with $6$ teams involved, such that for any $2$ teams $A$ and $B$, $A$ plays with $B$ and $B$ plays with $A$ ($2$ such games are distinct). After every match, the winning teams gains $3$ points, the loosing team gains $0$ points and if there is a draw, both teams gain $1$ point each.\\ \\ In the end, the team standing on the last place has $12$ points and there are no $2$ teams that scored the same amount of points.\\ \\ For all the remaining teams, find their final scores and provide an example with the outcomes of all matches for at least one of the possible final situations. $\textit{(Andrei Bâra)}$

Let $T$ be the set of all 2-digit numbers whose digits are in $\{1,2,3,4,5,6\}$ and the tens digit is strictly smaller than the units digit. Suppose $S$ is a subset of $T$ such that it contains all six digits and no three numbers in $S$ use all six digits. If the cardinality of $S$ is $n$, find all possible values of $n$.

Given a set $M$ of $1985$ positive integers, none of which has a prime divisor larger than $26$, prove that the set has four distinct elements whose geometric mean is an integer.

We color each unit square of a $8\times 8$ table into green or blue such that there are $a$ green unit squares in each $3 \times 3$ square and $b$ green unit squares in each $2 \times 4$ rectangle. Find all possible values of $(a, b)$. (Le Anh Vinh)

There are $m$ pieces of candy held in $n$ trays($n,m\ge 4$). An [i]operation[/i] is defined as follow: take out one piece of candy from any two trays respectively, then put them in a third tray. Determine, with proof, if we can move all candies to a single tray by finite [i]operations[/i].

The PMO Magician has a special party game. There are $n$ chairs, labelled $1$ to $n$. There are $n$ sheets of paper, labelled $1$ to $n$. [list] [*] On each chair, she attaches exactly one sheet whose number does not match the number on the chair. [*] She then asks $n$ party guests to sit on the chairs so that each chair has exactly one occupant. [*] Whenever she claps her hands, each guest looks at the number on the sheet attached to their current chair, and moves to the chair labelled with that number. [/list] Show that if $1 < m \leq n$, where $m$ is not a prime power, it is always possible for the PMO Magician to choose which sheet to attach to each chair so that everyone returns to their original seats after exactly $m$ claps.

The numbers $1$ through $4^{n}$ are written on a board. In each step, Pedro erases two numbers $a$ and $b$ from the board, and writes instead the number $\frac{ab}{\sqrt{2a^2+2b^2}}$. Pedro repeats this procedure until only one number remains. Prove that this number is less than $\frac{1}{n}$, no matter what numbers Pedro chose in each step.

Caltech's 900 students are evenly spaced along the circumference of a circle. How many equilateral triangles can be formed with at least two Caltech students as vertices?

[b]p1.[/b] What is the largest number of five dollar footlongs Jimmy can buy with 88 dollars? [b]p2.[/b] Austin, Derwin, and Sylvia are deciding on roles for BMT $2021$. There must be a single Tournament Director and a single Head Problem Writer, but one person cannot take on both roles. In how many ways can the roles be assigned to Austin, Derwin, and Sylvia? [b]p3.[/b] Sofia has$ 7$ unique shirts. How many ways can she place $2$ shirts into a suitcase, where the order in which Sofia places the shirts into the suitcase does not matter? [b]p4.[/b] Compute the sum of the prime factors of $2021$. [b]p5.[/b] A sphere has volume $36\pi$ cubic feet. If its radius increases by $100\%$, then its volume increases by $a\pi$ cubic feet. Compute $a$. [b]p6.[/b] The full price of a movie ticket is $\$10$, but a matinee ticket to the same movie costs only $70\%$ of the full price. If $30\%$ of the tickets sold for the movie are matinee tickets, and the total revenue from movie tickets is $\$1001$, compute the total number of tickets sold. [b]p7.[/b] Anisa rolls a fair six-sided die twice. The probability that the value Anisa rolls the second time is greater than or equal to the value Anisa rolls the first time can be expressed as $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Compute $m + n$. [b]p8.[/b] Square $ABCD$ has side length $AB = 6$. Let point $E$ be the midpoint of $\overline{BC}$. Line segments $\overline{AC}$ and $\overline{DE}$ intersect at point $F$. Compute the area of quadrilateral ABEF. [b]p9.[/b] Justine has a large bag of candy. She splits the candy equally between herself and her $4$ friends, but she needs to discard three candies before dividing so that everyone gets an equal number of candies. Justine then splits her share of the candy between herself and her two siblings, but she needs to discard one candy before dividing so that she and her siblings get an equal number of candies. If Justine had instead split all of the candy that was originally in the large bag between herself and $14$ of her classmates, what is the fewest number of candies that she would need to discard before dividing so that Justine and her $14$ classmates get an equal number of candies? [b]p10.[/b] For some positive integers $a$ and $b$, $a^2 - b^2 = 400$. If $a$ is even, compute $a$. [b]p11.[/b] Let $ABCDEFGHIJKL$ be the equilateral dodecagon shown below, and each angle is either $90^o$ or $270^o$. Let $M$ be the midpoint of $\overline{CD}$, and suppose $\overline{HM}$ splits the dodecagon into two regions. The ratio of the area of the larger region to the area of the smaller region can be expressed as $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Compute $m + n$. [img]https://cdn.artofproblemsolving.com/attachments/3/e/387bcdf2a6c39fcada4f21f24ceebd18a7f887.png[/img] [b]p12.[/b] Nelson, who never studies for tests, takes several tests in his math class. Each test has a passing score of $60/100$. Since Nelson's test average is at least $60/100$, he manages to pass the class. If only nonnegative integer scores are attainable on each test, and Nelson gets a dierent score on every test, compute the largest possible ratio of tests failed to tests passed. Assume that for each test, Nelson either passes it or fails it, and the maximum possible score for each test is 100. [b]p13.[/b] For each positive integer $n$, let $f(n) = \frac{n}{n+1} + \frac{n+1}{n}$ . Then $f(1)+f(2)+f(3)+...+f(10)$ can be expressed as $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Compute $m + n$. [b]p14.[/b] Triangle $\vartriangle ABC$ has point $D$ lying on line segment $\overline{BC}$ between $B$ and $C$ such that triangle $\vartriangle ABD$ is equilateral. If the area of triangle $\vartriangle ADC$ is $\frac14$ the area of triangle $\vartriangle ABC$, then $\left( \frac{AC}{AB}\right)^2$ can be expressed as $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Compute $m + n$. [b]p15.[/b] In hexagon $ABCDEF$, $AB = 60$, $AF = 40$, $EF = 20$, $DE = 20$, and each pair of adjacent edges are perpendicular to each other, as shown in the below diagram. The probability that a random point inside hexagon $ABCDEF$ is at least $20\sqrt2$ units away from point $D$ can be expressed in the form $\frac{a-b\pi}{c}$ , where $a$, $b$, $c$ are positive integers such that gcd$(a, b, c) = 1$. Compute $a + b + c$. [img]https://cdn.artofproblemsolving.com/attachments/3/c/1b45470265d10a73de7b83eff1d3e3087d6456.png[/img] [b]p16.[/b] The equation $\sqrt{x} +\sqrt{20-x} =\sqrt{20 + 20x - x^2}$ has $4$ distinct real solutions, $x_1$, $x_2$, $x_3$, and $x_4$. Compute $x_1 + x_2 + x_3 + x_4$. [b]p17.[/b] How many distinct words with letters chosen from $B, M, T$ have exactly $12$ distinct permutations, given that the words can be of any length, and not all the letters need to be used? For example, the word $BMMT$ has $12$ permutations. Two words are still distinct even if one is a permutation of the other. For example, $BMMT$ is distinct from $TMMB$. [b]p18.[/b] We call a positive integer binary-okay if at least half of the digits in its binary (base $2$) representation are $1$'s, but no two $1$s are consecutive. For example, $10_{10} = 1010_2$ and $5_{10} = 101_2$ are both binary-okay, but $16_{10} = 10000_2$ and $11_{10} = 1011_2$ are not. Compute the number of binary-okay positive integers less than or equal to $2020$ (in base $10$). [b]p19.[/b] A regular octahedron (a polyhedron with $8$ equilateral triangles) has side length $2$. An ant starts on the center of one face, and walks on the surface of the octahedron to the center of the opposite face in as short a path as possible. The square of the distance the ant travels can be expressed as $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Compute $m + n$. [img]https://cdn.artofproblemsolving.com/attachments/f/8/3aa6abe02e813095e6991f63fbcf22f2e0431a.png[/img] [b]p20.[/b] The sum of $\frac{1}{a}$ over all positive factors $a$ of the number $360$ can be expressed in the form $\frac{m}{n}$ ,where $m$ and $n$ are relatively prime positive integers. Compute $m + n$. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

We are given a positive integer $ r$ and a rectangular board $ ABCD$ with dimensions $ AB \equal{} 20, BC \equal{} 12$. The rectangle is divided into a grid of $ 20 \times 12$ unit squares. The following moves are permitted on the board: one can move from one square to another only if the distance between the centers of the two squares is $ \sqrt {r}$. The task is to find a sequence of moves leading from the square with $ A$ as a vertex to the square with $ B$ as a vertex. (a) Show that the task cannot be done if $ r$ is divisible by 2 or 3. (b) Prove that the task is possible when $ r \equal{} 73$. (c) Can the task be done when $ r \equal{} 97$?

Positive integers $1,2,...,n$ are written on а blackboard ($n >2$ ). Every minute two numbers are erased and the least prime divisor of their sum is written. In the end only the number 97 remains. Find the least $n$ for which it is possible.

Given is a convex polygon $ P$ with $ n$ vertices. Triangle whose vertices lie on vertices of $ P$ is called [i]good [/i] if all its sides are unit length. Prove that there are at most $ \frac {2n}{3}$ [i]good[/i] triangles. [i]Author: Vyacheslav Yasinskiy, Ukraine[/i]