The cells of the $11 \times 111 \times11$ cube contain the numbers $ 1, 2, , . .. . . 1331$, once each number. Two worms are sent from one corner cube to the opposite corner. Each of them can crawl into a cube adjacent to the edge, while the first can crawl if the number in the adjacent cube differs by $8$, the second - if they differ by $ 9$. Is there such an arrangement of numbers that both worms can get to the opposite corner cube?

A circle is divided by $2018$ points into equal parts. Two players delete these points in turns. A player loses, if after his turn it is possible to draw a diameter of the circle such that there are no undeleted points on one side of it. Which player has a winning strategy?

$\forall n \in \mathbb{N}$, $P(n)$ denotes the number of the partition of $n$ as the sum of positive integers (disregarding the order of the parts), e.g. since $4 = 1+1+1+1 = 1+1+2 = 1+3 = 2+2 = 4$, so $P(4)=5$. "Dispersion" of a partition denotes the number of different parts in that partitation. And denote $q(n)$ is the sum of all the dispersions, e.g. $q(4)=1+2+2+1+1=7$. $n \geq 1$. Prove that (1) $q(n) = 1 + \sum^{n-1}_{i=1} P(i).$ (2) $1 + \sum^{n-1}_{i=1} P(i) \leq \sqrt{2} \cdot n \cdot P(n)$.

We say that $A$$=${$a_1,a_2,a_3\cdots a_n$} consisting $n>2$ distinct positive integers is $good$ if for every $i=1,2,3\cdots n$ the number ${a_i}^{2015}$ is divisible by the product of all numbers in $A$ except $a_i$. Find all integers $n>2$ such that exists a $good$ set consisting of $n$ positive integers.

In the following illustration, starting from point X, we move one square along the segment until we arrive at point Y. Calculate the number of times a point has passed once and does not pass again, from X to Y. (However, starting point X is considered to have passed.)

Alexey and Bogdan play a game with two piles of stones. In the beginning, one of the piles contains $2021$ stones, and the second is empty. In one move, each of the guys has to pick up an even number of stones (more than zero) from an arbitrary pile, then transfer half of the stones taken to another pile, and the other half - to remove from the game. Loses the one who cannot make a move. Who will win this game if both strive to win, and Bogdan begins? (Oleksii Masalitin)

Let $M = \{(a,b,c,d)|a,b,c,d \in \{1,2,3,4\} \text{ and } abcd > 1\}$. For each $n\in \{1,2,\dots, 254\}$, the sequence $(a_1, b_1, c_1, d_1)$, $(a_2, b_2, c_2, d_2)$, $\dots$, $(a_{255}, b_{255},c_{255},d_{255})$ contains each element of $M$ exactly once and the equality \[|a_{n+1} - a_n|+|b_{n+1} - b_n|+|c_{n+1} - c_n|+|d_{n+1} - d_n| = 1\] holds. If $c_1 = d_1 = 1$, find all possible values of the pair $(a_1,b_1)$.

Suppose $n$ lines in plane are such that no two are parallel and no three are concurrent. For each two lines their angle is a real number in $[0,\frac{\pi}2]$. Find the largest value of the sum of the $\binom n2$ angles between line. [i]By Aliakbar Daemi[/i]

Can we put intengers $1,2,\cdots,12$ on a circle, number them $a_1,a_2,\cdots,a_{12}$ in order. For any $1\leq i<j\leq12$, $|a_i-a_j|\neq|i-j|$?

A rectangle formed by the lines of checkered paper is divided into figures of three kinds: isosceles right triangles (1) with base of two units, squares (2) with unit side, and parallelograms (3) formed by two sides and two diagonals of unit squares (figures may be oriented in any way). Prove that the number of figures of the third kind is even. [img]http://up.iranblog.com/Files7/dda310bab8b6455f90ce.jpg[/img]

Four frogs sit on the vertices of a square, one on each vertex. They jump in arbitrary order but not simultaneously. Each frog jumps to the point symmetrical to its take-off position with respect to the centre of gravity of the three other frogs. Can one of them jump (at a given time) exactly on to one of the others (considering their locations as points)? (A Andzans)

Prove that there exists a positive integer $n$, such that if an equilateral triangle with side lengths $n$ is split into $n^2$ triangles with side lengths 1 with lines parallel to its sides, then among the vertices of the small triangles it is possible to choose $1993n$ points so that no three of them are vertices of an equilateral triangle.

[b]p1.[/b] Compute the degree of the least common multiple of the polynomials $x - 1$, $x^2 - 1$, $x^3 - 1$,$...$, $x^{10} -1$. [b]p2.[/b] A line in the $xy$ plane is called wholesome if its equation is $y = mx+b$ where $m$ is rational and $b$ is an integer. Given a point with integer coordinates $(x,y)$ on a wholesome line $\ell$, let $r$ be the remainder when $x$ is divided by $7$, and let $s$ be the remainder when y is divided by $7$. The pair $(r, s)$ is called an [i]ingredient[/i] of the line $\ell$. The (unordered) set of all possible ingredients of a wholesome line $\ell$ is called the [i]recipe [/i] of $\ell$. Compute the number of possible recipes of wholesome lines. [b]p3.[/b] Let $\tau (n)$ be the number of distinct positive divisors of $n$. Compute $\sum_{d|15015} \tau (d)$, that is, the sum of $\tau (d)$ for all $d$ such that $d$ divides $15015$. [b]p4.[/b] Suppose $2202010_b - 2202010_3 = 71813265_{10}$. Compute $b$. ($n_b$ denotes the number $n$ written in base $b$.) [b]p5.[/b] Let $x = (3 -\sqrt5)/2$. Compute the exact value of $x^8 + 1/x^8$. [b]p6.[/b] Compute the largest integer that has the same number of digits when written in base $5$ and when written in base $7$. Express your answer in base $10$. [b]p7.[/b] Three circles with integer radii $a$, $b$, $c$ are mutually externally tangent, with $a \le b \le c$ and $a < 10$. The centers of the three circles form a right triangle. Compute the number of possible ordered triples $(a, b, c)$. [b]p8.[/b] Six friends are playing informal games of soccer. For each game, they split themselves up into two teams of three. They want to arrange the teams so that, at the end of the day, each pair of players has played at least one game on the same team. Compute the smallest number of games they need to play in order to achieve this. [b]p9.[/b] Let $A$ and $B$ be points in the plane such that $AB = 30$. A circle with integer radius passes through $A$ and $B$. A point $C$ is constructed on the circle such that $AC$ is a diameter of the circle. Compute all possible radii of the circle such that $BC$ is a positive integer. [b]p10.[/b] Each square of a $3\times 3$ grid can be colored black or white. Two colorings are the same if you can rotate or reflect one to get the other. Compute the total number of unique colorings. [b]p11.[/b] Compute all positive integers $n$ such that the sum of all positive integers that are less than $n$ and relatively prime to $n$ is equal to $2n$. [b]p12.[/b] The distance between a point and a line is defined to be the smallest possible distance between the point and any point on the line. Triangle $ABC$ has $AB = 10$, $BC = 21$, and $CA = 17$. Let $P$ be a point inside the triangle. Let $x$ be the distance between $P$ and $\overleftrightarrow{BC}$, let $y$ be the distance between $P$ and $\overleftrightarrow{CA}$, and let $z$ be the distance between $P$ and $\overleftrightarrow{AB}$. Compute the largest possible value of the product $xyz$. [b]p13.[/b] Alice, Bob, David, and Eve are sitting in a row on a couch and are passing back and forth a bag of chips. Whenever Bob gets the bag of chips, he passes the bag back to the person who gave it to him with probability $\frac13$ , and he passes it on in the same direction with probability $\frac23$ . Whenever David gets the bag of chips, he passes the bag back to the person who gave it to him with probability $\frac14$ , and he passes it on with probability $\frac34$ . Currently, Alice has the bag of chips, and she is about to pass it to Bob when Cathy sits between Bob and David. Whenever Cathy gets the bag of chips, she passes the bag back to the person who gave it to her with probability $p$, and passes it on with probability $1-p$. Alice realizes that because Cathy joined them on the couch, the probability that Alice gets the bag of chips back before Eve gets it has doubled. Compute $p$. [b]p14.[/b] Circle $O$ is in the plane. Circles $A$, $B$, and $C$ are congruent, and are each internally tangent to circle $O$ and externally tangent to each other. Circle $X$ is internally tangent to circle $O$ and externally tangent to circles $A$ and $B$. Circle $X$ has radius $1$. Compute the radius of circle $O$. [img]https://cdn.artofproblemsolving.com/attachments/f/d/8ddab540dca0051f660c840c0432f9aa5fe6b0.png[/img] [b]p15.[/b] Compute the number of primes $p$ less than 100 such that $p$ divides $n^2 +n+1$ for some integer $n$. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

$234$ viewers came to the cinema. Determine for which$ n \ge 4$ the viewers could be can be arranged in $n$ rows so that every viewer in $i$-th row gets to know just $j$ viewers in $j$-th row for any $i, j \in \{1, 2,... , n\}, i\ne j$. (The relationship of acquaintance is mutual.) (Tomáš Jurík)

On the planet Automory, there are infinitely many inhabitants. Every Automorian loves exactly one Automorian and honours exactly one Automorian. Additionally, the following can be noticed: $\bullet$ each Automorian is loved by some Automorian; $\bullet$ if Automorian $A$ loves Automorian $B$, then also all Automorians honouring $A$ love $B$, $\bullet$if Automorian $A$ honours Automorian $B$, then also all Automorians loving $A$ honour $B$. Is it correct to claim that every Automorian honours and loves the same Automorian?

What is the maximum number of non-overlapping $ 2\times 1$ dominoes that can be placed on a $ 8\times 9$ checkerboard if six of them are placed as shown? Each domino must be placed horizontally or vertically so as to cover two adjacent squares of the board.

A bear is in the center of the left down corner of a $100*100$ square .we call a cycle in this grid a bear cycle if it visits each square exactly ones and gets back to the place it started.Removing a row or column with compose the bear cycle into number of pathes.Find the minimum $k$ so that in any bear cycle we can remove a row or column so that the maximum length of the remaining pathes is at most $k$.

The point $O{}$ is given in the plane. Find all natural numbers $n{}$ for which $n{}$ points in the plane can be colored red, so that for any two red points $A{}$ and $B{}$ there is a third red point $C{}$ is such that $O{}$ lies strictly inside the triangle $ABC$. [i]From the folklore[/i]

[i]25 problems for 30 minutes[/i] [b]p1.[/b] Compute the sum $2019 + 201 + 20 + 2$. [b]p2.[/b] The sequence $100, 102, 104,..., 996$ and $998$ is the sequence of all three-digit even numbers. How many three digit even numbers are there? [b]p3.[/b] Find the units digit of $25\times 37\times 113\times 22$. [b]p4.[/b] Samuel has a number in his head. He adds $4$ to the number and then divides the result by $2$. After doing this, he ends up with the same number he had originally. What is his original number? [b]p5.[/b] According to Shay's Magazine, every third president is terrible (so the third, sixth, ninth president and so on were all terrible presidents). If there have been $44$ presidents, how many terrible presidents have there been in total? [b]p6.[/b] In the game Tic-Tac-Toe, a player wins by getting three of his or her pieces in the same row, column, or diagonal of a $3\times 3$ square. How many configurations of $3$ pieces are winning? Rotations and reflections are considered distinct. [b]p7.[/b] Eddie is a sad man. Eddie is cursed to break his arm $4$ times every $20$ years. How many times would he break his arm by the time he reaches age $100$? [b]p8. [/b]The figure below is made from $5$ congruent squares. If the figure has perimeter $24$, what is its area? [img]https://cdn.artofproblemsolving.com/attachments/1/9/6295b26b1b09cacf0c32bf9d3ba3ce76ddb658.png[/img] [b]p9.[/b] Sancho Panza loves eating nachos. If he eats $3$ nachos during the first minute, $4$ nachos during the second, $5$ nachos during the third, how many nachos will he have eaten in total after $15$ minutes? [b]p10.[/b] If the day after the day after the day before Wednesday was two days ago, then what day will it be tomorrow? [b]p11.[/b] Neetin the Rabbit and Poonam the Meerkat are in a race. Poonam can run at $10$ miles per hour, while Neetin can only hop at $2$ miles per hour. If Neetin starts the race $2$ miles ahead of Poonam, how many minutes will it take for Poonam to catch up with him? [b]p12.[/b] Dylan has a closet with t-shirts: $3$ gray, $4$ blue, $2$ orange, $7$ pink, and $2$ black. Dylan picks one shirt at random from his closet. What is the probability that Dylan picks a pink or a gray t-shirt? [b]p13.[/b] Serena's brain is $200\%$ the size of Eric's brain, and Eric's brain is $200\%$ the size of Carlson's. The size of Carlson's brain is what percent the size of Serena's? [b]p14.[/b] Find the sum of the coecients of $(2x + 1)^3$ when it is fully expanded. [b]p15. [/b]Antonio loves to cook. However, his pans are weird. Specifically, the pans are rectangular prisms without a top. What is the surface area of the outside of one of Antonio's pans if their volume is $210$, and their length and width are $6$ and $5$, respectively? [b]p16.[/b] A lattice point is a point on the coordinate plane with $2$ integer coordinates. For example, $(3, 4)$ is a lattice point since $3$ and $4$ are both integers, but $(1.5, 2)$ is not since $1.5$ is not an integer. How many lattice points are on the graph of the equation $x^2 + y^2 = 625$? [b]p17.[/b] Jonny has a beaker containing $60$ liters of $50\%$ saltwater ($50\%$ salt and $50\%$ water). Jonny then spills the beaker and $45$ liters pour out. If Jonny adds $45$ liters of pure water back into the beaker, what percent of the new mixture is salt? [b]p18.[/b] There are exactly 25 prime numbers in the set of positive integers between $1$ and $100$, inclusive. If two not necessarily distinct integers are randomly chosen from the set of positive integers from $1$ to $100$, inclusive, what is the probability that at least one of them is prime? [b]p19.[/b] How many consecutive zeroes are at the end of $12!$ when it is expressed in base $6$? [b]p20.[/b] Consider the following figure. How many triangles with vertices and edges from the following figure contain exactly $1$ black triangle? [img]https://cdn.artofproblemsolving.com/attachments/f/2/a1c400ff7d06b583c1906adf8848370e480895.png[/img] [b]p21.[/b] After Akshay got kicked o the school bus for rowdy behavior, he worked out a way to get home from school with his dad. School ends at $2:18$ pm, but since Akshay walks slowly he doesn't get to the front door until $2:30$. His dad doesn't like to waste time, so he leaves home everyday such that he reaches the high school at exactly $2:30$ pm, instantly picks up Akshay and turns around, then drives home. They usually get home at $3:30$ pm. However, one day Akshay left school early at exactly $2:00$ pm because he was expelled. Trying to delay telling his dad for as long as possible, Akshay starts jogging home. His dad left home at the regular time, saw Akshay on the way, picked him up and turned around instantly. They then drove home while Akshay's dad yelled at him for being a disgrace. They reached home at $3:10$ pm. How long had Akshay been walking before his dad picked him up? [b]p22.[/b] In quadrilateral $ABCD$, diagonals $AC$ and $BD$ intersect at $O$. Then $\angle BOC = \angle BCD$, $\angle COD =\angle BAD$, $AB = 4$, $DC = 6$, and $BD = 5$. What is the length of $BO$? [b]p23.[/b] A standard six-sided die is rolled. The number that comes up first determines the number of additional times the die will be rolled (so if the first number is $3$, then the die will be rolled $3$ more times). Each time the die is rolled, its value is recorded. What is the expected value of the sum of all the rolls? [b]p24.[/b] Dora has a peculiar calculator that can only perform $2$ operations: either adding $1$ to the current number or squaring the current number. Each minute, Dora randomly chooses an operation to apply to her number. She starts with $0$. What is the expected number of minutes it takes Dora's number to become greater than or equal to $10$? [b]p25.[/b] Let $\vartriangle ABC$ be such that $AB = 2$, $BC = 1$, and $\angle ACB = 90^o$. Let points $D$ and $E$ be such that $\vartriangle ADE$ is equilateral, $D$ is on segment $\overline{BC}$, and $D$ and $E$ are not on the same side of $\overline{AC}$. Segment $\overline{BE}$ intersects the circumcircle of $\vartriangle ADE$ at a second point $F$. If $BE =\sqrt{6}$, find the length of $\overline{BF}$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Tanya and Serezha take turns putting chips in empty squares of a chessboard. Tanya starts with a chip in an arbitrary square. At every next move, Serezha must put a chip in the column where Tanya put her last chip, while Tanya must put a chip in the row where Serezha put his last chip. The player who cannot make a move loses. Which of the players has a winning strategy? [i]Proposed by A. Golovanov[/i]

Let $a_1,a_2,\ldots$ be a sequence of integers with infinitely many positive and negative terms. Suppose that for every positive integer $n$ the numbers $a_1,a_2,\ldots,a_n$ leave $n$ different remainders upon division by $n$. Prove that every integer occurs exactly once in the sequence $a_1,a_2,\ldots$.

A particle moves from $(0, 0)$ to $(n, n)$ directed by a fair coin. For each head it moves one step east and for each tail it moves one step north. At $(n, y), y < n$, it stays there if a head comes up and at $(x, n), x < n$, it stays there if a tail comes up. Let$ k$ be a fixed positive integer. Find the probability that the particle needs exactly $2n+k$ tosses to reach $(n, n).$

At Durant University, an A grade corresponds to raw scores between $90$ and $100$, and a B grade corresponds to raw scores between $80$ and $90$. Travis has $3$ equally weighted exams in his math class. Given that Travis earned an A on his first exam and a B on his second (but doesn't know his raw score for either), what is the minimum score he needs to have a $90\%$ chance of getting an A in the class? Note that scores on exams do not necessarily have to be integers.

Let $n$ and $k$ be positive integers, where $n > 1$ is odd. Suppose $n$ voters are to elect one of the $k$ cadidates from a set $A$ according to the rule of "majoritarian compromise" described below. After each voter ranks the candidates in a column according to his/her preferences, these columns are concatenated to form a $k$ x $n$ voting matrix. We denote the number of ccurences of $a \in A$ in the $i$-th row of the voting matrix by $a_{i}$ . Let $l_{a}$ stand for the minimum integer $l$ for which $\sum^{l}_{i=1}{a_{i}}> \frac{n}{2}$. Setting $l'= min \{l_{a} | a \in A\}$, we will regard the voting matrices which make the set $\{a \in A | l_{a} = l' \}$ as admissible. For each such matrix, the single candidate in this set will get elected according to majoritarian compromise. Moreover, if $w_{1} \geq w_{2} \geq ... \geq
w_{k} \geq 0$ are given, for each admissible voting matrix, $\sum^{k}_{i=1}{w_{i}a_{i}}$ is called the total weighted score of $a \in A$. We will say that the system $(w_{1},w_{2}, . . . , w_{k})$ of weights represents majoritarian compromise if the total score of the elected candidate is maximum among the scores of all candidates. (a) Determine whether there is a system of weights representing majoritarian compromise if $k = 3$. (b) Show that such a system of weights does not exist for $k > 3$.

Let $A = (a_1, a_2, \ldots, a_{2001})$ be a sequence of positive integers. Let $m$ be the number of 3-element subsequences $(a_i,a_j,a_k)$ with $1 \leq i < j < k \leq 2001$, such that $a_j = a_i + 1$ and $a_k = a_j + 1$. Considering all such sequences $A$, find the greatest value of $m$.