[b]4.[/b] We see, in Figures 1 and 2, examples of lock screens from a cellphone that only works with a password that is not typed but drawn with straight line segments. Those segments form a polygonal line with vertices in a lattice. When drawing the pattern that corresponds to a password, the finger can't lose contact with the screen. Every polygonal line corresponds to a sequence of digits and this sequence is, in fact, the password. The tracing of the polygonal obeys the following rules: [i]i.[/i] The tracing starts at some of the detached points which correspond to the digits from $1$ to $9$ (Figure 3). [i]ii.[/i] Each segment of the pattern must have as one of its extremes (on which we end the tracing of the segment) a point that has not been used yet. [i]iii.[/i] If a segment connects two points and contains a third one (its middle point), then the corresponding digit to this third point is included in the password. That does not happen if this point/digit has already been used. [i]iv.[/i] Every password has at least four digits. Thus, every polygonal line is associated to a sequence of four or more digits, which appear in the password in the same order that they are visited. In Figure 1, for instance, the password is 218369, if the first point visited was $2$. Notice how the segment connecting the points associated with $3$ and $9$ includes the points associated to digit $6$. If the first visited point were the $9$, then the password would be $963812$. If the first visited point were the $6$, then the password would be $693812$. In this case, the $6$ would be skipped, because it can't be repeated. On the other side, the polygonal line of Figure 2 is associated to a unique password. Determine the smallest $n (n \geq 4)$ such that, given any subset of $n$ digits from $1$ to $9$, it's possible to elaborate a password that involves exactly those digits in some order.

There is a board with $2010$ rows and $2001$ columns, on it there is a token located in the upper left box that can perform one of the following operations: (A) Walk 3 steps horizontally or vertically. (B) Walk 2 steps to the right and 3 steps down. (C) Walk 2 steps to the left and 2 steps up. With the condition that immediately after carrying out an operation on (B) or (C) it is mandatory to take a step to the right before perform the following operation. It is possible to exit the board, so count the number of steps necessary, entering through the other end of the row or column from which it exits, as if the board outside circular (example: from the beginning you can walk to the square located in row $1$ and column $1999$). Will it be possible that after $2011$ operations allowed the checker to land exactly on the bottom square right?

16 students took part in a competition. All problems were multiple choice style. Each problem had four choices. It was said that any two students had at most one answer in common, find the maximum number of problems.

In a certain game, the set $\{1, 2, \dots, 10\}$ is partitioned into equally-sized sets $A$ and $B$. In each of five consecutive rounds, Alice and Bob simultaneously choose an element from $A$ or $B$, respectively, that they have not yet chosen; whoever chooses the larger number receives a point, and whoever obtains three points wins the game. Determine the probability that Alice is guaranteed to win immediately after the set is initially partitioned.

Let $n \ge 3$ be an integer. On a circle, there are $n$ points. Each of them is labelled with a real number at most $1$ such that each number is the absolute value of the difference of the two numbers immediately preceding it in clockwise order. Determine the maximal possible value of the sum of all numbers as a function of $n$. (Walther Janous)

Let $n$ be an integer such that $n > 3$. Suppose that we choose three numbers from the set $\{1, 2, \ldots, n\}$. Using each of these three numbers only once and using addition, multiplication, and parenthesis, let us form all possible combinations. (a) Show that if we choose all three numbers greater than $\frac{n}{2}$, then the values of these combinations are all distinct. (b) Let $p$ be a prime number such that $p \leq \sqrt{n}$. Show that the number of ways of choosing three numbers so that the smallest one is $p$ and the values of the combinations are not all distinct is precisely the number of positive divisors of $p - 1$.

Suppose that every point in the plane is coloured either black or white. Must there be an equilateral triangle such that all of its vertices are the same colour?

Decompose a $6\times 6$ square into unit squares and consider the $49$ vertices of these unit squares. We call a square good if its vertices are among the $49$ points and if its sides and diagonals do not lie on the gridlines of the $6\times 6$ square. [list=a] [*]Find the total number of good squares. [*]Prove that there exist two good disjoint squares such that the smallest distance between their vertices is $1/\sqrt{5}.$ [/list]

In a game, the first player paints a point on the plane red; the second player paints 10 uncoloured points on the plane green; then the first player paints an uncoloured point on the plane red; the second player paints 10 uncoloured points on the plane green; and so on. The first player wins if there are three red points which form an equilateral triangle. Can the second player prevent the first player from winning? (A Kanel)

The positive integers are colored with black and white such that: - There exists a bijection from the black numbers to the white numbers, - The sum of three black numbers is a black number, and - The sum of three white numbers is a white number. Find the number of possible colorings that satisfies the above conditions.

Let $ABCDEF$ regular hexagon of side length $1$ and $O$ is its center. In addition to the sides of the hexagon, line segments from $O$ to the every vertex are drawn, making as total of $12$ unit segments. Find the number paths of length $2003$ along these segments that star at $O$ and terminate at $O$.

The $T_0$ set consists of all the numbers, representable as $(2k)!, k = 0, 1, 2, ... , n, ...$. The $T_m$ set is obtained from $T_{m-1}$ by adding all the finite sums of different numbers, that belong to $T_{m-1}$. Prove that there is a natural number, that doesn't belong to $T_{1987}$.

Let $k$ and $s$ be positive integers such that $s<(2k + 1)^2$. Initially, one cell out of an $n \times n$ grid is coloured green. On each turn, we pick some green cell $c$ and colour green some $s$ out of the $(2k + 1)^2$ cells in the $(2k + 1) \times (2k + 1)$ square centred at $c$. No cell may be coloured green twice. We say that $s$ is $k-sparse$ if there exists some positive number $C$ such that, for every positive integer $n$, the total number of green cells after any number of turns is always going to be at most $Cn$. Find, in terms of $k$, the least $k$-sparse integer $s$. [I]Proposed by Nikolai Beluhov.[/i]

Naomi has a class of $100$ students who will compete with each other in five teams. Once the teams are made, each student will shake hands with every other student, except the students in his or her own team. Naomi chooses to partition the students into teams so as to maximize the number of handshakes. How many handshakes will there be?

[b]p1.[/b] Let $U = \{-2, 0, 1\}$ and $N = \{1, 2, 3, 4, 5\}$. Let $f$ be a function that maps $U$ to $N$. For any $x \in U$, $x + f(x) + xf(x)$ is an odd number. How many $f$ satisfy the above statement? [b]p2.[/b] Around a circle are written all of the positive integers from $ 1$ to $n$, $n \ge 2$ in such a way that any two adjacent integers have at least one digit in common in their decimal expressions. Find the smallest $n$ for which this is possible. [b]p3.[/b] Michael loses things, especially his room key. If in a day of the week he has $n$ classes he loses his key with probability $n/5$. After he loses his key during the day he replaces it before he goes to sleep so the next day he will have a key. During the weekend(Saturday and Sunday) Michael studies all day and does not leave his room, therefore he does not lose his key. Given that on Monday he has 1 class, on Tuesday and Thursday he has $2$ classes and that on Wednesday and Friday he has $3$ classes, what is the probability that loses his key at least once during a week? [b]p4.[/b] Given two concentric circles one with radius $8$ and the other $5$. What is the probability that the distance between two randomly chosen points on the circles, one from each circle, is greater than $7$ ? [b]p5.[/b] We say that a positive integer $n$ is lucky if $n^2$ can be written as the sum of $n$ consecutive positive integers. Find the number of lucky numbers strictly less than $2015$. [b]p6.[/b] Let $A = \{3^x + 3^y + 3^z|x, y, z \ge 0, x, y, z \in Z, x < y < z\}$. Arrange the set $A$ in increasing order. Then what is the $50$th number? (Express the answer in the form $3^x + 3^y + 3^z$). [b]p7.[/b] Justin and Oscar found $2015$ sticks on the table. I know what you are thinking, that is very curious. They decided to play a game with them. The game is, each player in turn must remove from the table some sticks, provided that the player removes at least one stick and at most half of the sticks on the table. The player who leaves just one stick on the table loses the game. Justin goes first and he realizes he has a winning strategy. How many sticks does he have to take off to guarantee that he will win? [b]p8.[/b] Let $(x, y, z)$ with $x \ge y \ge z \ge 0$ be integers such that $\frac{x^3+y^3+z^3}{3} = xyz + 21$. Find $x$. [b]p9.[/b] Let $p < q < r < s$ be prime numbers such that $$1 - \frac{1}{p} -\frac{1}{q} -\frac{1}{r}- \frac{1}{s}= \frac{1}{pqrs}.$$ Find $p + q + r + s$. [b]p10.[/b] In ”island-land”, there are $10$ islands. Alex falls out of a plane onto one of the islands, with equal probability of landing on any island. That night, the Chocolate King visits Alex in his sleep and tells him that there is a mountain of chocolate on one of the islands, with equal probability of being on each island. However, Alex has become very fat from eating chocolate his whole life, so he can’t swim to any of the other islands. Luckily, there is a teleporter on each island. Each teleporter will teleport Alex to exactly one other teleporter (possibly itself) and each teleporter gets teleported to by exactly one teleporter. The configuration of the teleporters is chosen uniformly at random from all possible configurations of teleporters satisfying these criteria. What is the probability that Alex can get his chocolate? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

[b]p1.[/b] The clock is $2$ hours $20$ minutes ahead of the correct time each week. The clock is set to the correct time at midnight Sunday to Monday. What time does this clock show at 6pm correct time on Thursday? [b]p2.[/b] Five cities $A,B,C,D$, and $E$ are located along the straight road in the alphabetical order. The sum of distances from $B$ to $A,C,D$ and $E$ is $20$ miles. The sum of distances from $C$ to the other four cities is $18$ miles. Find the distance between $B$ and $C$. [b]p3.[/b] Does there exist distinct digits $a, b, c$, and $d$ such that $\overline{abc}+\overline{c} = \overline{bda}$? Here $\overline{abc}$ means the three digit number with digits $a, b$, and $c$. [b]p4.[/b] Kuzya, Fyokla, Dunya, and Senya participated in a mathematical competition. Kuzya solved $8$ problems, more than anybody else. Senya solved $5$ problem, less than anybody else. Each problem was solved by exactly $3$ participants. How many problems were there? [b]p5.[/b] Mr Mouse got to the cellar where he noticed three heads of cheese weighing $50$ grams, $80$ grams, and $120$ grams. Mr. Mouse is allowed to cut simultaneously $10$ grams from any two of the heads and eat them. He can repeat this procedure as many times as he wants. Can he make the weights of all three pieces equal? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Do there exist positive integers $k$ and $n$ such that for any finite graph $G$ with diameter $k+1$ there exists a set $S$ of at most $n$ vertices such that for any $v\in V(G)\setminus S$, there exists a vertex $u\in S$ of distance at most $k$ from $v$? [i]David Yang.[/i]

[u]Round 1[/u] [b]p1.[/b] Alec rated the movie Frozen $1$ out of $5$ stars. At least how many ratings of $5$ out of $5$ stars does Eric need to collect to make the average rating for Frozen greater than or equal to $4$ out of $5$ stars? [b]p2.[/b] Bessie shuffles a standard $52$-card deck and draws five cards without replacement. She notices that all five of the cards she drew are red. If she draws one more card from the remaining cards in the deck, what is the probability that she draws another red card? [b]p3.[/b] Find the value of $121 \cdot 1020304030201$. [u]Round 2[/u] [b]p4.[/b] Find the smallest positive integer $c$ for which there exist positive integers $a$ and $b$ such that $a \ne b$ and $a^2 + b^2 = c$ [b]p5.[/b] A semicircle with diameter $AB$ is constructed on the outside of rectangle $ABCD$ and has an arc length equal to the length of $BC$. Compute the ratio of the area of the rectangle to the area of the semicircle. [b]p6.[/b] There are $10$ monsters, each with $6$ units of health. On turn $n$, you can attack one monster, reducing its health by $n$ units. If a monster's health drops to $0$ or below, the monster dies. What is the minimum number of turns necessary to kill all of the monsters? [u]Round 3[/u] [b]p7.[/b] It is known that $2$ students make up $5\%$ of a class, when rounded to the nearest percent. Determine the number of possible class sizes. [b]p8.[/b] At $17:10$, Totoro hopped onto a train traveling from Tianjin to Urumuqi. At $14:10$ that same day, a train departed Urumuqi for Tianjin, traveling at the same speed as the $17:10$ train. If the duration of a one-way trip is $13$ hours, then how many hours after the two trains pass each other would Totoro reach Urumuqi? [b]p9.[/b] Chad has $100$ cookies that he wants to distribute among four friends. Two of them, Jeff and Qiao, are rivals; neither wants the other to receive more cookies than they do. The other two, Jim and Townley, don't care about how many cookies they receive. In how many ways can Chad distribute all $100$ cookies to his four friends so that everyone is satisfied? (Some of his four friends may receive zero cookies.) [u]Round 4[/u] [b]p10.[/b] Compute the smallest positive integer with at least four two-digit positive divisors. [b]p11.[/b] Let $ABCD$ be a trapezoid such that $AB$ is parallel to $CD$, $BC = 10$ and $AD = 18$. Given that the two circles with diameters $BC$ and $AD$ are tangent, find the perimeter of $ABCD$. [b]p12.[/b] How many length ten strings consisting of only $A$s and Bs contain neither "$BAB$" nor "$BBB$" as a substring? PS. You should use hide for answers. Rounds 5-8 have been posted [url=https://artofproblemsolving.com/community/c3h2934037p26256063]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

We are given an $n \times n$ board. Rows are labeled with numbers $1$ to $n$ downwards and columns are labeled with numbers $1$ to $n$ from left to right. On each field we write the number $x^2 + y^2$ where $(x, y)$ are its coordinates. We are given a figure and can initially place it on any field. In every step we can move the figure from one field to another if the other field has not already been visited and if at least one of the following conditions is satisfied:[list] [*] the numbers in those $2$ fields give the same remainders when divided by $n$, [*] those fields are point reflected with respect to the center of the board.[/list]Can all the fields be visited in case: [list=a][*] $n = 4$, [*] $n = 5$?[/list] [i]Josip Pupić[/i]

There are a board with $2n \cdot 2n \ (= 4n^2)$ squares and $4n^2-1$ cards numbered with different natural numbers. These cards are put one by one on each of the squares. One square is empty. We can move a card to an empty square from one of the adjacent squares (two squares are adjacent if they have a common edge). Is it possible to exchange two cards on two adjacent squares of a column (or a row) in a finite number of movements?

Janson found $2025$ dogs on a circle. Janson wants to select some (possibly none) of the dogs to take home, such that no two selected dogs have exactly two dogs (whether selected or not) in between them. Let $S_{1}$ be the number of ways for him to do so. Ivan also found $2025$ cats on a circle. Ivan wants to select some (possibly none) of the cats to take home, such that no two selected cats have exactly five cats (whether selected or not) in between them. Let $S_{2}$ be the number of ways for him to do so. a) Prove that $S_{1}=S_{2}$. b) Prove that $S_{1}$ and $S_{2}$ are both perfect cubes.

A little boy wrote the numbers $1,2,\cdots,2011$ on a blackboard. He picks any two numbers $x,y$, erases them with a sponge and writes the number $|x-y|$. This process continues until only one number is left. Prove that the number left is even.

Let $m$ and $n$ be natural numbers. Professor Mubariz has $m$ folders and Professor Nazim has $n$ folders; initially, all folders are empty. Every day, where the day numbers are marked as $d = 1,2,3 ....,$ Prof. Mubariz is given $2018$ blue papers, and Prof. Nazim is given $2018$ orange papers. On day $d ( d = 1, 2, 3, ...),$ they both perform the following operations: [list] [*] If the $2018$ papers given to this professor are not enough to place $d$ papers in each of his folders, then he distributes all the $2018$ papers given to him to his students. If the $2018$ papers given to this professor are enough to place $d$ papers in each of his folders, firstly, he places $d$ papers in each of his folders. [*] If this professor still has papers left after the first step, he places them in the other professor's folders, with the same number in each folder and as many as possible. [*] If this professor still has papers left after the second step, he distributes them to his students. [/list] Prove that after $6$ years, the number of blue papers in one folder of Prof. Nazim will be equal to the number of orange papers in one folder of Prof. Mubariz.

In the Olympic school the exams are graded with whole numbers, the lowest possible grade is $0$, and the highest is $10$. In the arithmetic class the teacher takes two exams. This year he has $15$ students. When one of his students gets less than $3$ on the first exam and more than $7$ on the second exam, he calls him an overachieving student. The teacher, at the end of correcting the exams, averaged the $30$ grades and obtained $8$. What is the largest number of students who passed this class could have had?

A magical castle has $n$ identical rooms, each of which contains $k$ doors arranged in a line. In room $i, 1 \leq i \leq n - 1$ there is one door that will take you to room $i + 1$, and in room $n$ there is one door that takes you out of the castle. All other doors take you back to room $1$. When you go through a door and enter a room, you are unable to tell what room you are entering and you are unable to see which doors you have gone through before. You begin by standing in room $1$ and know the values of $n$ and $k$. Determine for which values of $n$ and $k$ there exists a strategy that is guaranteed to get you out of the castle and explain the strategy. For such values of $n$ and $k$, exhibit such a strategy and prove that it will work.