A rectangle with the side lengths $a$ and $b$ with $a <b$ should be placed in a right-angled coordinate system so that there is no point with integer coordinates in its interior or on its edge. Under what necessary and at the same time sufficient conditions for $a$ and $b$ is this possible?

A circle is divided into $n$ sectors. Pawns stand on some of the sectors; the total number of pawns equals $n + 1$. This configuration is changed as follows. Any two of the pawns standing on the same sector move simultaneously to the neighbouring sectors in different directions. Prove that after several such transformations a configuration in which no less than half of the sectors are occupied by pawns, will inevitably appear. (D. Fomin, St Petersburg)

In a country with $5$ distinct cities, there may or may not be a road between each pair of cities. It’s possible to get from any city to any other city through a series of roads, but there is no set of three cities $\{A,B,C\}$ such that there are roads between $A$ and $B$, $B$ and $C$, and $C$ and $A$. How many road systems between the five cities are possible?

Ten gamblers started playing with the same amount of money. Each turn they cast (threw) five dice. At each stage the gambler who had thrown paid to each of his 9 opponents $\frac{1}{n}$ times the amount which that opponent owned at that moment. They threw and paid one after the other. At the 10th round (i.e. when each gambler has cast the five dice once), the dice showed a total of 12, and after payment it turned out that every player had exactly the same sum as he had at the beginning. Is it possible to determine the total shown by the dice at the nine former rounds ?

[u]Round 5[/u] [b]p13.[/b] Initially, the three numbers $20$, $201$, and $2016$ are written on a blackboard. Each minute, Zhuo selects two of the numbers on the board and adds $1$ to each. Find the minimum $n$ for which Zhuo can make all three numbers equal to $n$. [b]p14.[/b] Call a three-letter string rearrangeable if, when the first letter is moved to the end, the resulting string comes later alphabetically than the original string. For example, $AAA$ and $BAA$ are not rearrangeable, while $ABB$ is rearrangeable. How many three-letters strings with (not necessarily distinct) uppercase letters are rearrangeable? [b]p15.[/b] Triangle $ABC$ is an isosceles right triangle with $\angle C = 90^o$ and $AC = 1$. Points $D$, $E$ and $F$ are chosen on sides $BC$,$CA$ and $AB$, respectively, such that $AEF$, $BFD$, $CDE$, and $DEF$ are isosceles right triangles. Find the sum of all distinct possible lengths of segment $DE$. [u]Round 6[/u] [b]p16.[/b] Let $p, q$, and $r$ be prime numbers such that $pqr = 17(p + q + r)$. Find the value of the product $pqr$. [b]p17.[/b] A cylindrical cup containing some water is tilted $45$ degrees from the vertical. The point on the surface of the water closest to the bottom of the cup is $6$ units away. The point on the surface of the water farthest from the bottom of the cup is $10$ units away. Compute the volume of the water in the cup. [b]p18.[/b] Each dot in an equilateral triangular grid with $63$ rows and $2016 = \frac12 \cdot 63 \cdot 64$ dots is colored black or white. Every unit equilateral triangle with three dots has the property that exactly one of its vertices is colored black. Find all possible values of the number of black dots in the grid. [u]Round 7[/u] [b]p19.[/b] Tomasz starts with the number $2$. Each minute, he either adds $2$ to his number, subtracts $2$ from his number, multiplies his number by $2$, or divides his number by $2$. Find the minimum number of minutes he will need in order to make his number equal $2016$. [b]p20.[/b] The edges of a regular octahedron $ABCDEF$ are painted with $3$ distinct colors such that no two edges with the same color lie on the same face. In how many ways can the octahedron be painted? Colorings are considered different under rotation or reflection. [b]p21.[/b] Jacob is trapped inside an equilateral triangle $ABC$ and must visit each edge of triangle $ABC$ at least once. (Visiting an edge means reaching a point on the edge.) His distances to sides $AB$, $BC$, and $CA$ are currently $3$, $4$, and $5$, respectively. If he does not need to return to his starting point, compute the least possible distance that Jacob must travel. [u]Round 8[/u] [b]p22.[/b] Four integers $a, b, c$, and $d$ with a $\le b \le c \le d$ satisfy the property that the product of any two of them is equal to the sum of the other two. Given that the four numbers are not all equal, determine the $4$-tuple $(a, b, c, d)$. [b]p23.[/b] In equilateral triangle $ABC$, points $D$,$E$, and $F$ lie on sides $BC$,$CA$ and $AB$, respectively, such that $BD = 4$ and $CD = 5$. If $DEF$ is an isosceles right triangle with right angle at $D$, compute $EA + FA$. [b]p24.[/b] On each edge of a regular tetrahedron, four points that separate the edge into five equal segments are marked. There are sixteen planes that are parallel to a face of the tetrahedron and pass through exactly three of the marked points. When the tetrahedron is cut along each of these sixteen planes, how many new tetrahedrons are produced? PS. You should use hide for answers. Rounds 1-4 have been posted [url=https://artofproblemsolving.com/community/c3h2934049p26256220]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

[u]Round 5[/u] [b]p13.[/b] Mandy is baking cookies. Her recipe calls for $N$ grams of flour, where $N$ is the number of perfect square divisors of $20! + 24!$. Find $N$. [b]p14.[/b] Consider a circular table with center $R$. Beef-loving Bryan places a steak at point $I$ on the circumference of the table. Then he places a bowl of rice at points $C$ and $E$ on the circumference of the table such that $CE \parallel IR$ and $\angle ICE = 25^o$. Find $\angle CIE$. [b]p15.[/b] Enya writes the $4$-letter words $LEEK$, $BEAN$, $SOUP$, $PEAS$, $HAMS$, and $TACO$ on the board. She then thinks of one of these words and gives Daria, Ava, Harini, and Tiffany a slip of paper containing exactly one letter from that word such that if they ordered the letters on their slips correctly, they would form the word. Each person announces at the same time whether they know the word or not. Ava, Harini, and Tiffany all say they do not know the word, while Daria says she knows the word. After hearing this, Ava, Harini, and Tiffany all know the word. Assuming all four girls are perfect logicians and they all thought of the same correct word, determine Daria’s letter. [u]Round 6[/u] [b]p16.[/b] Michael receives a cheese cube and a chocolate octahedron for his 5th birthday. On every day after, he slices off each corner of his cheese and chocolate with a knife. Each slice cuts off exactly one corner. He then eats each corner sliced off. Find the difference between the total number of cheese and chocolate pieces he has eaten by the end of his $6$th birthday. (Michael’s $5$th and $6$th birthdays do not occur on leap years.) [b]p17.[/b] Let $D$ be the average of all positive integers n satisfying $$lcm (gcd (n, 2000), gcd (n, 24)) = gcd (lcm (n, 2000), lcm (n, 24)).$$ Find $3D$. [b]p18.[/b] The base $\vartriangle ABC$ of the triangular pyramid $PABC$ is an equilateral triangle with a side length of $3$. Given that $PA = 3$, $PB = 4$, and $PC = 5$, find the circumradius of $PABC$. [u]Round 7[/u] [b]p19.[/b] $2049300$ points are arranged in an equilateral triangle point grid, a smaller version of which is shown below, such that the sides contain $2024$ points each. Peter starts at the topmost point of the grid. At $9:00$ am each day, he moves to an adjacent point in the row below him. Derrick wants to prevent Peter from reaching the bottom row, so at $12:00$ pm each day, he selects a point on the bottom row and places a rock at that point. Peter stops moving as soon as he is guaranteed to end up at a point with a rock on it. At least how many moves will Peter complete, no matter how Derrick places the rocks? [img]https://cdn.artofproblemsolving.com/attachments/f/a/346d25a5d7bb7a5fbefae7edad727965312b25.png[/img] [b]p20.[/b] There are $N$ stones in a pile, where $N$ is a positive integer. Ava and Anika take turns playing a game, with Ava moving first. If there are n stones in the pile, a move consists of removing $x$ stones, where $1 < gcd(x, n) \le x < n$. Whoever first has no possible moves on their turn wins. Both Ava and Anika play optimally. Find the $2024$th smallest value of $N$ for which Ava wins. [b]p21.[/b] Alan is bored and alone, so he plays a fun game with himself. He writes down all quadratic polynomials with leading coefficient $1$ whose coefficients are integers between $-10$ and $10$, inclusive, on a blackboard. He then erases all polynomials which have a non-integer root. Alan defines the size of a polynomial $P(x)$ to be $P(1)$ and spends an hour adding up the sizes of all the polynomials remaining on the blackboard. Assuming Alan does computation perfectly, find the sum Alan obtains. [u]Round 8[/u] [b]p22.[/b] A prime number is a positive integer with exactly two distinct divisors. You must submit a prime number for this problem. If you do not submit a prime number, you gain $0$ points, and your submission will not be considered valid. The median of all valid submitted numbers is $M$ (duplicates are counted). Estimate $2M$. If your team’s absolute difference between $2M$ and your submission is the $i$th smallest absolute difference among all teams, you gain max$(23 - 2i, 0)$ points. All teams who did not submit any number gain $0$ points. (In the case of a tie, all teams that tied gain the same amount of points.) [b]p23.[/b] Ribbotson the Frog is at the point $(0, 0)$ and wants to reach the point $(18, 18)$ in $36$ steps. Each step, he either moves one unit in the $+x$ direction or one unit in the $+y$ direction. However, Ribbotson hates turning, so he must make at least two steps in any direction before switching directions. If $m$ is the number of different paths Ribbotson the Frog can make, estimate $m$. If $N$ is your team’s submitted number, your team earns points equal to the closest integer to $21\left(1 -\left|\log_{10}\frac{N}{m} \right|^2\right)$. [b]p24.[/b] Let $M = \pi^{\pi^{\pi^{\pi}}}$. Estimate $k$, where $M = 10^{10^{k}}$. If $N$ is your team’s submitted number, your team earns points equal to the closest integer to $21 \cdot 1.01^{(-|N-k|^3)}$. PS. You should use hide for answers. Rounds 1-4 have been posted [url=https://artofproblemsolving.com/community/c3h3248729p29808138]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Jorge writes a list with an even number of integers, not all equal to $0$ (there may be repeated numbers). Show that Martin can cross out a number from the list, of his choice, so that it is impossible for Jorge to separate the remaining numbers into two groups in such a way that the sum of all the numbers in one group is equal to the sum of all the others. numbers from the other group.

In a dance party there are $n$ girls and $n$ boys, and some $m$ songs are played. Each song is danced to by at least one pair of a boy and a girl, who both receive a [i]malai [/i] each. Prove that for all positive integers $k \le n$, it is possible to select $k$ boys and $n - k$ girls so that the $n$ selected people received at least $m$ malai in total.

Given two positive integers $a$ and $b$, an [i]legal move[/i] consists in choosing a proper divisor of one of them and adding it to $a$ or adding it to $b$. Two players, Agustin and Ian, take turns making an legal move; Agustin plays first. Whoever gets a number greater than or equal to $2015$ wins the game. [list=a] [*]Determine which of the players has a winning strategy if $a=3, b=5$. [*]Determine which of the players has a winning strategy if $a=6, b=7$. [/list]

Given two positive integers $m$ and $n$, find the largest $k$ in terms of $m$ and $n$ such that the following condition holds: Any tree graph $G$ with $k$ vertices has two (possibly equal) vertices $u$ and $v$ such that for any other vertex $w$ in $G$, either there is a path of length at most $m$ from $u$ to $w$, or there is a path of length at most $n$ from $v$ to $w$. [i]Proposed by Ivan Chan Kai Chin[/i]

$(HUN 3)$ In the plane $4000$ points are given such that each line passes through at most $2$ of these points. Prove that there exist $1000$ disjoint quadrilaterals in the plane with vertices at these points.

Zoli wants to fill the given $4 \times 4$ table with the digits $1$, $2$, $3$ and $4$, such that in every row and column, and also in the diagonal going from the top left cell to the bottom right, each digit appears exactly once. What is the highest possible value of the sum of the digits in the six grey cells? [img]https://cdn.artofproblemsolving.com/attachments/7/0/498e652cd7ce556d8a638f3d51b65b13154ee5.png[/img]

A disc is divided into $30$ segments which are labelled by $50,100,150,\ldots,1500$ in some order. Show that there always exist three successive segments, the sum of whose labels is at least $2350$.

1990 mathematicians attend a meeting, every mathematician has at least 1327 friends (the relation of friend is reciprocal). Prove that there exist four mathematicians among them such that any two of them are friends.

A subset $E$ of the set $\{1,2,3,\ldots,50\}$ is said to be [i]special[/i] if it does not contain any pair of the form $\{x,3x\}.$ A special set $E$ is [i]superspecial[/i] if it contains as many elements as possible. How many element there are in a superspecial set and how many superspecial sets there are?

Let $n \ge 2$ and $k \ge1$ be positive integers. In a country there are $n$ cities and between each pair of cities there is a bus connection in both directions. Let $A$ and $B$ be two different cities. Prove that the number of ways in which you can travel from $A$ to $B$ by using exactly $k$ buses is equal to $\frac{(n - 1)^k - (-1)^k}{n}$ .

Seventeen people correspond by mail with one another-each one with all the rest. In their letters only three different topics are discussed. each pair of correspondents deals with only one of these topics. Prove that there are at least three people who write to each other about the same topic.

$ A_1 , A_2 , \cdots , A_n $ are given subsets. Let $ S = \left\{ 1, 2, \cdots , n \right\} $. For any $ X \subset S $, let \[ N(X)= \left\{ i \in S-X \ | \ \forall j \in X, \ A_i \cap A_j \ne \emptyset \right\} \] Let $ m $ be an integer such that $ 3 \le m \le n-2 $. Prove that there exist $ X \subset S $ such that $ |X|=m $ and $ |N(X)| \ne 1 $.

In some country several pairs of cities are connected by direct two-way flights. It is possible to go from any city to any other by a sequence of flights. The distance between two cities is defined to be the least possible numbers of flights required to go from one of them to the other. It is known that for any city there are at most $100$ cities at distance exactly three from it. Prove that there is no city such that more than $2550$ other cities have distance exactly four from it.

Let $n\ge2$ be a positive integer. The numbers $1,2,..., n^2$ are consecutively placed into squares of an $n\times n$, so the first row contains $1,2,...,n$ from left to right, the second row contains $n+1,n+2,...,2n$ from left to right, and so on. The [i]magic square value[/i] of a grid is defined to be the number of rows, columns, and main diagonals whose elements have an average value of $\frac{n^2 + 1}{2}$. Show that the magic-square value of the grid stays constant under the following two operations: (1) a permutation of the rows; and (2) a permutation of the columns. (The operations can be used multiple times, and in any order.) [i]Proposed by Ray Li[/i]

We say that a group of $25$ students is a [i]team[/i] if any two students in this group are friends. It is known that in the school any student belongs to at least one team but if any two students end their friendships at least one student does not belong to any team. We say that a team is [i]special[/i] if at least one student of the team has no friend outside of this team. Show that any two friends belong to some special team.

Starting from the number $ 1$ we write down a sequence of numbers where the next number in the sequence is obtained from the previous one either by doubling it, or by rearranging its digits (not allowing the first digit of the rearranged number to be $0$). For instance we might begin: $$1, 2, 4, 8, 16, 61, 122, 212, 424,...$$ Is it possible to construct such a sequence that ends with the number $1,000,000,000$? Is it possible to construct one that ends with the number $9,876,543,210$?

Let $n\geq 3$ be a fixed integer. Each side and each diagonal of a regular $n$-gon is labelled with a number from the set $\left\{1;\;2;\;...;\;r\right\}$ in a way such that the following two conditions are fulfilled: [b]1.[/b] Each number from the set $\left\{1;\;2;\;...;\;r\right\}$ occurs at least once as a label. [b]2.[/b] In each triangle formed by three vertices of the $n$-gon, two of the sides are labelled with the same number, and this number is greater than the label of the third side. [b](a)[/b] Find the maximal $r$ for which such a labelling is possible. [b](b)[/b] [i]Harder version (IMO Shortlist 2005):[/i] For this maximal value of $r$, how many such labellings are there? [hide="Easier version (5th German TST 2006) - contains answer to the harder version"] [i]Easier version (5th German TST 2006):[/i] Show that, for this maximal value of $r$, there are exactly $\frac{n!\left(n-1\right)!}{2^{n-1}}$ possible labellings.[/hide] [i]Proposed by Federico Ardila, Colombia[/i]

A father and two sons went to visit their grandmother, who Raya lives $33$ km from the city. My father has a motor roller, the speed of which $25$ km/h, and with a passenger - $20$ km/h (with two passengers on a scooter It’s impossible to move). Each of the brothers walks along the road at a speed of $5$ km/h. Prove that all three can get to grandma's in $3$ hours

There are $15$ magazines on a table, and they cover the surface of the table entirely. Prove that one can always take away $7$ magazines in such a way that the remaining ones cover at least $\dfrac{8}{15}$ of the area of the table surface