$A$ and $B$ are two identical convex polygons, each with an area of $2015$. The polygon $A$ is divided into polygons $A_1, A_2,...,A_{2015}$, while $B$ is divided into polygons $B_1, B_2,...,B_{2015}$. Each of these smaller polygons has a positive area. Furthermore, $A_1, A_2,...,A_{2015}$ and $B_1, B_2,...,B_{2015}$ are colored in $2015$ distinct colors, such that $A_i$ and $A_j$ are differently colored for every distinct $i$ and $j$ and $B_i$ and $B_j$ are also differently colored for every distinct $i$ and $j$. After $A$ and $B$ overlap, we calculate the sum of the areas with the same colors. Prove that we can color the polygons such that this sum is at least $1$.

For $n$ an odd positive integer, the unit squares of an $n\times n$ chessboard are coloured alternately black and white, with the four corners coloured black. A it tromino is an $L$-shape formed by three connected unit squares. For which values of $n$ is it possible to cover all the black squares with non-overlapping trominos? When it is possible, what is the minimum number of trominos needed?

[b]p1.[/b] Let $a$ be an integer. How many fractions $\frac{a}{100}$ are greater than $\frac17$ and less than $\frac13$ ?. [b]p2.[/b] Justin Bieber invited Justin Timberlake and Justin Shan to eat sushi. There were $5$ different kinds of fish, $3$ different rice colors, and $11$ different sauces. Justin Shan insisted on a spicy sauce. If the probability of a sushi combination that pleased Justin Shan is $6/11$, then how many non-spicy sauces were there? [b]p3.[/b] A palindrome is any number that reads the same forward and backward (for example, $99$ and $50505$ are palindromes but $2020$ is not). Find the sum of all three-digit palindromes whose tens digit is $5$. [b]p4.[/b] Isaac is given an online quiz for his chemistry class in which he gets multiple tries. The quiz has $64$ multiple choice questions with $4$ choices each. For each of his previous attempts, the computer displays Isaac's answer to that question and whether it was correct or not. Given that Isaac is too lazy to actually read the questions, the maximum number of times he needs to attempt the quiz to guarantee a $100\%$ can be expressed as $2^{2^k}$. Find $k$. [b]p5.[/b] Consider a three-way Venn Diagram composed of three circles of radius $1$. The area of the entire Venn Diagram is of the form $\frac{a}{b}\pi +\sqrt{c}$ for positive integers $a$, $b$, $c$ where $a$, $b$ are relatively prime. Find $a+b+c$. (Each of the circles passes through the center of the other two circles) [b]p6.[/b] The sum of two four-digit numbers is $11044$. None of the digits are repeated and none of the digits are $0$s. Eight of the digits from $1-9$ are represented in these two numbers. Which one is not? [b]p7.[/b] Al wants to buy cookies. He can buy cookies in packs of $13$, $15$, or $17$. What is the maximum number of cookies he can not buy if he must buy a whole number of packs of each size? [b]p8.[/b] Let $\vartriangle ABC$ be a right triangle with base $AB = 2$ and hypotenuse $AC = 4$ and let $AD$ be a median of $\vartriangle ABC$. Now, let $BE$ be an altitude in $\vartriangle ABD$ and let $DF$ be an altitude in $\vartriangle ADC$. The quantity $(BE)^2 - (DF)^2$ can be expressed as a common fraction $\frac{a}{b}$ in lowest terms. Find $a + b$. [b]p9.[/b] Let $P(x)$ be a monic cubic polynomial with roots $r$, $s$, $t$, where $t$ is real. Suppose that $r + s + 2t = 8$, $2rs + rt + st = 12$ and $rst = 9$. Find $|P(2)|$. [b]p10.[/b] Let S be the set $\{1, 2,..., 21\}$. How many $11$-element subsets $T$ of $S$ are there such that there does not exist two distinct elements of $T$ such that one divides the other? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $ a_1, a_2, \ldots , a_n$ be distinct positive integers and let $ M$ be a set of $ n \minus{} 1$ positive integers not containing $ s \equal{} a_1 \plus{} a_2 \plus{} \ldots \plus{} a_n.$ A grasshopper is to jump along the real axis, starting at the point $ 0$ and making $ n$ jumps to the right with lengths $ a_1, a_2, \ldots , a_n$ in some order. Prove that the order can be chosen in such a way that the grasshopper never lands on any point in $ M.$ [i]Proposed by Dmitry Khramtsov, Russia[/i]

Dragoons take up $1\times 1$ squares in the plane with sides parallel to the coordinate axes such that the interiors of the squares do not intersect. A dragoon can fire at another dragoon if the difference in the $x$-coordinates of their centers and the difference in the y-coordinates of their centers are both at most $6$, regardless of any dragoons in between. For example, a dragoon centered at $(4, 5)$ can re at a dragoon centered at the origin, but a dragoon centered at $(7, 0)$ can not. A dragoon cannot fire at itself. What is the maximum number of dragoons that can fire at a single dragoon simultaneously?

Consider the set of all strictly decreasing sequences of $n$ natural numbers having the property that in each sequence no term divides any other term of the sequence. Let $A = (a_j)$ and $B = (b_j)$ be any two such sequences. We say that $A$ precedes $B$ if for some $k$, $a_k < b_k$ and $a_i = b_i$ for $i < k$. Find the terms of the first sequence of the set under this ordering.

This ISL 2005 problem has not been used in any TST I know. A pity, since it is a nice problem, but in its shortlist formulation, it is absolutely incomprehensible. Here is a mathematical restatement of the problem: Let $k$ be a nonnegative integer. A forest consists of rooted (i. e. oriented) trees. Each vertex of the forest is either a leaf or has two successors. A vertex $v$ is called an [i]extended successor[/i] of a vertex $u$ if there is a chain of vertices $u_{0}=u$, $u_{1}$, $u_{2}$, ..., $u_{t-1}$, $u_{t}=v$ with $t>0$ such that the vertex $u_{i+1}$ is a successor of the vertex $u_{i}$ for every integer $i$ with $0\leq i\leq t-1$. A vertex is called [i]dynastic[/i] if it has two successors and each of these successors has at least $k$ extended successors. Prove that if the forest has $n$ vertices, then there are at most $\frac{n}{k+2}$ dynastic vertices.

Determine for which positive integers $ k$ the set \[ X \equal{} \{1990, 1990 \plus{} 1, 1990 \plus{} 2, \ldots, 1990 \plus{} k\}\] can be partitioned into two disjoint subsets $ A$ and $ B$ such that the sum of the elements of $ A$ is equal to the sum of the elements of $ B.$

A polyhedron has an even number of edges. Show that we can place an arrow on each edge so that each vertex has an even number of arrows pointing towards it (on adjacent edges).

Let $ p_1,\dots,p_k$ be prime numbers, and let $ S$ be the set of those integers whose all prime divisors are among $ p_1,\dots,p_k$. For a finite subset $ A$ of the integers let us denote by $ \mathcal G(A)$ the graph whose vertices are the elements of $ A$, and the edges are those pairs $ a,b\in A$ for which $ a \minus{} b\in S$. Does there exist for all $ m\geq 3$ an $ m$-element subset $ A$ of the integers such that (i) $ \mathcal G(A)$ is complete? (ii) $ \mathcal G(A)$ is connected, but all vertices have degree at most 2?

Numbers from $1$ to $8$ are placed on the vertices of a cube, one on each of the eight vertices, so that the sum of the numbers on any three vertices of the same face is greater than $9$. Determines the minimum value that the sum of the numbers on one side can have.

Let $\Lambda$ denote the set of points $(x,y)$ in 2D space with integer coordinates such that $0\leq x\leq 4$ and $0\leq y\leq 2$. That is, \[ \Lambda=\{ (x,y) \in \mathbb{Z}^2: 0\leq x\leq 4, \ 0\leq y\leq 2 \}. \] Find the number of ways to connect points of $\Lambda$ with segments of length $\sqrt{2}$ or $\sqrt{5}$ such that the interior of any unit square with vertices in $\Lambda$ contains part of exactly one segment; an example is shown below (connections that differ by reflections are distinct). [asy] unitsize(1cm); dot((0,0)); dot((1,0)); dot((2,0)); dot((3,0)); dot((4,0)); dot((0,1)); dot((1,1)); dot((2,1)); dot((3,1)); dot((4,1)); dot((0,2)); dot((1,2)); dot((2,2)); dot((3,2)); dot((4,2)); draw((0,0)--(1,1)); draw((0,2)--(2,1)); draw((1,1)--(2,0)); draw((2,0)--(3,2)); draw((3,1)--(4,2)); draw((3,0)--(4,1)); [/asy]

Let $n$ be a positive integer and $S = \{ m \mid 2^n \le m < 2^{n+1} \}$. We call a pair of non-negative integers $(a, b)$ [i]fancy[/i] if $a + b$ is in $S$ and is a palindrome in binary. Find the number of [i]fancy[/i] pairs $(a, b)$.

Prove that from any finite set of points on the plane, you can remove a point from the bottom in such a way that the remaining set can be split into two parts of smaller diameter. (Diameter is the maximum distance between points of the set.) [hide=original wording]Докажите, что из любого конечного множества точек на плоскости можно так удалитьо дну точку, что оставшееся множество можно разбить на две части меньшего диаметра. (Диаметр—это максимальное расстояние между точками множества.)[/hide]

There is $n$ black points in the plane.We do the following algorithm: Start from any point from those $n$ points and colour it red. Then connect this point to the nearest black point available and colour this new point red. Then do the same with this point but at any step , but you are never allowed to draw a line which intersects on of the current drawn segments. If you reach an intersection , the algorithm is over. Is it true that for any $n$ and at any initial position , we can start from a point st in the algorithm , we reach all the points?

Suppose that we have a set $S$ of $756$ arbitrary integers between $1$ and $2008$ ($1$ and $2008$ included). Prove that there are two distinct integers $a$ and $b$ in $S$ such that their sum $a + b$ is divisible by $8$.

[u]Round 1[/u] [b]p1.[/b] Alice and Bob played $25$ games of rock-paper-scissors. Alice played rock $12$ times, scissors $6$ times, and paper $7$ times. Bob played rock $13$ times, scissors $9$ times, and paper $3$ times. If there were no ties, who won the most games? (Remember, in each game each player picks one of rock, paper, or scissors. Rock beats scissors, scissors beat paper, and paper beats rock. If they choose the same object, the result is a tie.) [b]p2.[/b] On the planet Vulcan there are eight big volcanoes and six small volcanoes. Big volcanoes erupt every three years and small volcanoes erupt every two years. In the past five years, there were $30$ eruptions. How many volcanoes could erupt this year? [b]p3.[/b] A tangle is a sequence of digits constructed by picking a number $N\ge 0$ and writing the integers from $0$ to $N$ in some order, with no spaces. For example, $010123459876$ is a tangle with $N = 10$. A palindromic sequence reads the same forward or backward, such as $878$ or $6226$. The shortest palindromic tangle is $0$. How long is the second-shortest palindromic tangle? [b]p4.[/b] Balls numbered $1$ to $N$ have been randomly arranged in a long input tube that feeds into the upper left square of an $8 \times 8$ board. An empty exit tube leads out of the lower right square of the board. Your goal is to arrange the balls in order from $1$ to $N$ in the exit tube. As a move, you may 1. move the next ball in line from the input tube into the upper left square of the board, 2. move a ball already on the board to an adjacent square to its right or below, or 3. move a ball from the lower right square into the exit tube. No square may ever hold more than one ball. What is the largest number $N$ for which you can achieve your goal, no matter how the balls are initially arranged? You can see the order of the balls in the input tube before you start. [img]https://cdn.artofproblemsolving.com/attachments/1/8/bbce92750b01052db82d58b96584a36fb5ca5b.png[/img] [b]p5.[/b] A $2018 \times 2018$ board is covered by non-overlapping $2 \times 1$ dominoes, with each domino covering two squares of the board. From a given square, a robot takes one step to the other square of the domino it is on and then takes one more step in the same direction. Could the robot continue moving this way forever without falling off the board? [img]https://cdn.artofproblemsolving.com/attachments/9/c/da86ca4ff0300eca8e625dff891ed1769d44a8.png[/img] [u]Round 2[/u] [b]p6.[/b] Seventeen teams participated in a soccer tournament where a win is worth $1$ point, a tie is worth $0$ points, and a loss is worth $-1$ point. Each team played each other team exactly once. At least $\frac34$ of all games ended in a tie. Show that there must be two teams with the same number of points at the end of the tournament. [b]p7.[/b] The city of Old Haven is known for having a large number of secret societies. Any person may be a member of multiple societies. A secret society is called influential if its membership includes at least half the population of Old Haven. Today, there are $2018$ influential secret societies. Show that it is possible to form a council of at most $11$ people such that each influential secret society has at least one member on the council. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Each integer in $\{1, 2, 3, . . . , 2020\}$ is coloured in such a way that, for all positive integers $a$ and $b$ such that $a + b \leq 2020$, the numbers $a$, $b$ and $a + b$ are not coloured with three different colours. Determine the maximum number of colours that can be used. [i]Massimiliano Foschi, Italy[/i]

Let $n \geq 2$ be a positive integer. Let $\mathcal{R}$ be a connected set of unit squares on a grid. A [i]bar[/i] is a rectangle of length or width $1$ which is fully contained in $\mathcal{R}$. A bar is [i]special[/i] if it is not fully contained within any larger bar. Given that $\mathcal{R}$ contains special bars of sizes $1 \times 2,1 \times 3,\ldots,1 \times n$, find the smallest possible number of unit squares in $\mathcal{R}$. [i]Srinivas Arun[/i]

A participant is given a deck of thirteen cards numerated from 1 to 13, from which he chooses seven and gives them to the assistant. Then the assistant chooses three of these seven cards and the participant – one of the remaining six in his hand. The magician then takes the chosen four cards (arranged by the participant) and guesses which one is chosen from the participant. What should the magician and assistant do so that the magic trick always happens?

A number is written in each square of a chessboard, so that each number not on the border is the mean of the $4$ neighboring numbers. Show that if the largest number is $N$, then there is a number equal to $N$ in the border squares.

Given the finite set $M$ with $m$ elements and $1986$ further sets $M_1,M_2,M_3,...,M_{1986}$, each of which contains more than $\frac{m}{2}$ elements from $M$ . Show that no more than ten elements need to be marked in order for any set $M_i$ ($i =1, 2, 3,..., 1986$) contains at least one marked element.

Consider a $ 2018\times 2018$. board. An "LC-tile" is a tile consisting of $9$ unit squares, having the shape as in the gure below. What is the maximum number of "LC-tiles" that can be placed on the board without superposing them? (Each of the $9$ unit squares of the tile must cover one of the unit squares of the board; a tile may be rotated, turned upside down, etc.) [img]https://cdn.artofproblemsolving.com/attachments/7/4/a2f992bc0341def1a6e5e26ba8a9eb3384698a.png [/img] Alexandru Girban

There are a million numbered chairs at a large round table. The Sultan has seated a million wise men on them. Each of them sees the thousand people following him in clockwise order. Each of them was given a cap of black or white color, and they must simultaneously write down on their own piece of paper a guess about the color of their cap. Those who do not guess will be executed. The wise men had the opportunity to agree on a strategy before the test. What is the largest number of survivors that they can guarantee?

Consider $n$ points, some of them connected by segments. These segments do not intersect each other. You can reach every point from any every other one in exactly one way by traveling along the segments. Prove that the total number of segments is $n-1$.