The King called two wizards. He ordered First Wizard to write down $100$ positive integers (not necessarily distinct) on cards without revealing them to Second Wizard. Second Wizard must correctly determine all these integers, otherwise both wizards will lose their heads. First Wizard is allowed to provide Second Wizard with a list of distinct integers, each of which is either one of the integers on the cards or a sum of some of these integers. He is not allowed to tell which integers are on the cards and which integers are their sums. If Second Wizard correctly determines all $100$ integers the King tears as many hairs from each wizard's beard as the number of integers in the list given to Second Wizard. What is the minimal number of hairs each wizard should sacrice to stay alive?

Given $120$ unit squares arbitrarily situated in the $20\times 25$ rectangle. Prove that you can place a circle with the unit diameter without intersecting any of the squares.

Let $ n$ and $ k$ be positive integers with $ k \geq n$ and $ k \minus{} n$ an even number. Let $ 2n$ lamps labelled $ 1$, $ 2$, ..., $ 2n$ be given, each of which can be either [i]on[/i] or [i]off[/i]. Initially all the lamps are off. We consider sequences of steps: at each step one of the lamps is switched (from on to off or from off to on). Let $ N$ be the number of such sequences consisting of $ k$ steps and resulting in the state where lamps $ 1$ through $ n$ are all on, and lamps $ n \plus{} 1$ through $ 2n$ are all off. Let $ M$ be number of such sequences consisting of $ k$ steps, resulting in the state where lamps $ 1$ through $ n$ are all on, and lamps $ n \plus{} 1$ through $ 2n$ are all off, but where none of the lamps $ n \plus{} 1$ through $ 2n$ is ever switched on. Determine $ \frac {N}{M}$. [i]Author: Bruno Le Floch and Ilia Smilga, France[/i]

Two players $A$ and $B$ take turns playing the following game: There is a pile of $2003$ stones. In his first turn, $A$ selects a divisor of $2003$ and removes this number of stones from the pile. $B$ then chooses a divisor of the number of remaining stones, and removes that number of stones from the new pile, and so on. The player who has to remove the last stone loses. Show that one of the two players has a winning strategy and describe the strategy.

It is proposed to partition a set of positive integers into two disjoint subsets $ A$ and $ B$ subject to the conditions [b]i.)[/b] 1 is in $ A$ [b]ii.)[/b] no two distinct members of $ A$ have a sum of the form $ 2^k \plus{} 2, k \equal{} 0,1,2, \ldots;$ and [b]iii.)[/b] no two distinct members of B have a sum of that form. Show that this partitioning can be carried out in unique manner and determine the subsets to which 1987, 1988 and 1989 belong.

[u]Round 1[/u] [b]p1.[/b] Jom and Terry both flip a fair coin. What is the probability both coins show the same side? [b]p2.[/b] Under the same standard air pressure, when measured in Fahrenheit, water boils at $212^o$ F and freezes at $32^o$ F. At thesame standard air pressure, when measured in Delisle, water boils at $0$ D and freezes at $150$ D. If x is today’s temperature in Fahrenheit and y is today’s temperature expressed in Delisle, we have $y = ax + b$. What is the value of $a + b$? (Ignore units.) [b]p3.[/b] What are the last two digits of $5^1 + 5^2 + 5^3 + · · · + 5^{10} + 5^{11}$? [u]Round 2[/u] [b]p4.[/b] Compute the average of the magnitudes of the solutions to the equation $2x^4 + 6x^3 + 18x^2 + 54x + 162 = 0$. [b]p5.[/b] How many integers between $1$ and $1000000$ inclusive are both squares and cubes? [b]p6.[/b] Simon has a deck of $48$ cards. There are $12$ cards of each of the following $4$ suits: fire, water, earth, and air. Simon randomly selects one card from the deck, looks at the card, returns the selected card to the deck, and shuffles the deck. He repeats the process until he selects an air card. What is the probability that the process ends without Simon selecting a fire or a water card? [u]Round 3[/u] [b]p7.[/b] Ally, Beth, and Christine are playing soccer, and Ally has the ball. Each player has a decision: to pass the ball to a teammate or to shoot it. When a player has the ball, they have a probability $p$ of shooting, and $1 - p$ of passing the ball. If they pass the ball, it will go to one of the other two teammates with equal probability. Throughout the game, $p$ is constant. Once the ball has been shot, the game is over. What is the maximum value of $p$ that makes Christine’s total probability of shooting the ball $\frac{3}{20}$ ? [b]p8.[/b] If $x$ and $y$ are real numbers, then what is the minimum possible value of the expression $3x^2 - 12xy + 14y^2$ given that $x - y = 3$? [b]p9.[/b] Let $ABC$ be an equilateral triangle, let $D$ be the reflection of the incenter of triangle $ABC$ over segment $AB$, and let $E$ be the reflection of the incenter of triangle $ABD$ over segment $AD$. Suppose the circumcircle $\Omega$ of triangle $ADE$ intersects segment $AB$ again at $X$. If the length of $AB$ is $1$, find the length of $AX$. [u]Round 4[/u] [b]p10.[/b] Elaine has $c$ cats. If she divides $c$ by $5$, she has a remainder of $3$. If she divides $c$ by $7$, she has a remainder of $5$. If she divides $c$ by $9$, she has a remainder of $7$. What is the minimum value $c$ can be? [b]p11.[/b] Your friend Donny offers to play one of the following games with you. In the first game, he flips a fair coin and if it is heads, then you win. In the second game, he rolls a $10$-sided die (its faces are numbered from $1$ to $10$) $x$ times. If, within those $x$ rolls, the number $10$ appears, then you win. Assuming that you like winning, what is the highest value of $x$ where you would prefer to play the coin-flipping game over the die-rolling game? [b]p12.[/b] Let be the set $X = \{0, 1, 2, ..., 100\}$. A subset of $X$, called $N$, is defined as the set that contains every element $x$ of $X$ such that for any positive integer $n$, there exists a positive integer $k$ such that n can be expressed in the form $n = x^{a_1}+x^{a_2}+...+x^{a_k}$ , for some integers $a_1, a_2, ..., a_k$ that satisfy $0 \le a_1 \le a_2 \le ... \le a_k$. What is the sum of the elements in $N$? PS. You should use hide for answers. Rounds 5-7 have be posted [url=https://artofproblemsolving.com/community/c4h2782880p24446580]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

The numbers $1$ to $500$ are written on a board. Two pupils $A$ and $B$ play the following game: A player in turn deletes one of the numbers from the board. The game is over when only two numbers remain. Player $B$ wins if the sum of the two remaining numbers is divisible by $3,$ otherwise $A$ wins. If $A$ plays first, show that $B$ has a winning strategy.

Given is a regular polygon. Volodya wants to mark $k$ points on its perimeter so that any another regular polygon (maybe having a different number of sides) doesn’t contain all marked points on its perimeter. Find the minimal $k$ sufficient for any given polygon.

Let $ Z $ be a set of $ n $ elements. Find the number of such pairs of sets $ (A, B) $ such that $ A $ is contained in $ B $ and $ B $ is contained in $ Z $. We assume that every set also contains itself and the empty set.

In the set of $20$ elements $\{1, 2, 3, 4, 5, 6, 7, 8, 9, 0, A, B, C, D, J, K, L, U, X, Y , Z\}$ we have made a random sequence of $28$ throws. What is the probability that the sequence $CUBA \ JULY \ 1987$ appears in this order in the sequence already thrown?

In a sequence of positive integers, a inversion is a pair of positions, where the number in left is greater than the number in right. For example in the sequence $2, 5, 3, 1, 3$ has $5$ inversions{(5,1),(3,1),(5,3),(2,1),(5,3)}. Find the greatest number of inversions in a sequence where the sum of elements is $n$ a) where $n=7$ b) where $n=2019$

Let $a_0, a_1, \ldots, a_9$ and $b_1 , b_2, \ldots,b_9$ be positive integers such that $a_9<b_9$ and $a_k \neq b_k, 1 \leq k \leq 8.$ In a cash dispenser/automated teller machine/ATM there are $n\geq a_9$ levs (Bulgarian national currency) and for each $1 \leq i \leq 9$ we can take $a_i$ levs from the ATM (if in the bank there are at least $a_i$ levs). Immediately after that action the bank puts $b_i$ levs in the ATM or we take $a_0$ levs. If we take $a_0$ levs from the ATM the bank doesn’t put any money in the ATM. Find all possible positive integer values of $n$ such that after finite number of takings money from the ATM there will be no money in it.

Let $M$ be the set of positive integers which do not have a prime divisor greater than 3. For any infinite family of subsets of $M$, say $A_1,A_2,\ldots $, prove that there exist $i\ne j$ such that for each $x\in A_i$ there exists some $y\in A_j $ such that $y\mid x$.

Let $A$ be a set consisting of 6 points in the plane. denoted $n(A)$ as the number of the unit circles which meet at least three points of $A$. Find the maximum of $n(A)$

A table with $m$ rows and $n$ columns is given. At any move one chooses some empty cells such that any two of them lie in different rows and columns, puts a white piece in any of those cells and then puts a black piece in the cells whose rows and columns contain white pieces. The game is over if it is not possible to make a move. Find the maximum possible number of white pieces that can be put on the table.

Let a [i]triplet [/i] be some set of three distinct pairwise parallel lines. $20$ triplets are drawn on a plane. Find the maximum number of regions these $60$ lines can divide the plane into.

Suppose that $1985$ points are given inside a unit cube. Show that one can always choose $32$ of them in such a way that every (possibly degenerate) closed polygon with these points as vertices has a total length of less than $8 \sqrt 3.$

The base of a prism is an $n$-gon. We wish to colour its $2n$ vertices in three colours in such a way that every vertex is connected by edges to vertices of all three colours. (a) Prove that if $n$ is divisible by $3$, then the task is possible. {b) Prove that if the task is possible, then $n$ is divisible by $3$. (A Shapovalov)

The numbers $ 1, 2,\ldots, 50 $ are written on a blackboard. Each minute any two numbers are erased and their positive difference is written instead. At the end one number remains. Which values can take this number?

The kingdom of Anisotropy consists of $n$ cities. For every two cities there exists exactly one direct one-way road between them. We say that a [i]path from $X$ to $Y$[/i] is a sequence of roads such that one can move from $X$ to $Y$ along this sequence without returning to an already visited city. A collection of paths is called [i]diverse[/i] if no road belongs to two or more paths in the collection. Let $A$ and $B$ be two distinct cities in Anisotropy. Let $N_{AB}$ denote the maximal number of paths in a diverse collection of paths from $A$ to $B$. Similarly, let $N_{BA}$ denote the maximal number of paths in a diverse collection of paths from $B$ to $A$. Prove that the equality $N_{AB} = N_{BA}$ holds if and only if the number of roads going out from $A$ is the same as the number of roads going out from $B$. [i]Proposed by Warut Suksompong, Thailand[/i]

Olja writes down $n$ positive integers $a_1, a_2, \ldots, a_n$ smaller than $p_n$ where $p_n$ denotes the $n$-th prime number. Oleg can choose two (not necessarily different) numbers $x$ and $y$ and replace one of them with their product $xy$. If there are two equal numbers Oleg wins. Can Oleg guarantee a win? [i]Proposed by Matko Ljulj.[/i]

Some cities of a country consisting of $n$ cities are connected by round trip flights so that there are at least $k$ flights from any city and any city is reachable from any city. Prove that for any such flight organization these flights can be distributed among $n-k$ air companies so that one can reach any city from any city by using of at most one flight of each air company.

There are 2000 cities in a country and no roads. Prove that some cities can be connected by a road such that there would be 2 cities with 1 road passing through them, there would be 2 cities with 2 roads passim through them,...,there would be 2 cities with 1000 roads passing through them. [I]Proposed by F. Bakharev[/i]

Find the maximum possible number of edges of a simple graph with $8$ vertices and without any quadrilateral. (a simple graph is an undirected graph that has no loops (edges connected at both ends to the same vertex) and no more than one edge between any two different vertices.)

$4n$ first grade students at Songkhla Primary School, including $2n$ boys and $2n$ girls, participate in a taekwondo tournament where every pair of students compete against each other exactly once. The tournament is scored as follows: $\bullet$ In a match between two boys or between two girls, a win is worth $3$ points, a draw $1$ point, and a loss $0$ points. $\bullet$ In a math between a boy and a girl, if the boy wins, he receives $2$ points, else he receives $0$ points. If the girl wins, she receives $3$ points, if she draws, she receives $2$ points, and if she loses, she receives $0$ points. After the tournament, the total score of each student is calculated. Let $P$ be the number of matches ending in a draw, and let $Q$ be the total number of matches. Suppose that the maximum total score is $4n - 1$. Find $P/Q$.