For finite sets $A,M$ such that $A \subseteq M \subset \mathbb{Z}^+$, we define $$f_M(A)=\{x\in M \mid x\text{ is divisible by an odd number of elements of }A\}.$$ Given a positive integer $k$, we call $M$ [i]k-colorable[/i] if it is possible to color the subsets of $M$ with $k$ colors so that for any $A \subseteq M$, if $f_M(A)\neq A$ then $f_M(A)$ and $A$ have different colors. Determine the least positive integer $k$ such that every finite set $M \subset\mathbb{Z}^+$ is k-colorable.

A sequence $a_1, a_2, \ldots $ consisting of $1$'s and $0$'s satisfies for all $k>2016$ that \[ a_k=0 \quad \Longleftrightarrow \quad a_{k-1}+a_{k-2}+\cdots+a_{k-2016}>23. \] Prove that there exist positive integers $N$ and $T$ such that $a_k=a_{k+T}$ for all $k>N$.

$4n$ 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 $P/Q$.

In each square of a $5 \times 5$ board one of the numbers $2, 3, 4$ or $5$ is written so that the the sum of all the numbers in each row, in each column and on each diagonal is always even. How many ways can we fill the board? Clarification. A $5\times 5$ board has exactly $18$ diagonals of different sizes. In particular, the corners are size $ 1$ diagonals.

A table consisting of $5$ columns and $32$ rows, which are filled with zero and one numbers, are "varied", if no two lines are filled in the same way.\\ On the exterior of a cylinder, a table with $32$ rows and $16$ columns is constructed. Is it possible to fill the numbers cells of the table with numbers zero and one, such that any five consecutive columns, table $32\times5$ created by these columns, is a varied one? [i]Proposed by Morteza Saghafian[/i]

Set $A=\{1,2,\cdots,n\}$. If there exists nonempty sets $B,C$, such that $B\cap C=\varnothing,B\cup C=A$. Sum of Squares of all elements in $B$ is $M$, Sum of Squares of all elements in $C$ is $N$, $M-N=2016$. Find the minimum value of $n$.

The numbers $1, 2,\ldots, 2006$ are written around the circumference of a circle. A [i]move[/i] consists of exchanging two adjacent numbers. After a sequence of such moves, each number ends up $13$ positions to the right of its initial position. lf the numbers $1, 2,\ldots, 2006$ are partitioned into $1003$ distinct pairs, then show that in at least one of the moves, the two numbers of one of the pairs were exchanged.

Cross-shaped tiles are to be placed on a $8\times8$ square grid without overlapping. Find the largest possible number of tiles that can be placed. [img]https://services.artofproblemsolving.com/download.php?id=YXR0YWNobWVudHMvMy8zL2EyY2Q4MDcyMWZjM2FmZGFhODkxYTk5ZmFiMmMwNzk0MzZmYmVjLnBuZw==&rn=U2NyZWVuIFNob3QgMjAyMS0wNC0wNyBhdCA2LjIzLjU4IEFNLnBuZw[/img]

A clock has an hour, minute, and second hand, all of length $1$. Let $T$ be the triangle formed by the ends of these hands. A time of day is chosen uniformly at random. What is the expected value of the area of $T$? [i]Proposed by Dylan Toh[/i]

$n(\geq 4)$ islands are connected by bridges to satisfy the following conditions: [list] [*]Each bridge connects only two islands and does not go through other islands. [*]There is at most one bridge connecting any two different islands. [*]There does not exist a list $A_1, A_2, \ldots, A_{2k}(k \geq 2)$ of distinct islands that satisfy the following: [center]For every $i=1, 2, \ldots, 2k$, the two islands $A_i$ and $A_{i+1}$ are connected by a bridge. (Let $A_{2k+1}=A_1$)[/center] [/list] Prove that the number of the bridges is at most $\frac{3(n-1)}{2}$.

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.

[b]p1.[/b] a) There are barrels weighing $1, 2, 3, 4, ..., 19, 20$ pounds. Is it possible to distribute them equally (by weight) into three trucks? b) The same question for barrels weighing $1, 2, 3, 4, ..., 9, 10$ pounds. [b]p2.[/b] There are apples and pears in the basket. If you add the same number of apples there as there are now pears (in pieces), then the percentage of apples will be twice as large as what you get if you add as many pears to the basket as there are now apples. What percentage of apples are in the basket now? [b]p3.[/b] What is the smallest number of integers from $1000$ to $1500$ that must be marked so that any number $x$ from $1000$ to $1500$ differs from one of the marked numbers by no more than $10\% $of the value of $x$? [b]p4.[/b] Draw a perpendicular from a given point to a given straight line, having a compass and a short ruler (the length of the ruler is significantly less than the distance from the point to the straight line; the compass reaches from the point to the straight line “with a margin”). [b]p5.[/b] There is a triangle on the chessboard (left figure). It is allowed to roll it around the sides (in this case, the triangle is symmetrically reflected relative to the side around which it is rolled). Can he, after a few steps, take the position shown in right figure? [img]https://cdn.artofproblemsolving.com/attachments/f/5/eeb96c92f30b837e7ed2cdf7cf77b0fbb8ceda.png[/img] [b]p6.[/b] The natural number $a$ is less than the natural number $b$. In this case, the sum of the digits of number $a$ is $100$ less than the sum of the digits of number $b$. Prove that between the numbers $ a$ and $b$ there is a number whose sum of digits is $43$ more than the sum of the digits of $a$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics]here.[/url]

A convex $N\text{-gon}$ is divided by diagonals into triangles so that no two diagonals intersect inside the polygon. The triangles are painted in black and white so that any two triangles are painted in black and white so that any two triangles with a common side are painted in different colors. For each $N$ find the maximal difference between the numbers of black and white triangles.

A quadruple of integers $(a, b, c, d)$ is said good if $ad-bc=2020$. Two good quadruplets are said to be dissimilar if it is not possible to obtain one from the other using a finite number of applications of the following operations: $$(a,b,c,d) \rightarrow (-c,-d,a,b)$$ $$(a,b,c,d) \rightarrow (a,b,c+a,d+b)$$ $$(a,b,c,d) \rightarrow (a,b,c-a,d-b)$$ Let $A$ be a set of $k$ good quadruples, two by two dissimilar. Show that $k \leq 4284$.

Given a permutation $\sigma = (a_1,a_2,a_3,...a_n)$ of $(1,2,3,...n)$ , an ordered pair $(a_j,a_k)$ is called an inversion of $\sigma$ if $a \leq j < k \leq n$ and $a_j > a_k$. Let $m(\sigma)$ denote the no. of inversions of the permutation $\sigma$. Find the average of $m(\sigma)$ as $\sigma$ varies over all permutations.

Show that given any nine integers, we can find four, $a, b, c, d$ such that $a + b - c - d$is divisible by $20$. Show that this is not always true for eight integers.

How many distinct positive integers can be formed by choosing their digits from the string $04072019$?

a) Is it possible to place $2024$ checkers on a board $70 \times 70$ so that any square $2 \times 2$ contains even number of checkers? b) Is it possible to place $2023$ checkers on a board $70 \times 70$ so that any square $2 \times 2$ contains odd number of checkers?

There are $n\ge 3$ girls in a class sitting around a circular table, each having some apples with her. Every time the teacher notices a girl having more apples than both of her neighbours combined, the teacher takes away one apple from that girl and gives one apple each to her neighbours. Prove that, this process stops after a finite number of steps. (Assume that, the teacher has an abundant supply of apples.)

Square table $5 \times 5$ is filled with numbers in a following way. [img]https://services.artofproblemsolving.com/download.php?id=YXR0YWNobWVudHMvYi8zLzQ0Y2M1NjdiNjQ3NjhlYTAwMWQ0MTg2ZjIwZWE4NzkwYzcwYWFkLnBuZw==&rn=dGFiZWxpY2EucG5n[/img] We can change the table in a way we take two arbitrary numbers from the table and we decrease both of them with value of smaller of those two. Can we get to the table with all zeros?

There are $4n$ pebbles of weights $1, 2, 3, \dots, 4n.$ Each pebble is coloured in one of $n$ colours and there are four pebbles of each colour. Show that we can arrange the pebbles into two piles so that the following two conditions are both satisfied: [list] [*]The total weights of both piles are the same. [*] Each pile contains two pebbles of each colour. [/list] [i]Proposed by Milan Haiman, Hungary and Carl Schildkraut, USA[/i]

Let $T$ denote the $15$-element set $\{10a+b:a,b\in\mathbb{Z},1\le a<b\le 6\}$. Let $S$ be a subset of $T$ in which all six digits $1,2,\ldots ,6$ appear and in which no three elements together use all these six digits. Determine the largest possible size of $S$.

The identifier of a book is an n-tuple of numbers 0, 1, .... , 9, followed by a checksum. The checksum is computed by a fixed rule that satisfies the following property: whenever one increases a single number in the n-tuple (without modifying the other numbers), the checksum also increases. Find the smallest possible number of required checksums if all possible n-tuples are in use.

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.

What is the largest number of cells that can be marked on a $100 \times 100$ board in such a way that a chess king from any cell attacks no more than two marked ones? (The cell on which a king stands is also considered to be attacked by this king.)