There are $4$ piles of stones with the following quantities: $1004$, $1005$, $2009$ and $2010$. A legitimate move is to remove a stone from each from $3$ different piles. Two players $A$ and $B$ play in turns. $A$ begins the game . The player who, on his turn, cannot make a legitimate move, loses. Determine which of the players has a winning strategy and give a strategy for that player.

A grid is called $k$-special if in each cell is written a distinct integer such that the set of integers in the grid is precisely the set of positive divisors of $k$. A grid is called $k$-awesome if it is $k$-special and for each positive divisor $m$ of $k$, there exists an $m$-special grid within this $k$-special grid (within meaning you could draw a box in this grid to obtain the new grid). Find the sum of the $4$ smallest integers $k$ for which no $k$-awesome grid exists. [i]Proposed by Oliver Hayman[/i]

It is given set $A=\{1,2,3,...,2n-1\}$. From set $A$, at least $n-1$ numbers are expelled such that: $a)$ if number $a \in A$ is expelled, and if $2a \in A$ then $2a$ must be expelled $b)$ if $a,b \in A$ are expelled, and $a+b \in A$ then $a+b$ must be also expelled Which numbers must be expelled such that sum of numbers remaining in set stays minimal

We are given a fixed point on the circle of radius $1$, and going from this point along the circumference in the positive direction on curved distances $0, 1, 2, \ldots $ from it we obtain points with abscisas $n = 0, 1, 2, .\ldots$ respectively. How many points among them should we take to ensure that some two of them are less than the distance $\frac 15$ apart ?

Let $n>1$ be an integer. Find the number of permutations $(a_1,a_2,\ldots,a_n)$ of the numbers $1,2,\ldots,n$ such that $a_i>a_{i+1}$ holds for exactly one $i\in\{1,2,\ldots,n-1\}$.

There are $1000$ cities in the country of Euleria, and some pairs of cities are linked by dirt roads. It is possible to get from any city to any other city by traveling along these roads. Prove that the government of Euleria may pave some of the roads so that every city will have an odd number of paved roads leading out of it.

Four drivers took part in the round-robin racing. Their cars started simultaneously from one point and moved at constant speeds. It is known that after the start of the race, for any three cars there was a moment when they met. Prove that after the start of the race there will be a moment when all 4 cars meet. (We consider races to be infinitely long in time.)

Let $n$ be a positive integer relatively prime to $6$. We paint the vertices of a regular $n$-gon with three colours so that there is an odd number of vertices of each colour. Show that there exists an isosceles triangle whose three vertices are of different colours.

[color=darkred]On a table there are $k\ge 2$ piles having $n_1,n_2,\ldots,n_k$ pencils respectively. A [i]move[/i] consists in choosing two piles having $a$ and $b$ pencils respectively, $a\ge b$ and transferring $b$ pencils from the first pile to the second one. Find the necessary and sufficient condition for $n_1,n_2,\ldots,n_k$ , such that there exists a succession of moves through which all pencils are transferred to the same pile.[/color]

How many ways are there to place the numbers $2, 3, . . . , 10$ in a $3 \times 3$ grid, such that any two numbers that share an edge are mutually prime?

There are $n$ parallel lines on a plane, and there is a set $S$ of distinct points. Each point in $S$ lies on one of the $n$ lines and is colored either red or blue. Determine the minimum value of $n$ such that if $S$ satisfies the following condition, it is guaranteed that there are infinitely many red points and infinitely many blue points. [list] [*] Each line contains at least one red point and at least one blue point from $S$. [*] Consider a triangle formed by three elements of $S$ located on three distinct lines. If two of the vertices of the triangle are red, there must exist a blue point, not one of the vertices, either inside or on the boundary of the triangle. Similarly, if two of the vertices are blue, there must exist a red point, not one of the vertices, either inside or on the boundary of the triangle. [/list]

[b]p1.[/b] James has $8$ Instagram accounts, $3$ Facebook accounts, $4$ QQ accounts, and $3$ YouTube accounts. If each Instagram account has $19$ pictures, each Facebook account has $5$ pictures and $9$ videos, each QQ account has a total of $17$ pictures, and each YouTube account has $13$ videos and no pictures, how many pictures in total does James have in all these accounts? [b]p2.[/b] If Poonam can trade $7$ shanks for $4$ shinks, and she can trade $10$ shinks for $17$ shenks. How many shenks can Poonam get if she traded all of her $105$ shanks? [b]p3.[/b] Jerry has a bag with $3$ red marbles, $5$ blue marbles and $2$ white marbles. If Jerry randomly picks two marbles from the bag without replacement, the probability that he gets two different colors can be expressed as a fraction $\frac{m}{n}$ in lowest terms. What is $m + n$? [b]p4.[/b] Bob's favorite number is between $1200$ and $4000$, divisible by $5$, has the same units and hundreds digits, and the same tens and thousands digits. If his favorite number is even and not divisible by $3$, what is his favorite number? [b]p5.[/b] Consider a unit cube $ABCDEFGH$. Let $O$ be the center of the face $EFGH$. The length of $BO$ can be expressed in the form $\frac{\sqrt{a}}{b}$, where $a$ and $b$ are simplified to lowest terms. What is $a + b$? [b]p6.[/b] Mr. Eddie Wang is a crazy rich boss who owns a giant company in Singapore. Even though Mr. Wang appears friendly, he finds great joy in firing his employees. His immediately fires them when they say "hello" and/or "goodbye" to him. It is well known that $1/2$ of the total people say "hello" and/or "goodbye" to him everyday. If Mr. Wang had $2050$ employees at the end of yesterday, and he hires $2$ new employees at the beginning of each day, in how many days will Mr. Wang first only have $6$ employees left? [b]p7.[/b] In $\vartriangle ABC$, $AB = 5$, $AC = 6$. Let $D,E,F$ be the midpoints of $\overline{BC}$, $\overline{AC}$, $\overline{AB}$, respectively. Let $X$ be the foot of the altitude from $D$ to $\overline{EF}$. Let $\overline{AX}$ intersect $\overline{BC}$ at $Y$ . Given $DY = 1$, the length of $BC$ is $\frac{p}{q}$ for relatively prime positive integers $p, q$: Find $p + q$. [b]p8.[/b] Given $\frac{1}{2006} = \frac{1}{a} + \frac{1}{b}$ where $a$ is a $4$ digit positive integer and $b$ is a $6$ digit positive integer, find the smallest possible value of $b$. [b]p9.[/b] Pocky the postman has unlimited stamps worth $5$, $6$ and $7$ cents. However, his post office has two very odd requirements: On each envelope, an odd number of $7$ cent stamps must be used, and the total number of stamps used must also be odd. What is the largest amount of postage money Pocky cannot make with his stamps, in cents? [b]p10.[/b] Let $ABCDEF$ be a regular hexagon with side length $2$. Let $G$ be the midpoint of side $DE$. Now let $O$ be the intersection of $BG$ and $CF$. The radius of the circle inscribed in triangle $BOC$ can be expressed in the form $\frac{a\sqrt{b}-\sqrt{c}}{d} $ where $a$, $b$, $c$, $d$ are simplified to lowest terms. What is $a + b + c + d$? [b]p11.[/b] Estimation (Tiebreaker): What is the total number of characters in all of the participants' email addresses in the Accuracy Round? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

For a positive integer $n$, there is a school with $n$ people. For a set $X$ of students in this school, if any two students in $X$ know each other, we call $X$ [i]well-formed[/i]. If the maximum number of students in a well-formed set is $k$, show that the maximum number of well-formed sets is not greater than $3^{(n+k)/3}$. Here, an empty set and a set with one student is regarded as well-formed as well.

There are 2012 chameleons sitting at a round table. At the beginning each has the color red or green. After every full minute, each chamaleon, which has two neighbors of the same color, changes its color from red to green or from green to red. All others keep their color. Show that after $2012$ minutes there are at least $2$ chameleons that have the same often changed color. [hide=original wording]Es sitzen 2012 Chamaleons an einem runden Tisch. Am Anfang besitzt jedes die Farbe rot oder grun. Nach jeder vollen Minute wechselt jedes Cham aleon, welches zwei gleichfarbige Nachbarn hat, seine Farbe von rot zu grun respektive von gr un zu rot. Alle anderen behalten ihre Farbe. Zeige, dass es nach 2012 Minuten mindestens 2 Chamaleons gibt, welche gleich oft die Farbe gewechselt haben.[/hide]

Find the number of permutations of the letters $ABCDE$ where the letters $A$ and $B$ are not adjacent and the letters $C$ and $D$ are not adjacent. For example, count the permutations $ACBDE$ and $DEBCA$ but not $ABCED$ or $EDCBA$.

A test contains $5$ multiple choice questions which have $4$ options in each. Suppose each examinee chose one option for each question. There exists a number $n$, such that for any $n$ sheets among $2000$ sheets of answer papers, there are $4$ sheets of answer papers such that any two of them have at most $3$ questions with the same answers. Find the minimum value of $n$.

$n$ ($n \ge 4$) green points are in a data space and no $4$ green points lie on one plane. Some segments which connect green points have been colored red. Number of red segments is even. Each two green points are connected with polyline which is build from red segments. Show that red segments can be split on pairs, such that segments from one pair have common end.

Let $\,S\,$ be a finite set of points in three-dimensional space. Let $\,S_{x},\,S_{y},\,S_{z}\,$ be the sets consisting of the orthogonal projections of the points of $\,S\,$ onto the $yz$-plane, $zx$-plane, $xy$-plane, respectively. Prove that \[ \vert S\vert^{2}\leq \vert S_{x} \vert \cdot \vert S_{y} \vert \cdot \vert S_{z} \vert, \] where $\vert A \vert$ denotes the number of elements in the finite set $A$. [hide="Note"] Note: The orthogonal projection of a point onto a plane is the foot of the perpendicular from that point to the plane. [/hide]

Let $S$ be a set of $n$ points $P_1,P_2,\ldots,P_n$ in a plane such that no three of the points are collinear. Let $\alpha$ be the smallest of the angles $\angle P_iP_jP_k$ ($i\ne j\ne k\ne i,i,j,k\in\{1,2,\ldots,n\}$). Find $\max_S\alpha$ and determine those sets $S$ for which this maximal value is attained.

All arrangements of letters $VNNWHTAAIE$ are listed in lexicographic (dictionary) order. If $AAEHINNTVW$ is the first entry, what entry number is $VANNAWHITE$?

$5$ people stand in a line facing one direction. In every round, the person at the front moves randomly to any position in the line, including the front or the end. Suppose that $\frac{m}{n}$ is the expected number of rounds needed for the last person of the initial line to appear at the front of the line, where $m$ and $n$ are relatively prime positive integers. What is $m + n$?

Let $f$ be an increasing mapping from the family of subsets of a given finite set $H$ into itself, i.e. such that for every $X \subseteq Y\subseteq H$ we have $f (X )\subseteq f (Y )\subseteq H .$ Prove that there exists a subset $H_{0}$ of $H$ such that $f (H_{0}) = H_{0}.$

Show that a unit square can be covered with three equal disks with radius less than $\frac{1}{\sqrt{2}}$. What is the smallest possible radius?

We want to choose telephone numbers for a city. The numbers have $10$ digits and $0$ isn’t used in the numbers. Our aim is: We don’t choose some numbers such that every $2$ telephone numbers are different in more than one digit OR every $2$ telephone numbers are different in a digit which is more than $1$. What is the maximum number of telephone numbers which can be chosen? In how many ways, can we choose the numbers in this maximum situation?

$11$ students take a test. For any two question in a test, there are at least $6$ students who solved exactly one of those two questions. Prove that there are no more than $12$ questions in this test. Showing the equality case is not needed.