Let $r>1$ be a rational number. Alice plays a solitaire game on a number line. Initially there is a red bead at $0$ and a blue bead at $1$. In a move, Alice chooses one of the beads and an integer $k \in \mathbb{Z}$. If the chosen bead is at $x$, and the other bead is at $y$, then the bead at $x$ is moved to the point $x'$ satisfying $x'-y=r^k(x-y)$. Find all $r$ for which Alice can move the red bead to $1$ in at most $2021$ moves.

[u]Set 1[/u] [b]p1.[/b] Arnold is currently stationed at $(0, 0)$. He wants to buy some milk at $(3, 0)$, and also some cookies at $(0, 4)$, and then return back home at $(0, 0)$. If Arnold is very lazy and wants to minimize his walking, what is the length of the shortest path he can take? [b]p2.[/b] Dilhan selects $1$ shirt out of $3$ choices, $1$ pair of pants out of $4$ choices, and $2$ socks out of $6$ differently-colored socks. How many outfits can Dilhan select? All socks can be worn on both feet, and outfits where the only difference is that the left sock and right sock are switched are considered the same. [b]p3.[/b] What is the sum of the first $100$ odd positive integers? [b]p4.[/b] Find the sum of all the distinct prime factors of $1591$. [b]p5.[/b] Let set $S = \{1, 2, 3, 4, 5, 6\}$. From $S$, four numbers are selected, with replacement. These numbers are assembled to create a $4$-digit number. How many such $4$-digit numbers are multiples of $3$? [u]Set 2[/u] [b]p6.[/b] What is the area of a triangle with vertices at $(0, 0)$, $(7, 2)$, and $(4, 4)$? [b]p7.[/b] Call a number $n$ “warm” if $n - 1$, $n$, and $n + 1$ are all composite. Call a number $m$ “fuzzy” if $m$ may be expressed as the sum of $3$ consecutive positive integers. How many numbers less than or equal to $30$ are warm and fuzzy? [b]p8.[/b] Consider a square and hexagon of equal area. What is the square of the ratio of the side length of the hexagon to the side length of the square? [b]p9.[/b] If $x^2 + y^2 = 361$, $xy = -40$, and $x - y$ is positive, what is $x - y$? [b]p10.[/b] Each face of a cube is to be painted red, orange, yellow, green, blue, or violet, and each color must be used exactly once. Assuming rotations are indistinguishable, how many ways are there to paint the cube? [u]Set 3[/u] [b]p11.[/b] Let $D$ be the midpoint of side $BC$ of triangle $ABC$. Let $P$ be any point on segment $AD$. If $M$ is the maximum possible value of $\frac{[PAB]}{[PAC]}$ and $m$ is the minimum possible value, what is $M - m$? Note: $[PQR]$ denotes the area of triangle $PQR$. [b]p12.[/b] If the product of the positive divisors of the positive integer $n$ is $n^6$, find the sum of the $3$ smallest possible values of $n$. [b]p13.[/b] Find the product of the magnitudes of the complex roots of the equation $(x - 4)^4 +(x - 2)^4 + 14 = 0$. [b]p14.[/b] If $xy - 20x - 16y = 2016$ and $x$ and $y$ are both positive integers, what is the least possible value of $\max (x, y)$? [b]p15.[/b] A peasant is trying to escape from Chyornarus, ruled by the Tsar and his mystical faith healer. The peasant starts at $(0, 0)$ on a $6 \times 6$ unit grid, the Tsar’s palace is at $(3, 3)$, the healer is at $(2, 1)$, and the escape is at $(6, 6)$. If the peasant crosses the Tsar’s palace or the mystical faith healer, he is executed and fails to escape. The peasant’s path can only consist of moves upward and rightward along the gridlines. How many valid paths allow the peasant to escape? PS. You should use hide for answers. Rest sets have been posted [url=https://artofproblemsolving.com/community/c3h2784259p24464954]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

[b]p1.[/b] What is the smallest positive integer divisible by $20$, $12$, and $13$? [b]p2.[/b] Two circles of radius $5$ are placed in the plane such that their centers are $7$ units apart. What is the largest possible distance between a point on one circle and a point on the other? [b]p3.[/b] In a magic square, all the numbers in the rows, columns, and diagonals sum to the same value. How many $2\times 2$ magic squares containing the integers $\{1, 2, 3, 4\}$ are there? [b]p4.[/b] Ethan's sock drawer contains two pairs of white socks and one pair of red socks. Ethan picks two socks at random. What is the probability that he picks two white socks? [b]p5.[/b] The sum of the time on a digital clock is the sum of the digits displayed on the screen. For example, the sum of the time at $10:23$ would be $6$. Assuming the clock is a $12$ hour clock, what is the greatest possible positive difference between the sum of the time at some time and the sum of the time one minute later? [b]p6.[/b] Given the expression $1 \div 2 \div 3 \div 4$, what is the largest possible resulting value if one were to place parentheses $()$ somewhere in the expression? [b]p7.[/b] At a convention, there are many astronomers, astrophysicists, and cosmologists. At $first$, all the astronomers and astrophysicists arrive. At this point, $\frac35$ of the people in the room are astronomers. Then, all the cosmologists come, so now, $30\%$ of the people in the room are astrophysicists. What fraction of the scientists are cosmologists? [b]p8.[/b] At $10:00$ AM, a minuteman starts walking down a $1200$-step stationary escalator at $40$ steps per minute. Halfway down, the escalator starts moving up at a constant speed, while the minuteman continues to walk in the same direction and at the same pace that he was going before. At $10:55$ AM, the minuteman arrives back at the top. At what speed is the escalator going up, in steps per minute? [b]p9.[/b] Given that $x_1 = 57$, $x_2 = 68$, and $x_3 = 32$, let $x_n = x_{n-1} -x_{n-2} +x_{n-3}$ for $n \ge 4$. Find $x_{2013}$. [b]p10.[/b] Two squares are put side by side such that one vertex of the larger one coincides with a vertex of the smaller one. The smallest rectangle that contains both squares is drawn. If the area of the rectangle is $60$ and the area of the smaller square is $24$, what is the length of the diagonal of the rectangle? [b]p11.[/b] On a dield trip, $2$ professors, $4$ girls, and $4$ boys are walking to the forest to gather data on butterflies. They must walk in a line with following restrictions: one adult must be the first person in the line and one adult must be the last person in the line, the boys must be in alphabetical order from front to back, and the girls must also be in alphabetical order from front to back. How many such possible lines are there, if each person has a distinct name? [b]p12.[/b] Flatland is the rectangle with vertices $A, B, C$, and $D$, which are located at $(0, 0)$, $(0, 5)$, $(5, 5)$, and $(5, 0)$, respectively. The citizens put an exact map of Flatland on the rectangular region with vertices $(1, 2)$, $(1, 3)$, $(2, 3)$, and $(2, 2)$ in such a way so that the location of $A$ on the map lies on the point $(1, 2)$ of Flatland, the location of $B$ on the map lies on the point $(1, 3)$ of Flatland, the location of C on the map lies on the point $(2, 3)$ of Flatland, and the location of D on the map lies on the point $(2, 2)$ of Flatland. Which point on the coordinate plane is thesame point on the map as where it actually is on Flatland? [b]p13.[/b] $S$ is a collection of integers such that any integer $x$ that is present in $S$ is present exactly $x$ times. Given that all the integers from $1$ through $22$ inclusive are present in $S$ and no others are, what is the average value of the elements in $S$? [b]p14.[/b] In rectangle $PQRS$ with $PQ < QR$, the angle bisector of $\angle SPQ$ intersects $\overline{SQ}$ at point $T$ and $\overline{QR }$ at $U$. If $PT : TU = 3 : 1$, what is the ratio of the area of triangle $PTS$ to the area of rectangle $PQRS$? [b]p15.[/b] For a function $f(x) = Ax^2 + Bx + C$, $f(A) = f(B)$ and $A + 6 = B$. Find all possible values of $B$. [b]p16.[/b] Let $\alpha$ be the sum of the integers relatively prime to $98$ and less than $98$ and $\beta$ be the sum of the integers not relatively prime to $98$ and less than $98$. What is the value of $\frac{\alpha}{\beta}$ ? [b]p17.[/b] What is the value of the series $\frac{1}{3} + \frac{3}{9} + \frac{6}{27} + \frac{10}{81} + \frac{15}{243} + ...$? [b]p18.[/b] A bug starts at $(0, 0)$ and moves along lattice points restricted to $(i, j)$, where $0 \le i, j \le 2$. Given that the bug moves $1$ unit each second, how many different paths can the bug take such that it ends at $(2, 2)$ after $8$ seconds? [b]p19.[/b] Let $f(n)$ be the sum of the digits of $n$. How many different values of $n < 2013$ are there such that $f(f(f(n))) \ne f(f(n))$ and $f(f(f(n))) < 10$? [b]p20.[/b] Let $A$ and $B$ be points such that $\overline{AB} = 14$ and let $\omega_1$ and $\omega_2$ be circles centered at $A$ and $B$ with radii $13$ and $15$, respectively. Let $C$ be a point on $\omega_1$ and $D$ be a point on $\omega_2$ such that $\overline{CD}$ is a common external tangent to $\omega_1$ and $\omega_2$. Let $P$ be the intersection point of the two circles that is closer to $\overline{CD}$. If $M$ is the midpoint of $\overline{CD}$, what is the length of segment $\overline{PM}$? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $n$ be a positive integer. $n$ people take part in a certain party. For any pair of the participants, either the two are acquainted with each other or they are not. What is the maximum possible number of the pairs for which the two are not acquainted but have a common acquaintance among the participants?

Square $ABCD$ is divided into $n^2$ equal small squares by lines parallel to its sides.A spider starts from $A$ and moving only rightward or upwards,tries to reach $C$.Every "movement" of the spider consists of $k$ steps rightward and $m$ steps upwards or $m$ steps rightward and $k$ steps upwards(it can follow any possible order for the steps of each "movement").The spider completes $l$ "movements" and afterwards it moves without limitation (it still moves rightwards and upwards only).If $n=m\cdot l$,find the number of the possible paths the spider can follow to reach $C$.Note that $n,m,k,l\in \mathbb{N^{*}}$ with $k<m$.

$101$ blue and $101$ red points are selected on the plane, and no three lie on one straight line. The sum of the pairwise distances between the red points is $1$ (that is, the sum of the lengths of the segments with ends at red points), the sum of the pairwise distances between the blue ones is also $1$, and the sum of the lengths of the segments with the ends of different colors is $400$. Prove that you can draw a straight line separating everything red dots from all blue ones.

Let $n\ge 3$ be a positive integer. Amy wrote all the integers from $1$ to $n^2$ on the $n\times n$ grid, so that each cell contains exactly one number. For $i=1,2,\cdots ,n^2-1$, the cell containing $i$ shares a common side with the cell containing $i+1$. Each turn, Bred can choose one cell, and check what number is written. Bred wants to know where $1$ is written by less than $3n$ turns. Determine whether $n$ such that Bred can always achieve his goal is infinite.

Each subset of $97$ out of $1997$ given real numbers has positive sum. Show that the sum of all the $1997$ numbers is positive.

Let $P$ is a regular $(2n+1)$-gon in the plane, where $n$ is a positive integer. We say that a point $S$ on one of the sides of $P$ can be seen from a point $E$ that is external to $P$ , if the line segment $\overline{ES}$ contains no other points that lie on the sides of $P$ except $S$ . We want to color the sides of $P$ in $3$ colors, such that every side is colored in exactly one color, and each color must be used at least once. Moreover, from every point in the plane external to $P$ , at most $2$ different colors on $P$ can be seen (ignore the vertices of $P$ , we consider them colorless). Find the largest positive integer $n$ for which such a coloring is possible.

Given a regular tetrahedron with edge $a$, its edges are divided into $n$ equal segments, thus obtaining $n + 1$ points: $2$ at the ends and $n - 1$ inside. The following set of planes is considered: $\bullet$ those that contain the faces of the tetrahedron, and $\bullet$ each of the planes parallel to a face of the tetrahedron and containing at least one of the points determined above. Now all those points $P$ that belong (simultaneously) to four planes of that set are considered. Determine the smallest positive natural $n$ so that among those points $P$ the eight vertices of a square-based rectangular parallelepiped can be chosen.

[u]Round 1[/u] [b]p1.[/b] How many distinct ways are there to scramble the letters in $EXETER$? [b]p2.[/b] Given that $\frac{x - y}{x - z}= 3$, find $\frac{x - z}{y - z}$. [b]p3.[/b] When written in base $10$, $9^9 =\overline{ABC420DEF}.$ Find the remainder when $A + B + C + D + E + F$ is divided by $9$. [u]Round 2[/u] [b]p4.[/b] How many positive integers, when expressed in base $7$, have exactly $3$ digits, but don't contain the digit $3$? [b]p5.[/b] Pentagon $JAMES$ is such that its internal angles satisfy $\angle J = \angle A = \angle M = 90^o$ and $\angle E = \angle S$. If $JA = AM = 4$ and $ME = 2$, what is the area of $JAMES$? [b]p6.[/b] Let $x$ be a real number such that $x = \frac{1+\sqrt{x}}{2}$ . What is the sum of all possible values of $x$? [u]Round 3[/u] [b]p7.[/b] Farmer James sends his favorite chickens, Hen Hao and PEAcock, to compete at the Fermi Estimation All Star Tournament (FEAST). The first problem at the FEAST requires the chickens to estimate the number of boarding students at Eggs-Eater Academy given the number of dorms $D$ and the average number of students per dorm $A$. Hen Hao rounds both $D$ and $A$ down to the nearest multiple of $10$ and multiplies them, getting an estimate of $1200$ students. PEAcock rounds both $D$ and $A$ up to the nearest multiple of $10$ and multiplies them, getting an estimate of $N$ students. What is the maximum possible value of $N$? [b]p8.[/b] Farmer James has decided to prepare a large bowl of egg drop soup for the Festival of Eggs-Eater Annual Soup Tasting (FEAST). To flavor the soup, Hen Hao drops eggs into it. Hen Hao drops $1$ egg into the soup in the first hour, $2$ eggs into the soup in the second hour, and so on, dropping $k$ eggs into the soup in the $k$th hour. Find the smallest positive integer $n$ so that after exactly n hours, Farmer James finds that the number of eggs dropped in his egg drop soup is a multiple of $200$. [b]p9.[/b] Farmer James decides to FEAST on Hen Hao. First, he cuts Hen Hao into $2018$ pieces. Then, he eats $1346$ pieces every day, and then splits each of the remaining pieces into three smaller pieces. How many days will it take Farmer James to eat Hen Hao? (If there are fewer than $1346$ pieces remaining, then Farmer James will just eat all of the pieces.) [u]Round 4[/u] [b]p10.[/b] Farmer James has three baskets, and each basket has one magical egg. Every minute, each magical egg disappears from its basket, and reappears with probability $\frac12$ in each of the other two baskets. Find the probability that after three minutes, Farmer James has all his eggs in one basket. [b]p11.[/b] Find the value of $\frac{4 \cdot 7}{\sqrt{4 +\sqrt7} +\sqrt{4 -\sqrt7}}$. [b]p12.[/b] Two circles, with radius $6$ and radius $8$, are externally tangent to each other. Two more circles, of radius $7$, are placed on either side of this configuration, so that they are both externally tangent to both of the original two circles. Out of these $4$ circles, what is the maximum distance between any two centers? PS. You should use hide for answers. Rounds 5-8 have been posted [url=https://artofproblemsolving.com/community/c3h2949222p26406222]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

If Percy rolls a fair six-sided die until he rolls a $5$, what is his expected number of rolls, given that all of his rolls are prime?

There are $300$ participants to a mathematics competition. After the competition some of the contestants play some games of chess. Each two contestants play at most one game against each other. There are no three contestants, such that each of them plays against each other. Determine the maximum value of $n$ for which it is possible to satisfy the following conditions at the same time: each contestant plays at most $n$ games of chess, and for each $m$ with $1 \le m \le n$, there is a contestant playing exactly $m$ games of chess.

Does there exist a hexagon (not necessarily convex) with side lengths $1, 2, 3, 4, 5, 6$ (not necessarily in this order) that can be tiled with a) $31$ b) $32$ equilateral triangles with side length $1?$

In a $m\times n$ chessboard ($m,n\ge 2$), some dominoes are placed (without overlap) with each domino covering exactly two adjacent cells. Show that if no more dominoes can be added to the grid, then at least $2/3$ of the chessboard is covered by dominoes. [i]Proposed by DVDthe1st, mzy and jjax[/i]

Given a $2007\times 2007$ array of numbers $1$ and $-1$, let $A_{i}$ denote the product of the entries in the $i$th row, and $B_{j}$ denote the product of the entries in the $j$th column. Show that \[A_{1}+A_{2}+\cdots +A_{2007}+B_{1}+B_{2}+\cdots +B_{2007}\neq 0.\]

[u]Round 4[/u] [b]p10.[/b] How many divisors of $10^{11}$ have at least half as many divisors that $10^{11}$ has? [b]p11.[/b] Let $f(x, y) = \frac{x}{y}+\frac{y}{x}$ and $g(x, y) = \frac{x}{y}-\frac{y}{x} $. Then, if $\underbrace{f(f(... f(f(}_{2021 fs} f(f(1, 2), g(2,1)), 2), 2)... , 2), 2)$ can be expressed in the form $a + \frac{b}{c}$, where $a$, $b$,$c$ are nonnegative integers such that $b < c$ and $gcd(b,c) = 1$, find $a + b + \lceil (\log_2 (\log_2 c)\rceil $ [b]p12.[/b] Let $ABC$ be an equilateral triangle, and let$ DEF$ be an equilateral triangle such that $D$, $E$, and $F$ lie on $AB$, $BC$, and $CA$, respectively. Suppose that $AD$ and $BD$ are positive integers, and that $\frac{[DEF]}{[ABC]}=\frac{97}{196}$. The circumcircle of triangle $DEF$ meets $AB$, $BC$, and $CA$ again at $G$, $H$, and $I$, respectively. Find the side length of an equilateral triangle that has the same area as the hexagon with vertices $D, E, F, G, H$, and $I$. [u]Round 5 [/u] [b]p13.[/b] Point $X$ is on line segment $AB$ such that $AX = \frac25$ and $XB = \frac52$. Circle $\Omega$ has diameter $AB$ and circle $\omega$ has diameter $XB$. A ray perpendicular to $AB$ begins at $X$ and intersects $\Omega$ at a point $Y$. Let $Z$ be a point on $\omega$ such that $\angle YZX = 90^o$. If the area of triangle $XYZ$ can be expressed as $\frac{a}{b}$ for positive integers $a, b$ with $gcd(a, b) = 1$, find $a + b$. [b]p14.[/b] Andrew, Ben, and Clayton are discussing four different songs; for each song, each person either likes or dislikes that song, and each person likes at least one song and dislikes at least one song. As it turns out, Andrew and Ben don't like any of the same songs, but Clayton likes at least one song that Andrew likes and at least one song that Ben likes! How many possible ways could this have happened? [b]p15.[/b] Let triangle $ABC$ with circumcircle $\Omega$ satisfy $AB = 39$, $BC = 40$, and $CA = 25$. Let $P$ be a point on arc $BC$ not containing $A$, and let $Q$ and $R$ be the reflections of $P$ in $AB$ and $AC$, respectively. Let $AQ$ and $AR$ meet $\Omega$ again at $S$ and $T$, respectively. Given that the reflection of $QR$ over $BC$ is tangent to $\Omega$ , $ST$ can be expressed as $\frac{a}{b}$ for positive integers $a, b$ with $gcd(a,b)= 1$. Find $a + b$. PS. You should use hide for answers. Rounds 1-3 have been posted [url=https://artofproblemsolving.com/community/c4h3131401p28368159]here [/url] and 6-7 [url=https://artofproblemsolving.com/community/c4h3131434p28368604]here [/url],Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

[u]Round 1[/u] [b]p1.[/b] Alice and Bob compiled a list of movies that exactly one of them saw, then Cindy and Dale did the same. To their surprise, these two lists were identical. Prove that if Alice and Cindy list all movies that exactly one of them saw, this list will be identical to the one for Bob and Dale. [b]p2.[/b] Several whole rounds of cheese were stored in a pantry. One night some rats sneaked in and consumed $10$ of the rounds, each rat eating an equal portion. Some were satisfied, but $7$ greedy rats returned the next night to finish the remaining rounds. Their portions on the second night happened to be half as large as on the first night. How many rounds of cheese were initially in the pantry? [b]p3.[/b] You have $100$ pancakes, one with a single blueberry, one with two blueberries, one with three blueberries, and so on. The pancakes are stacked in a random order. Count the number of blueberries in the top pancake, and call that number N. Pick up the stack of the top N pancakes, and flip it upside down. Prove that if you repeat this counting-and-flipping process, the pancake with one blueberry will eventually end up at the top of the stack. [b]p4.[/b] There are two lemonade stands along the $4$-mile-long circular road that surrounds Sour Lake. $100$ children live in houses along the road. Every day, each child buys a glass of lemonade from the stand that is closest to her house, as long as she does not have to walk more than one mile along the road to get there. A stand's [u]advantage [/u] is the difference between the number of glasses it sells and the number of glasses its competitor sells. The stands are positioned such that neither stand can increase its advantage by moving to a new location, if the other stand stays still. What is the maximum number of kids who can't buy lemonade (because both stands are too far away)? [b]p5.[/b] Merlin uses several spells to move around his $64$-room castle. When Merlin casts a spell in a room, he ends up in a different room of the castle. Where he ends up only depends on the room where he cast the spell and which spell he cast. The castle has the following magic property: if a sequence of spells brings Merlin from some room $A$ back to room $A$, then from any other room $B$ in the castle, that same sequence brings Merlin back to room $B$. Prove that there are two different rooms $X$ and $Y$ and a sequence of spells that both takes Merlin from $X$ to $Y$ and from $Y$ to $X$. [u]Round 2[/u] [b]p6.[/b] Captains Hook, Line, and Sinker are deciding where to hide their treasure. It is currently buried at the $X$ in the map below, near the lairs of the three pirates. Each pirate would prefer that the treasure be located as close to his own lair as possible. You are allowed to propose a new location for the treasure to the pirates. If at least two out of the three pirates prefer the new location (because it moves closer to their own lairs), then the treasure will be moved there. Assuming the pirates’ lairs form an acute triangle, is it always possible to propose a sequence of new locations so that the treasure eventually ends up in your backyard (wherever that is)? [img]https://cdn.artofproblemsolving.com/attachments/c/c/a9e65624d97dec612ef06f8b30be5540cfc362.png[/img] [b]p7.[/b] Homer went on a Donut Diet for the month of May ($31$ days). He ate at least one donut every day of the month. However, over any stretch of $7$ consecutive days, he did not eat more than $13$ donuts. Prove that there was some stretch of consecutive days over which Homer ate exactly $30$ donuts. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

In a city at every square exactly three roads meet, one is called street, one is an avenue, and one is a crescent. Most roads connect squares but three roads go outside of the city. Prove that among the roads going out of the city one is a street, one is an avenue and one is a crescent.

Georg plays the following game. He chooses two positive integers $n$ and $k$. On an $n \times n$ - board where all the tiles are white, Georg paints k of the tiles black. Then he counts the number of black tiles in each row, forms the square of each of these n numbers and adds up the squares. He calls the result $S$. In the same way he counts the number of white tiles in each row, forms the square of each of these n numbers and adds up those squares. He calls the result $H$. Georg would like to achieve $S - H = 49$. Determine all possible values of n and k for which this is possible. Example: If Georg chooses $n = 5$ and$ k = 14$, he could for example paint the board as shown. [img]https://cdn.artofproblemsolving.com/attachments/f/2/d3c778f603f0a43c9aa877a4564734eab50058.png[/img] Then $S = 1^2 + 2^2 + 3^2 + 3^2 + 5^2 = 1 + 4 + 9 + 9 + 25 = 48$, $H = 4^2 + 3^2 + 2^2 + 2^2 + 0^2 = 16 + 9 + 4 + 4 + 0 = 33$, so in this case $S - H = 48 - 33 = 15$.

Graph $G$ is defined in 3d space. It has $e$ edges and every vertex are connected if the distance between them is $1.$ Given that there exists the Hamilton cycle, prove that for $e>1,$ we have $$\min d(v)\le 1+2\left(\frac{e}{2}\right)^{0.4}.$$

Player $A$ places an odd number of boxes around a circle and distributes $2013$ balls into some of these boxes. Then the player $B$ chooses one of these boxes and takes the balls in it. After that the player $A$ chooses half of the remaining boxes such that none of two are consecutive and take the balls in them. If player $A$ guarantees to take $k$ balls, find the maximum possible value of $k$.

The game of $pebbles$ is played on an infinite board of lattice points $(i,j)$. Initially there is a $pebble$ at $(0,0)$. A move consists of removing a $pebble$ from point $(i,j)$and placing a $pebble$ at each of the points $(i+1,j)$ and $(i,j+1)$ provided both are vacant. Show taht at any stage of the game there is a $pebble$ at some lattice point $(a,b)$ with $0 \leq a+b \leq 3$

There are distinct points $O, A, B, K_1, . . . , K_n, L_1, . . . , L_n$ on a plane such that no three points are collinear. The open line segments $K_1L_1, . . . , K_nL_n$ are coloured red, other points on the plane are left uncoloured. An allowed path from point $O$ to point $X$ is a polygonal chain with first and last vertices at points $O$ and $X$, containing no red points. For example, for $n = 1$, and $K_1 = (-1, 0)$, $L_1 = (1, 0)$, $O = (0,-1)$, and $X = (0,1)$, $OK_1X$ and $OL_1X$ are examples of allowed paths from $O$ to $X$, there are no shorter allowed paths. Find the least positive integer n such that it is possible that the first vertex that is not $O$ on any shortest possible allowed path from $O$ to $A$ is closer to $B$ than to $A$, and the first vertex that is not $O$ on any shortest possible allowed path from $O$ to $B$ is closer to $A$ than to $B$.

The six-pointed star in the figure is regular: all interior angles of the small triangles are equal. Each of the thirteen marked points is assigned a color, green or red. Prove that there are always three points of the same color, which are the vertices of an equilateral triangle.