Alice and Bob take turns alternatively on a $2020\times2020$ board with Alice starting the game. In every move each person colours a cell that have not been coloured yet and will be rewarded with as many points as the coloured cells in the same row and column. When the table is coloured completely, the points determine the winner. Who has the wining strategy and what is the maximum difference he/she can grantees? [i]Proposed by Seyed Reza Hosseini[/i]

Fernanda decided to decorate a square blanket with a ribbon and buttons, placing a button in the center of each square where the ribbon passes and forming the design indicated in the figure. If Fernanda sews the first button in the shaded square on line $0$, on which line does she sew the $2007$th button? [img]https://cdn.artofproblemsolving.com/attachments/2/9/0c9c85ec6448ee3f6f363c8f4bcdd5209f53f6.png[/img]

The table of dimensions $n\times n$, $n\in\mathbb{N}$, is filled with numbers from $1$ to $n^2$, but the difference any two numbers on adjacent fields is at most $n$, and that for every $k = 1, 2,\dots , n^2$ set of fields whose numbers are $1, 2,\dots , k$ is connected, as well as the set of fields whose numbers are $k, k + 1,\dots , n^2$. Neighboring fields are fields with a common side, while a set of fields is considered connected if from each field to every other field of that set can be reached going only to the neighboring fields within that set. We call a pair of adjacent numbers, ie. numbers on adjacent fields, good, if their absolute difference is exactly $n$ (one number can be found in several good pairs). Prove that the table has at least $2 (n - 1)$ good pairs.

Let $n$ be a positive integer. Define a chameleon to be any sequence of $3n$ letters, with exactly $n$ occurrences of each of the letters $a, b,$ and $c$. Define a swap to be the transposition of two adjacent letters in a chameleon. Prove that for any chameleon $X$ , there exists a chameleon $Y$ such that $X$ cannot be changed to $Y$ using fewer than $3n^2/2$ swaps.

There are 1997 pieces of medicine. Three bottles $A, B, C$ can contain at most 1997, 97, 19 pieces of medicine respectively. At first, all 1997 pieces are placed in bottle $A$, and the three bottles are closed. Each piece of medicine can be split into 100 part. When a bottle is opened, all pieces of medicine in that bottle lose a part each. A man wishes to consume all the medicine. However, he can only open each of the bottles at most once each day, consume one piece of medicine, move some pieces between the bottles, and close them. At least how many parts will be lost by the time he finishes consuming all the medicine?

Decide whether the integers $1,2,\ldots,100$ can be arranged in the cells $C(i, j)$ of a $10\times10$ matrix (where $1\le i,j\le 10$), such that the following conditions are fullfiled: i) In every row, the entries add up to the same sum $S$. ii) In every column, the entries also add up to this sum $S$. iii) For every $k = 1, 2, \ldots, 10$ the ten entries $C(i, j)$ with $i-j\equiv k\bmod{10}$ add up to $S$. [i](Proposed by Gerhard Woeginger, Austria)[/i]

Tireless Thomas and Jeremy construct a sequence. At the beginning there is one positive integer in the sequence. Then they successively write new numbers in the sequence in the following way: Thomas obtains the next number by adding to the previous number one of its (decimal) digits, while Jeremy obtains the next number by subtracting from the previous number one of its digits. Prove that there is a number in this sequence which will be repeated at least $100$ times. (A Shapovalov)

A triangle is divided into nine smaller triangles, where counters with the number zero are placed at each of the ten vertices. A [i]movement[/i] consists of choosing one of the nine triangles and applying the same operation to the three counters of that triangle: adding a unit or subtracting a unit. The figure illustrates a possible [i]movement[/i]. We shall call the integer number n [i]dominant [/i] if it is possible, after a few moves, to obtain a configuration in which the counter numbers are all consecutive and the largest of these numbers is $n$. Determine all [i]dominant [/i] numbers. [img]https://cdn.artofproblemsolving.com/attachments/7/3/731160e6e9a2b3292a31c4555d4adbc7028596.png[/img]

Let $ n \geq 3$ and consider a set $ E$ of $ 2n \minus{} 1$ distinct points on a circle. Suppose that exactly $ k$ of these points are to be colored black. Such a coloring is [b]good[/b] if there is at least one pair of black points such that the interior of one of the arcs between them contains exactly $ n$ points from $ E$. Find the smallest value of $ k$ so that every such coloring of $ k$ points of $ E$ is good.

Given an integer $n\ge3$, prove that the set $X=\{1,2,3,\ldots,n^2-n\}$ can be divided into two non-intersecting subsets such that neither of them contains $n$ elements $a_1,a_2,\ldots,a_n$ with $a_1<a_2<\ldots<a_n$ and $a_k\le\frac{a_{k-1}+a_{k+1}}2$ for all $k=2,\ldots,n-1$.

Eight red boxes and eight blue boxes are randomly placed in four stacks of four boxes each. The probability that exactly one of the stacks consists of two red boxes and two blue boxes is $\frac{m}{n}$ , where m and n are relatively prime positive integers. Find $m + n$.

There are lamps in every field of $n\times n$ table. At start all the lamps are off. A move consists of chosing $m$ consecutive fields in a row or a column and changing the status of that $m$ lamps. Prove that you can reach a state in which all the lamps are on only if $m$ divides $n.$

Find for every value of $n$ a set of numbers $p$ for which the following statement is true: Any convex $n$-gon can be divided into $p$ isosceles triangles.

Let $p, k$ and $x$ be positive integers such that $p \geq k$ and $x < \left[ \frac{p(p-k+1)}{2(k-1)} \right]$, where $[q]$ is the largest integer no larger than $q$. Prove that when $x$ balls are put into $p$ boxes arbitrarily, there exist $k$ boxes with the same number of balls.

Let $P_1, \ldots , P_s$ be arithmetic progressions of integers, the following conditions being satisfied: [b](i)[/b] each integer belongs to at least one of them; [b](ii)[/b] each progression contains a number which does not belong to other progressions. Denote by $n$ the least common multiple of the ratios of these progressions; let $n=p_1^{\alpha_1} \cdots p_k^{\alpha_k}$ its prime factorization. Prove that \[s \geq 1 + \sum^k_{i=1} \alpha_i (p_i - 1).\] [i]Proposed by Dierk Schleicher, Germany[/i]

[b]p1.[/b] Steven has just learned about polynomials and he is struggling with the following problem: expand $(1-2x)^7$ as $a_0 +a_1x+...+a_7x^7$ . Help Steven solve this problem by telling him what $a_1 +a_2 +...+a_7$ is. [b]p2.[/b] Each element of the set ${2, 3, 4, ..., 100}$ is colored. A number has the same color as any divisor of it. What is the maximum number of colors? [b]p3.[/b] Fuchsia is selecting $24$ balls out of $3$ boxes. One box contains blue balls, one red balls and one yellow balls. They each have a hundred balls. It is required that she takes at least one ball from each box and that the numbers of balls selected from each box are distinct. In how many ways can she select the $24$ balls? [b]p4.[/b] Find the perfect square that can be written in the form $\overline{abcd} - \overline{dcba}$ where $a, b, c, d$ are non zero digits and $b < c$. $\overline{abcd}$ is the number in base $10$ with digits $a, b, c, d$ written in this order. [b]p5.[/b] Steven has $100$ boxes labeled from $ 1$ to $100$. Every box contains at most $10$ balls. The number of balls in boxes labeled with consecutive numbers differ by $ 1$. The boxes labeled $1,4,7,10,...,100$ have a total of $301$ balls. What is the maximum number of balls Steven can have? [b]p6.[/b] In acute $\vartriangle ABC$, $AB=4$. Let $D$ be the point on $BC$ such that $\angle BAD = \angle CAD$. Let $AD$ intersect the circumcircle of $\vartriangle ABC$ at $X$. Let $\Gamma$ be the circle through $D$ and $X$ that is tangent to $AB$ at $P$. If $AP = 6$, compute $AC$. [b]p7.[/b] Consider a $15\times 15$ square decomposed into unit squares. Consider a coloring of the vertices of the unit squares into two colors, red and blue such that there are $133$ red vertices. Out of these $133$, two vertices are vertices of the big square and $32$ of them are located on the sides of the big square. The sides of the unit squares are colored into three colors. If both endpoints of a side are colored red then the side is colored red. If both endpoints of a side are colored blue then the side is colored blue. Otherwise the side is colored green. If we have $196$ green sides, how many blue sides do we have? [b]p8.[/b] Carl has $10$ piles of rocks, each pile with a different number of rocks. He notices that he can redistribute the rocks in any pile to the other $9$ piles to make the other $9$ piles have the same number of rocks. What is the minimum number of rocks in the biggest pile? [b]p9.[/b] Suppose that Tony picks a random integer between $1$ and $6$ inclusive such that the probability that he picks a number is directly proportional to the the number itself. Danny picks a number between $1$ and $7$ inclusive using the same rule as Tony. What is the probability that Tony’s number is greater than Danny’s number? [b]p10.[/b] Mike wrote on the board the numbers $1, 2, ..., n$. At every step, he chooses two of these numbers, deletes them and replaces them with the least prime factor of their sum. He does this until he is left with the number $101$ on the board. What is the minimum value of $n$ for which this is possible? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Tom Sawyer must whitewash a circular fence consisting of $N$ planks. He whitewashes the fence going clockwise and following the rule: He whitewashes the first plank, skips two planks, whitewashes one, skips three, and so on. Some planks may be whitewashed several times. Tom believes that all planks will be whitewashed sooner or later, but aunt Polly is sure that some planks will remain unwhitewashed forever. Prove that Tom is right if $N$ is a power of two, otherwise aunt Polly is right.

We will say that a natural number is [i]acceptable[/i] if it has at most $9$ distinct prime divisors. There is a stack of $100!=1\times2\times\cdots\times100$ stones. A [i]legal move[/i] consists in removing $k$ stones from the stack, where $k$ is an acceptable number. Two players, Lucas and Nicolas, take turns making legal moves; Lucas starts the game. The one who removes the last stone wins. Determine which of the players has a winning strategy and describe this strategy.

Thanos establishes $5$ settlements on a remote planet, randomly choosing one of them to stay in, and then he randomly builds a system of roads between these settlements such that each settlement has exactly one outgoing (unidirectional) road to another settlement. Afterwards, the Avengers randomly choose one of the $5$ settlements to teleport to. Then, they (the Avengers) must use the system of roads to travel from one settlement to another. The probability that the Avengers can find Thanos can be written as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Find $m+n$.

There is $3 \times 4 \times 5$ - box with its faces divided into $1 \times 1$ - squares. Is it possible to place numbers in these squares so that the sum of numbers in every stripe of squares (one square wide) circling the box, equals $120$?

The chess piece [i]sick rook[/i] can move along rows and columns as a regular rook, but at most by $2$ fields. We can place [i]sick rooks[/i] on a square board in such a way that no two of them attack each other and no field is attacked by more than one [i]sick rook[/i]. a) Prove that on $30\times 30$ board, we cannot place more than $100$ [i]sick rooks[/i]. b) Find the maximum number of [i]sick rooks[/i] which can be placed on $8\times 8$ board. c) Prove that on $32\times 32$ board, we cannot place more than $120$ [i]sick rooks[/i].

Is it possible to paint the plane in $4$ colors so that inside any circle are the dots of all four colors?

Let $A=\{a_1,a_2,\cdots,a_n,b_1,b_2,\cdots,b_n\}$ be a set with $2n$ distinct elements, and $B_i\subseteq A$ for any $i=1,2,\cdots,m.$ If $\bigcup_{i=1}^m B_i=A,$ we say that the ordered $m-$tuple $(B_1,B_2,\cdots,B_m)$ is an ordered $m-$covering of $A.$ If $(B_1,B_2,\cdots,B_m)$ is an ordered $m-$covering of $A,$ and for any $i=1,2,\cdots,m$ and any $j=1,2,\cdots,n,$ $\{a_j,b_j\}$ is not a subset of $B_i,$ then we say that ordered $m-$tuple $(B_1,B_2,\cdots,B_m)$ is an ordered $m-$covering of $A$ without pairs. Define $a(m,n)$ as the number of the ordered $m-$coverings of $A,$ and $b(m,n)$ as the number of the ordered $m-$coverings of $A$ without pairs. $(1)$ Calculate $a(m,n)$ and $b(m,n).$ $(2)$ Let $m\geq2,$ and there is at least one positive integer $n,$ such that $\dfrac{a(m,n)}{b(m,n)}\leq2021,$ Determine the greatest possible values of $m.$

Let $A_n$ be the number of arranged n-tuples of natural numbers $(a_1,a_2…a_n)$, such that $\frac{1}{a_1} +\frac{1}{a_2} +...+\frac{1}{a_n} =1$. Find the parity of $A_{68}$.

The points of a circle are colored in green and yellow, such that every equilateral triangle inscribed in the circle has exactly 2 vertices colored in yellow. Prove that there exist a square inscribed in the circle which has at least 3 vertices colored in yellow.