Let $n\ge 2$ be an integer. They are given $n + 1$ red points in the plane. Prove that there exist $2n$ circles $C_1 , C_2 , \ldots , C_n , D_1 , D_2 , \ldots , D_n$ such that: $\bullet$ $C_1 , C_2 ,\ldots , C_n$ are concentric. $\bullet$ $D_1 , D_2 ,\ldots , D_n$ are concentric. $\bullet$ For $k = 1, 2, 3,\ldots , n$ the circles $C_k$ and $D_k$ are disjoint. $\bullet$ For $k = 1, 2, 3,\ldots , n$ it is true that $C_k$ contains exactly $k$ red dots in its interior and $D_k$ contains exactly $n + 1 - k$ red dots in its interior.

A very well known family of mathematicians has three children called [i]Antonia, Bernhard[/i] and [i]Christian[/i]. Each evening one of the children has to do the dishes. One day, their dad decided to construct of plan that says which child has to do the dishes at which day for the following $55$ days. Let $x$ be the number of possible such plans in which Antonia has to do the dishes on three consecutive days at least once. Furthermore, let $y$ be the number of such plans in which there are three consecutive days in which Antonia does the dishes on the first, Bernhard on the second and Christian on the third day. Determine, whether $x$ and $y$ are different and if so, then decide which of those is larger.

Consider five segments $AB_1, AB_2, AB_3, AB_4, AB_5$. From each point $B_i$ there can exit either $5$ segments or no segments at all, so that the endpoints of any two segments of the resulting graph (system of segments) do not coincide. Can the number of free endpoints of the segments thus constructed be equal to $1001$? (A free endpoint is an endpoint from which no segment begins.)

In a chess tournament each player played one match with every other player. It is known that all players have different scores. The player who is on the last place got $k$ points. What is the smallest number of wins that the first placed player got? (For the win $1$ point is given, for loss $0$ and for a draw both players get $0,5$ points.)

There is a piece on each square of the solitaire board shown except for the central square. A move can be made when there are three adjacent squares in a horizontal or vertical line with two adjacent squares occupied and the third square vacant. The move is to remove the two pieces from the occupied squares and to place a piece on the third square. (One can regard one of the pieces as hopping over the other and taking it.) Is it possible to end up with a single piece on the board, on the square marked $X$?

To each pair $ (x, y)$ of distinct elements of a finite set $ X$ a number $ f(x, y)$ equal to 0 or 1 is assigned in such a way that $ f(x, y) \neq f(y, x)$ $ \forall x,y$ and $ x \neq y.$ Prove that exactly one of the following situations occurs: [b](i)[/b] $ X$ is the union of two disjoint nonempty subsets $ U, V$ such that $ f(u, v) \equal{} 1$ $ \forall u \in U, v \in V.$ [b](ii)[/b] The elements of $ X$ can be labeled $ x_1, \ldots , x_n$ so that \[ f(x_1, x_2) \equal{} f(x_2, x_3) \equal{} \cdots \equal{} f(x_{n\minus{}1}, x_n) \equal{} f(x_n, x_1) \equal{} 1.\] [i]Alternative formulation:[/i] In a tournament of n participants, each pair plays one game (no ties). Prove that exactly one of the following situations occurs: [b](i)[/b] The league can be partitioned into two nonempty groups such that each player in one of these groups has won against each player of the other. [b](ii)[/b] All participants can be ranked 1 through $ n$ so that $ i\minus{}$th player wins the game against the $ (i \plus{} 1)$st and the $ n\minus{}$th player wins against the first.

There are three colleges in a town. Each college has $n$ students. Any student of any college knows $n+1$ students of the other two colleges. Prove that it is possible to choose a student from each of the three colleges so that all three students would know each other.

Wanda the Worm likes to eat Pascal's triangle. One day, she starts at the top of the triangle and eats $\textstyle\binom{0}{0}=1$. Each move, she travels to an adjacent positive integer and eats it, but she can never return to a spot that she has previously eaten. If Wanda can never eat numbers $a,b,c$ such that $a+b=c$, prove that it is possible for her to eat 100,000 numbers in the first 2011 rows given that she is not restricted to traveling only in the first 2011 rows. (Here, the $n+1$st row of Pascal's triangle consists of entries of the form $\textstyle\binom{n}{k}$ for integers $0\le k\le n$. Thus, the entry $\textstyle\binom{n}{k}$ is considered adjacent to the entries $\textstyle\binom{n-1}{k-1}$, $\textstyle\binom{n-1}{k}$, $\textstyle\binom{n}{k-1}$, $\textstyle\binom{n}{k+1}$, $\textstyle\binom{n+1}{k}$, $\textstyle\binom{n+1}{k+1}$.) [i]Linus Hamilton.[/i]

[b]p1.[/b] A boy has as many sisters as brothers. How ever, his sister has twice as many brothers as sisters. How many boys and girls are there in the family? [b]p2.[/b] Solve each of the following problems. (1) Find a pair of numbers with a sum of $11$ and a product of $24$. (2) Find a pair of numbers with a sum of $40$ and a product of $400$. (3) Find three consecutive numbers with a sum of $333$. (4) Find two consecutive numbers with a product of $182$. [b]p3.[/b] $2008$ integers are written on a piece of paper. It is known that the sum of any $100$ numbers is positive. Show that the sum of all numbers is positive. [b]p4.[/b] Let $p$ and $q$ be prime numbers greater than $3$. Prove that $p^2 - q^2$ is divisible by $24$. [b]p5.[/b] Four villages $A,B,C$, and $D$ are connected by trails as shown on the map. [img]https://cdn.artofproblemsolving.com/attachments/4/9/33ecc416792dacba65930caa61adbae09b8296.png[/img] On each path $A \to B \to C$ and $B \to C \to D$ there are $10$ hills, on the path $A \to B \to D$ there are $22$ hills, on the path $A \to D \to B$ there are $45$ hills. A group of tourists starts from $A$ and wants to reach $D$. They choose the path with the minimal number of hills. What is the best path for them? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Odin and Evelyn are playing a game, Odin going first. There are initially $3k$ empty boxes, for some given positive integer $k$. On each player’s turn, they can write a non-negative integer in an empty box, or erase a number in a box and replace it with a strictly smaller non-negative integer. However, Odin is only ever allowed to write odd numbers, and Evelyn is only allowed to write even numbers. The game ends when either one of the players cannot move, in which case the other player wins; or there are exactly $k$ boxes with the number $0$, in which case Evelyn wins if all other boxes contain the number $1$, and Odin wins otherwise. Who has a winning strategy? $Agnijo \ Banerjee \ , United \ Kingdom$

[b]p1.[/b] A square is tiled by smaller squares as shown in the figure. Find the area of the black square in the middle if the perimeter of the square $ABCD$ is $14$ cm. [img]https://cdn.artofproblemsolving.com/attachments/1/1/0f80fc5f0505fa9752b5c9e1c646c49091b4ca.png[/img] [b]p2.[/b] If $a, b$, and $c$ are numbers so that $a + b + c = 0$ and $a^2 + b^2 + c^2 = 1$. Compute $a^4 + b^4 + c^4$. [b]p3.[/b] A given fraction $\frac{a}{b}$ ($a, b$ are positive integers, $a \ne b$) is transformed by the following rule: first, $1$ is added to both the numerator and the denominator, and then the numerator and the denominator of the new fraction are each divided by their greatest common divisor (in other words, the new fraction is put in simplest form). Then the same transformation is applied again and again. Show that after some number of steps the denominator and the numerator differ exactly by $1$. [b]p4.[/b] A goat uses horns to make the holes in a new $30\times 60$ cm large towel. Each time it makes two new holes. Show that after the goat repeats this $61$ times the towel will have at least two holes whose distance apart is less than $6$ cm. [b]p5.[/b] You are given $555$ weights weighing $1$ g, $2$ g, $3$ g, $...$ , $555$ g. Divide these weights into three groups whose total weights are equal. [b]p6.[/b] Draw on the regular $8\times 8$ chessboard a circle of the maximal possible radius that intersects only black squares (and does not cross white squares). Explain why no larger circle can satisfy the condition. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Graph $G$ has $n$ vertices and $mn$ edges, where $n>2m$, show that there exists a path with $m+1$ vertices. (A path is an open walk without repeating vertices )

Is it possible to write a natural number in every cell of an infinite chessboard in such a manner that for all integers $m, n > 100$, the sum of numbers in every $m\times n$ rectangle is divisible by $m + n \ ?$

$n \geq 4$ players participated in a tennis tournament. Any two players have played exactly one game, and there was no tie game. We call a company of four players $bad$ if one player was defeated by the other three players, and each of these three players won a game and lost another game among themselves. Suppose that there is no bad company in this tournament. Let $w_i$ and $l_i$ be respectively the number of wins and losses of the $i$-th player. Prove that \[\sum^n_{i=1} \left(w_i - l_i\right)^3 \geq 0.\] [i]Proposed by Sung Yun Kim, South Korea[/i]

$k,n$ are two arbitrary positive integers. Prove that there exists at least $(k-1)(n-k+1)$ positive integers that can be produced by $n$ number of $k$'s and using only $+,-,\times, \div$ operations and adding parentheses between them, but cannot be produced using $n-1$ number of $k$'s. [i]Proposed by Aryan Tajmir[/i]

Let be given $32$ positive integers with the sum $120$, none of which is greater than $60.$ Prove that these integers can be divided into two disjoint subsets with the same sum of elements.

[b]p1. [/b] A clock has a long hand for minutes and a short hand for hours. A placement of those hands is [i]natural [/i] if you will see it in a correctly functioning clock. So, having both hands pointing straight up toward $12$ is natural and so is having the long hand pointing toward $6$ and the short hand half-way between $2$ and $3$. A natural placement of the hands is symmetric if you get another natural placement by interchanging the long and short hands. One kind of symmetric natural placement is when the hands are pointed in exactly the same direction. Are there symmetric natural placements of the hands in which the two hands are not pointed in exactly the same direction? If so, describe one such placement. If not, explain why none are possible. [b]p2.[/b] Let $\frac{m}{n}$ be a fraction such that when you write out the decimal expansion of $\frac{m}{n}$ , it eventually ends up with the four digits $2001$ repeated over and over and over. Prove that $101$ divides $n$. [b]p3.[/b] Consider the following two questions: Question $1$: I am thinking of a number between $0$ and $15$. You get to ask me seven yes-or-no questions, and I am allowed to lie at most once in answering your questions. What seven questions can you ask that will always allow you to determine the number? Note: You need to come up with seven questions that are independent of the answers that are received. In other words, you are not allowed to say, "If the answer to question $1$ is yes, then question $2$ is XXX; but if the answer to question $1$ is no, then question $2$ is YYY." Question $2$: Consider the set $S$ of all seven-tuples of zeros and ones. What sixteen elements of $S$ can you choose so that every pair of your chosen seven-tuples differ in at least three coordinates? a. These two questions are closely related. Show that an answer to Question $1$ gives an answer to Question $2$. b. Answer either Question $1$ or Question $2$. [b]p4.[/b] You may wish to use the angle addition formulas for the sine and cosine functions: $\sin (\alpha + \beta) = \sin \alpha \cos \beta + \cos \alpha \sin \beta$ $\cos (\alpha + \beta) = \cos \alpha \cos \beta - \sin \alpha \sin \beta$ a) Prove the identity $(\sin x)(1 + 2 \cos 2x) = \sin (3x)$. b) For any positive integer $n$, prove the identity $$(sin x)(1 + 2 \cos 2x + 2\cos 4x + ... +2\cos 2nx) = \sin ((2n +1)x)$$ [b]p5.[/b] Define the set $\Omega$ in the $xy$-plane as the union of the regions bounded by the three geometric figures: triangle $A$ with vertices $(0.5, 1.5)$, $(1.5, 0.5)$ and $(0.5,-0.5)$, triangle $B$ with vertices $(-0.5,1.5)$, $(-1.5,-0.5)$ and $(-0.5, 0.5)$, and rectangle $C$ with corners $(0.5, 1.0)$, $(-0.5, 1.0)$, $(-0.5,-1.0)$, and $(0.5,-1.0)$. a. Explain how copies of $\Omega$ can be used to cover the $xy$-plane. The copies are obtained by translating $\Omega$ in the $xy$-plane, and copies can intersect only along their edges. b. We can define a transformation of the plane as follows: map any point $(x, y)$ to $(x + G, x + y + G)$, where $G = 1$ if $y < -2x$, $G = -1$ if $y > -2x$, and $G = 0$ if $y = -2x$. Prove that every point in $\Omega$ is transformed into another point in $\Omega$, and that there are at least two points in $\Omega$ that are transformed into the same point. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Decide if there is a permutation $a_1,a_2,\cdots,a_{6666}$ of the numbers $1,2,\cdots,6666$ with the property that the sum $k+a_k$ is a perfect square for all $k=1,2,\cdots,6666$

We are given the natural numbers $1=a_1,\, \, a_2,...,a_n$, for which $a_i\leq a_{i+1}\leq 2a_i$ for $i=1,2,...,n-1$ and the sum $\sum_{i=1}^n a_i$ is even. Prove that these numbers can be partitioned into two groups with equal sum.

Nine points are placed on a circle. Show that it is possible to colour the $36$ chords connecting them using four colours so that for any set of four points, each of the four colours is used for at least one of the six chords connecting the given points

Kobar and Borah are playing on a whiteboard with the following rules: They start with two distinct positive integers on the board. On each step, beginning with Kobar, each player takes turns changing the numbers on the board, either from $P$ and $Q$ to $2P-Q$ and $2Q-P$, or from $P$ and $Q$ to $5P-4Q$ and $5Q-4P$. The game ends if a player writes an integer that is not positive. That player is declared to lose, and the opponent is declared the winner. At the beginning of the game, the two numbers on the board are $2024$ and $A$. If it is known that Kobar does not lose on his first move, determine the largest possible value of $A$ so that Borah can win this game.

Patrick tosses four four-sided dice, each numbered $1$ through $4$. What's the probability their product is a multiple of four?

The sequence $0, 1, 1, 1, 1, 1,....,1$ where have $1$ number zero and $1995$ numbers one. If we choose two or more numbers in this sequence(but not the all $1996$ numbers) and substitute one number by arithmetic mean of the numbers selected, we obtain a new sequence with $1996$ numbers!!! Show that, we can repeat this operation until we have all $1996$ numbers are equal Note: It's not necessary to choose the same quantity of numbers in each operation!!!

Find all natural numbers \( n \) for which it is possible to construct an \( n \times n \) square using only tetrominoes like the one below:

An $m \times n$ chessboard where $m \le n$ has several black squares such that no two rows have the same pattern. Determine the largest integer $k$ such that we can always color $k$ columns red while still no two rows have the same pattern.