A simple polygon in the plane is a figure bounded by a closed nonself-intersecting broken line. (a) Do there exist two congruent simple $7$-gons in the plane such that all the seven vertices of one of the $7$-gons are the vertices of the other one and yet these two $7$-gons have no common sides? (b) Do there exist three such $7$-gons? (V Proizvolov)

Some cells of a checkered plane are marked so that figure $A$ formed by marked cells satisfies the following condition:$1)$ any cell of the figure $A$ has exactly two adjacent cells of $A$; and $2)$ the figure $A$ can be divided into isosceles trapezoids of area $2$ with vertices at the grid nodes (and acute angles of trapezoids are equal to $45$) . Prove that the number of marked cells is divisible by $8$.

We are given $n>1$ piles of coins. There are two different types of coins: real and fake coins; they all look alike, but coins of the same type have the same mass, while the coins from different types have different masses. Coins that belong to the same pile are of the same type. We know the mass of real coin. Find the minimal number of weightings on digital scale that we need in order to conclude: which piles consists of which type of coins and also the mass of fake coin. (We assume that every pile consists from infinite number of coins.)

Let $m$ and $n$ be positive integers greater than $1$. In each unit square of an $m\times n$ grid lies a coin with its tail side up. A [i]move[/i] consists of the following steps. [list=1] [*]select a $2\times 2$ square in the grid; [*]flip the coins in the top-left and bottom-right unit squares; [*]flip the coin in either the top-right or bottom-left unit square. [/list] Determine all pairs $(m,n)$ for which it is possible that every coin shows head-side up after a finite number of moves. [i]Thanasin Nampaisarn, Thailand[/i]

$ a)$ There are $ 25$ participants in a mathematics contest having four problems. Each problem is considered solved or not solved (that is, partial solutions are not possible). Show that either there are four contestants having solved the same problems (or not having solved any of them), or two contestants, one of which has solved exactly the problems that the other did not solve. $ b)$ There are $ k$ sport clubs for the students of a secondary school. The school has $ 100$ students, and for any selection of three of them, there exists a club having at least one of them, but not all, as a member. What is the least possible value of $ k$?

We call a permutation $\sigma$ of the first $n$ positive integers friendly if and only if the following conditions are fulfilled: (1) $\sigma(k + 1) \in \{2\sigma(k), 2\sigma(k) - 1, 2\sigma(k) - n, 2\sigma(k) - n - 1\}, \forall k \in \{1, 2, ..., n - 1\}$ (2) $\sigma(1) \in \{2 \sigma(n), 2\sigma(n) - 1, 2\sigma(n) - n, 2\sigma(n) - n - 1\}$. Find all positive integers $n$ for which there exists such a friendly permutation of the first $n$ positive integers.

When rolling two normal dice, the set of possible outcomes of the sum of the points is $2, 3, 3, 4,4, 4,..., 11, 11,12$. Notice that this sequence can be obtained from the identity $$(x + x^2 + x^3 + x^4 + x^5 + x^6) (x + x^2 + x^3 + x^4 + x^5 + x^6) = x^2 + 2x^3 + 3x^4 +... + 2x^{11} + x^{12}.$$ Design a crazy pair of dice, that is, two other cubes, not necessarily the same, with a natural number indicated on each face, such that the set of possible results of the sum of its points is equal to of two normal dice.

Given $n$ students in the plane such that the $\frac{n(n-1)}{2}$ distances are pairwise distinct. Each student gives a candy each to the $k$ students closest to him. Given that each student receives the same amount of candies, determine all possible values of $n$ in terms of $k$. [i]Proposed by Wong Jer Ren[/i]

Some towns are connected by roads, with at most one road between any two towns. Let $v$ be the number of towns and $e$ be the number of roads. Prove that $(a)$ if $e<v-1$, then there are two towns such that one cannot travel between them; $(b)$ if $2e>(v-1)(v-2)$, then one can travel between any two towns.

Write some positive integers in the following table such that $\cdot$ there is at most one number in each field $\cdot$ each number is equal to how many numbers there are in edge-adjacent fields, $\cdot$ edge-adjacent fields cannot have equal numbers. What is the sum of numbers in the resulting table? [img]https://cdn.artofproblemsolving.com/attachments/a/9/63a9c38762a4c895688fff049ed08c96b2c22c.png[/img]

Given a set $S$ of $2024$ natural numbers satisfying the following condition: if you select any $10$ (different) numbers from $S$, then you can select another number from $S$ so that the sum of all $11$ selected numbers is divisible by $10$. Prove that one of the numbers can be thrown out of $S$ so that the resulting set $S'$ of $2023$ numbers satisfies the condition: if you choose any $9$ (different) numbers from $S'$, then you can choose another number from $S'$ so that the sum of all $10$ selected numbers is divisible by $10$. [i]K.A.Sukhov[/i]

A cube with size $10\times 10\times 10$ consists of $1000$ unit cubes, all colored white. $A$ and $B$ play a game on this cube. $A$ chooses some pillars with size $1\times 10\times 10$ such that no two pillars share a vertex or side, and turns all chosen unit cubes to black. $B$ is allowed to choose some unit cubes and ask $A$ their colors. How many unit cubes, at least, that $B$ need to choose so that for any answer from $A$, $B$ can always determine the black unit cubes?

Let $ S$ be a set that contains $ n$ elements. Let $ A_{1},A_{2},\cdots,A_{k}$ be $ k$ distinct subsets of $ S$, where $ k\geq 2, |A_{i}| \equal{} a_{i}\geq 1 ( 1\leq i\leq k)$. Prove that the number of subsets of $ S$ that don't contain any $ A_{i} (1\leq i\leq k)$ is greater than or equal to $ 2^n\prod_{i \equal{} 1}^k(1 \minus{} \frac {1}{2^{a_{i}}}).$

Let $n$ be a positive integer. For each partition of the set $\{1,2,\dots,3n\}$ into arithmetic progressions, we consider the sum $S$ of the respective common differences of these arithmetic progressions. What is the maximal value that $S$ can attain? (An [i]arithmetic progression[/i] is a set of the form $\{a,a+d,\dots,a+kd\}$, where $a,d,k$ are positive integers, and $k\geqslant 2$; thus an arithmetic progression has at least three elements, and successive elements have difference $d$, called the [i]common difference[/i] of the arithmetic progression.)

Shakur and Tiham are playing a game. Initially, Shakur picks a positive integer not greater than $1000$. Then Tiham picks a positive integer strictly smaller than that.Then they keep on doing this taking turns to pick progressively smaller and smaller positive integers until some one picks $1$. After that, all the numbers that have been picked so far are added up. The person picking the number $1$ wins if and only if this sum is a perfect square. Otherwise, the other player wins. What is the sum of all possible values of $n$ such that if Shakur starts with the number $n$, he has a winning strategy?

A boy goes $n$ times at a merry-go-round with $n$ seats. After every time he moves in the clockwise direction and takes another seat, not making a full circle. The number of seats he passes by at each move is called the length of the move. For which $n$ can he sit at every seat, if the lengths of all the $n-1$ moves he makes have different lengths? [i]V. New[/i]

We are given $n$ discs in a plane, possibly overlapping, whose union has the area $1$. Prove that we can choose some of them which are mutually disjoint and have the total area greater than $1/9$.

Let $ \alpha < \frac {3 \minus{} \sqrt {5}}{2}$ be a positive real number. Prove that there exist positive integers $ n$ and $ p > \alpha \cdot 2^n$ for which one can select $ 2 \cdot p$ pairwise distinct subsets $ S_1, \ldots, S_p, T_1, \ldots, T_p$ of the set $ \{1,2, \ldots, n\}$ such that $ S_i \cap T_j \neq \emptyset$ for all $ 1 \leq i,j \leq p$ [i]Author: Gerhard Wöginger, Austria[/i]

Consider $n$ points, some of them connected by segments. These segments do not intersect each other. You can reach every point from any every other one in exactly one way by traveling along the segments. Prove that the total number of segments is $n-1$.

An international society has its members from six different countries. The list of members contain $1978$ names, numbered $1, 2, \dots, 1978$. Prove that there is at least one member whose number is the sum of the numbers of two members from his own country, or twice as large as the number of one member from his own country.

The space is filled in the usual way with unit cubes. (Each cube is adjacent to $6$ others that have a common face with it.) On three edges of one of the cubes emerging from one vertex, points are marked at a distance of $1/19$, $1/9$ and $1/7$ from it, respectively. A plane is drawn through these points. Let's consider the many different polygons formed when this plane intersects with the cubes filling the space. How many different (unequal) polygons are there in this set?

Let $n$ be natural number. Each of the numbers $\in\{1,2,\ldots ,n\}$ is coloured in black or white. When we choose a number, we flip it's colour and the colour of all the numbers which have at least one common divider with the chosen number. At the beginning all the numbers were coloured white. For which $n$ are all the numbers black after a finite number of changes?

Two guys are playing the game "Sea Battle-2000". On the board $ 1 \times 200 $, they take turns placing the letter "$ S $" or "$ O $" on the empty squares of the board. The winner is the one who gets the word "$ SOS $" first. Prove that the second player wins when played correctly.

On a chessboard $5 \times 9$ squares, the following game is played. Initially, a number of frogs are randomly placed on some of the squares, no square containing more than one frog. A turn consists of moving all of the frogs subject to the following rules: $\bullet$ Each frog may be moved one square up, down, left, or right; $\bullet$ If a frog moves up or down on one turn, it must move left or right on the next turn, and vice versa; $\bullet$ At the end of each turn, no square can contain two or more frogs. The game stops if it becomes impossible to complete another turn. Prove that if initially $33$ frogs are placed on the board, the game must eventually stop. Prove also that it is possible to place $32$ frogs on the board so that the game can continue forever.

Consider the set of all strictly decreasing sequences of $n$ natural numbers having the property that in each sequence no term divides any other term of the sequence. Let $A = (a_j)$ and $B = (b_j)$ be any two such sequences. We say that $A$ precedes $B$ if for some $k$, $a_k < b_k$ and $a_i = b_i$ for $i < k$. Find the terms of the first sequence of the set under this ordering.