Let $C$ be a finite set of chords in a circle such that each chord passes through the midpoint of some other chord. Prove that any two of these chords intersect inside the circle.

Find all pairs of natural numbers $(n, k)$ with the following property: Given a $k\times k$ array of cells, such that every cell contains one integer, there always exists a path from the left to the right edges such that the sum of the numbers on the path is a multiple of $n$. Note: A path from the left to the right edge is a sequence of cells of the array $a_1, a_2, ... , a_m$ so that $a_1$ is a cell of the leftmost column, $a_m$ is the cell of the rightmost column, and $a_{i}$, $a_{i+1}$ share an edge for all $i = 1, 2, ... , m -1$.

Let $t(A)$ denote the sum of elements of a nonempty set $A$ of integers, and define $t(\emptyset)=0$. Find a set $X$ of positive integers such that for every integers $k$ there is a unique ordered pair of disjoint subsets $(A_{k},B_{k})$ of $X$ such that $t(A_{k})-t(B_{k}) = k$.

A class consists of 26 students with two students sitting on each desk. Suddenly, the students decide to change seats, such that every two students that were previously sitting together are now apart. Find the maximum value of positive integer $N$ such that, regardless of the students' sitting positions, at the end there is a set $S$ consisting of $N$ students satisfying the following property: every two of them have never been sitting together.

Tomás is an avid domino player. One day, while playing with the tiles, he realized he could arrange all the tiles in a single row following the rules, meaning that the number on the right side of each tile matches the number on the left side of the next tile. If the number on the left side of the first tile is 5, what is the number on the right side of the last tile?

In the Flower-city there are $ n$ squares and $ m$ streets, where $ m \geq n \plus{} 1$. Each street connects two squares and does not pass through other squares. According to a tradition in the city, each street is named either blue or red. Every year, a square is selected and the names of all streets emanating from that square are changed. Show that the streets can be initially named in such a way that, no matter how the names will be changed, the streets will never all have the same name.

Let $n\ge 3$ be an integer. Let $s(n)$ be the number of (ordered) pairs $(a;b)$ consisting of positive integers $a,b$ from the set $\{1,2,\dots ,n\}$ which satisfy $\gcd (a,b,n)=1$. Prove that $s(n)$ is divisible by $4$ for all $n\ge 3$. [i] (Vlad Robu) [/i]

In a $99\times 101$ table , cubes of natural numbers, as shown in figure . Prove that the sum of all numbers in the table are divisible by $200$. [img]https://cdn.artofproblemsolving.com/attachments/3/e/dd3d38ca00a36037055acaaa0c2812ae635dcb.png[/img]

Let $n(\ge 2)$ be an integer. $2n^2$ contestants participate in a Chinese chess competition, where any two contestant play exactly once. There may be draws. It is known that (1)If A wins B and B wins C, then A wins C. (2)there are at most $\frac{n^3}{16}$ draws. Proof that it is possible to choose $n^2$ contestants and label them $P_{ij}(1\le i,j\le n)$, so that for any $i,j,i',j'\in \{1,2,...,n\}$, if $i<i'$, then $P_{ij}$ wins $P_{i'j'}$.

Let $k$ be a positive real. $A$ and $B$ play the following game: at the start, there are $80$ zeroes arrange around a circle. Each turn, $A$ increases some of these $80$ numbers, such that the total sum added is $1$. Next, $B$ selects ten consecutive numbers with the largest sum, and reduces them all to $0$. $A$ then wins the game if he/she can ensure that at least one of the number is $\geq k$ at some finite point of time. Determine all $k$ such that $A$ can always win the game.

A traffic accident involved three cars: one blue, one green, and one red. Three witnesses spoke to the police and gave the following statements: **Person 1:** The red car was guilty, and either the green or the blue one was involved. **Person 2:** Either the green car or the red car was guilty, but not both. **Person 3:** Only one of the cars was guilty, but it was not the blue one. The police know that at least one car was guilty and that at least one car was not. However, the police do not know if any of the three witnesses lied. Which car(s) were responsible for the accident?

Consider any graph with $50$ vertices and $225$ edges. We say that a triplet of its (mutually distinct) vertices is [i]connected[/i] if the three vertices determine at least two edges. Determine the smallest and the largest possible number of connected triples. [i](Jan Mazak, Josef Tkadlec)[/i]

Albert and Beatrice play a game. $2021$ stones lie on a table. Starting with Albert, they alternatively remove stones from the table, while obeying the following rule. At the $n$-th turn, the active player (Albert if $n$ is odd, Beatrice if $n$ is even) can remove from $1$ to $n$ stones. Thus, Albert first removes $1$ stone; then, Beatrice can remove $1$ or $2$ stones, as she wishes; then, Albert can remove from $1$ to $3$ stones, and so on. The player who removes the last stone on the table loses, and the other one wins. Which player has a strategy to win regardless of the other player's moves?

You wish to color every vertex, edge, face, and the interior of a cube one color each such that no two adjacent objects are the same color. Faces are adjacent if they share an edge. Edges are adjacent if they share a vertex. The interior is adjacent to all of its faces, edges, and vertices. Each face is adjacent to all of its edges and vertices, but is not adjacent to any other edges or vertices. Each edge is adjacent to both of its vertices, but is not adjacent to any other vertices. What is the minimum number of colors required for a coloring satisfying this property?

Given are $N$ points on the plane such that no three of them are collinear,which are coloured red,green and black.We consider all the segments between these points and give to each segment a [i]"value"[/i] according to the following conditions: [b]i.[/b]If at least one of the endpoints of a segment is black then the segment's [i]"value"[/i] is $0$. [b]ii.[/b]If the endpoints of the segment have the same colour,re or green,then the segment's [i]"value"[/i] is $1$. [b]iii.[/b]If the endpoints of the segment have different colours but none of them is black,then the segment's [i]"value"[/i] is $-1$. Determine the minimum possible sum of the [i]"values"[/i] of the segments.

$a)$ Prove that there exists $5$ nonnegative real numbers with sum equal to $1$, such that no matter how we arrange them on circle, two neighboring numbers exist with product not less than $\frac{1}{9}$ $a)$ Prove that for every $5$ nonnegative real numbers with sum equal to $1$, we can arrange them on circle, such that product of every two neighboring numbers is not greater than $\frac{1}{9}$

Let $n \geq 2$ be a positive integer. In a mathematics competition, there are $n+1$ students, with one of them being a hacker. The competition is conducted as follows: each receives the same problem with an open-ended answer, has 5 minutes to give their own answer, after which all answers are submitted simultaneously, the correct answer is announced, then they receive a new problem, and so on. The hacker cheats by using spy cameras to see the answers of the other participants. A correct answer gives 1 point, while a wrong answer gives -1 point to everyone except the hacker; for him, it's 0 points because he managed to hack the scoring system. Prove that regardless of the total number of problems, if at some point the hacker is ahead of the second-place contestant by at least $2^{n-2} + 1$ points, then he has a strategy to ensure he will be the sole winner by the end of the competition.

Alice and Bob want to play a game. In the beginning of the game, they are teleported to two random position on a train, whose length is $ 1 $ km. This train is closed and dark. So they dont know where they are. Fortunately, both of them have iPhone 133, it displays some information: 1. Your facing direction 2. Your total walking distance 3. whether you are at the front of the train 4. whether you are at the end of the train Moreover, one may see the information of the other one. Once Alice and Bob meet, the game ends. Alice and Bob can only discuss their strategy before the game starts. Find the least value $ x $ so that they are guarantee to end the game with total walking distance $ \le x $ km.

There are 5 points in a rectangle (including its boundary) with area 1, no three of them are in the same line. Find the minimum number of triangles with the area not more than $\frac 1{4}$, vertex of which are three of the five points.

A class has $n > 1$ boys and $n$ girls. For each arrangement $X$ of the class in a line let $f(X)$ be the number of ways of dividing the line into two non-empty segments, so that in each segment the number of boys and girls is equal. Let the number of arrangements with $f(X) = 0$ be $A$, and the number of arrangements with $f(X) = 1$ be $B$. Show that $B = 2A$.

Suppose a table with one row and infinite columns. We call each $1\times1$ square a [i]room[/i]. Let the table be finite from left. We number the rooms from left to $\infty$. We have put in some rooms some coins (A room can have more than one coin.). We can do $2$ below operations: $a)$ If in $2$ adjacent rooms, there are some coins, we can move one coin from the left room $2$ rooms to right and delete one room from the right room. $b)$ If a room whose number is $3$ or more has more than $1$ coin, we can move one of its coins $1$ room to right and move one other coin $2$ rooms to left. $i)$ Prove that for any initial configuration of the coins, after a finite number of movements, we cannot do anything more. $ii)$ Suppose that there is exactly one coin in each room from $1$ to $n$. Prove that by doing the allowed operations, we cannot put any coins in the room $n+2$ or the righter rooms.

Points $P_1,P_2,\ldots,P_n,Q$ are given in space $(n\ge4)$, no four of which are in a plane. Prove that if for any three distinct points $P_\alpha,P_\beta,P_\gamma$ there is a point $P_\delta$ such that the tetrahedron $P_\alpha P_\beta P_\gamma P_\delta$ contains the point $Q$, then $n$ is an even number.

A cube side $5$ is made up of unit cubes. Two small cubes are [i]adjacent [/i] if they have a common face. Can we start at a cube adjacent to a corner cube and move through all the cubes just once? (The path must always move from a cube to an adjacent cube).

For $i = 1,2, \ldots, 1991$, we choose $n_i$ points and write number $i$ on them (each point has only written one number on it). A set of chords are drawn such that: (i) They are pairwise non-intersecting. (ii) The endpoints of each chord have distinct numbers. If for all possible assignments of numbers the operation can always be done, find the necessary and sufficient condition the numbers $n_1, n_2, \ldots, n_{1991}$ must satisfy for this to be possible.

Ali Baba and the $40$ thieves want to cross Bosporus strait. They made a line so that any two people standing next to each other are friends. Ali Baba is the first, he is also a friend with the thief next to his neighbour. There is a single boat that can carry $2$ or $3$ people and these people must be friends. Can Ali Baba and the $40$ thieves always cross the strait if a single person cannot sail?