Prove that there exists an undirected graph having $ 2004 $ vertices such that for any $ \in\{ 1,2,\ldots ,1002 \} , $ there exists at least two vertices whose orders are $ n. $

The localities $P_1, P_2, \dots, P_{1983}$ are served by ten international airlines $A_1,A_2, \dots , A_{10}$. It is noticed that there is direct service (without stops) between any two of these localities and that all airline schedules offer round-trip flights. Prove that at least one of the airlines can offer a round trip with an odd number of landings.

There are $24$ participants attended a meeting. Each two of them shook hands once or not. A total of $216$ handshakes occured in the meeting. For any two participants who have shaken hands, at most $10$ among the rest $22$ participants have shaken hands with exactly one of these two persons. Define a [i]friend circle[/i] to be a group of $3$ participants in which each person has shaken hands with the other two. Find the minimum possible value of friend circles.

Let $n$ be an even positive integer. Show that there is a permutation $\left(x_{1},x_{2},\ldots,x_{n}\right)$ of $\left(1,\,2,\,\ldots,n\right)$ such that for every $i\in\left\{1,\ 2,\ ...,\ n\right\}$, the number $x_{i+1}$ is one of the numbers $2x_{i}$, $2x_{i}-1$, $2x_{i}-n$, $2x_{i}-n-1$. Hereby, we use the cyclic subscript convention, so that $x_{n+1}$ means $x_{1}$.

Each one of $2001$ children chooses a positive integer and writes down his number and names of some of other $2000$ children to his notebook. Let $A_c$ be the sum of the numbers chosen by the children who appeared in the notebook of the child $c$. Let $B_c$ be the sum of the numbers chosen by the children who wrote the name of the child $c$ into their notebooks. The number $N_c = A_c - B_c$ is assigned to the child $c$. Determine whether all of the numbers assigned to the children could be positive.

Consider the figure below, composed of 16 segments. Prove that there is no curve intersecting each segment exactly once. (The curve may be not closed, may intersect itself, but it is not allowed to touch the segments or to pass through the vertices.) [asy] draw((0,0)--(6,0)--(6,3)--(0,3)--(0,0)); draw((0,3/2)--(6,3/2)); draw((2,0)--(2,3/2)); draw((4,0)--(4,3/2)); draw((3,3/2)--(3,3)); [/asy]

Emperor Zorg wishes to found a colony on a new planet. Each of the $n$ cities that he will establish there will have to speak exactly one of the Empire's $2020$ official languages. Some towns in the colony will be connected by a direct air link, each link can be taken in both directions. The emperor fixed the cost of the ticket for each connection to $1$ galactic credit. He wishes that, given any two cities speaking the same language, it is always possible to travel from one to the other via these air links, and that the cheapest trip between these two cities costs exactly $2020$ galactic credits. For what values of $n$ can Emperor Zorg fulfill his dream?

Let $S$ be an infinite set of positive integers, such that there exist four pairwise distinct $a,b,c,d \in S$ with $\gcd(a,b) \neq \gcd(c,d)$. Prove that there exist three pairwise distinct $x,y,z \in S$ such that $\gcd(x,y)=\gcd(y,z) \neq \gcd(z,x)$.

There are 15 people at a party; each person has 10 friends. To greet each other each person hugs all their friends. How many hugs are exchanged at this party?

An airline operates flights between any two capital cities in the European Union. Each flight has a fixed price which is the same in both directions. Furthermore, the flight prices from any given city are pairwise distinct. Anna and Bella wish to visit each city exactly once, not necessarily starting from the same city. While Anna always takes the cheapest flight from her current city to some city she hasn't visited yet, Bella always continues her tour with the most expensive flight available. Is it true that Bella's tour will surely cost at least as much as Anna's tour? [i](Based on a Soviet problem)[/i]

A white plane is partitioned onto cells (in a usual way). A finite number of cells are coloured black. Each black cell has an even (0, 2 or 4) adjacent (by the side) white cells. Prove that one may colour each white cell in green or red such that every black cell will have equal number of red and green adjacent cells.

This ISL 2005 problem has not been used in any TST I know. A pity, since it is a nice problem, but in its shortlist formulation, it is absolutely incomprehensible. Here is a mathematical restatement of the problem: Let $k$ be a nonnegative integer. A forest consists of rooted (i. e. oriented) trees. Each vertex of the forest is either a leaf or has two successors. A vertex $v$ is called an [i]extended successor[/i] of a vertex $u$ if there is a chain of vertices $u_{0}=u$, $u_{1}$, $u_{2}$, ..., $u_{t-1}$, $u_{t}=v$ with $t>0$ such that the vertex $u_{i+1}$ is a successor of the vertex $u_{i}$ for every integer $i$ with $0\leq i\leq t-1$. A vertex is called [i]dynastic[/i] if it has two successors and each of these successors has at least $k$ extended successors. Prove that if the forest has $n$ vertices, then there are at most $\frac{n}{k+2}$ dynastic vertices.

A group consists of $n$ tourists. Among every three of them there are at least two that are not familiar. For any partition of the group into two busses, there are at least two familiar tourists in one bus. Prove that there is a tourist who is familiar with at most two fifth of all the tourists.

The $n$-element set of real numbers is given, where $n \geq 6$. Prove that there exist at least $n-1$ two-element subsets of this set, in which the arithmetic mean of elements is not less than the arithmetic mean of elements in the whole set.

There are $n$ people, and given that any $2$ of them have contacted with each other at most once. In any group of $n-2$ of them, any one person of the group has contacted with other people in this group for $3^k$ times, where $k$ is a non-negative integer. Determine all the possible value of $n.$

For a positive integer $n$, there is a school with $2n$ people. For a set $X$ of students in this school, if any two students in $X$ know each other, we call $X$ [i]well-formed[/i]. If the maximum number of students in a well-formed set is no more than $n$, find the maximum number of well-formed set. Here, an empty set and a set with one student is regarded as well-formed as well.

Let $G$ be a simple graph with 100 vertices such that for each vertice $u$, there exists a vertice $v \in N \left ( u \right )$ and $ N \left ( u \right ) \cap N \left ( v \right ) = \o $. Try to find the maximal possible number of edges in $G$. The $ N \left ( . \right )$ refers to the neighborhood.

In a simple graph, there are two disjoint set of vertices $A$ and $B$ where $A$ has $k$ and $B$ has $2016$ vertices. Four numbers are written to each vertex using the colors red, green, blue and black. There is no any edge at the beginning. For each vertex in $A$, we first choose a color and then draw all edges from this vertex to the vertices in $B$ having a larger number with the chosen color. It is known that for each vertex in $B$, the set of vertices in $A$ connected to this vertex are different. Find the minimal possible value of $k$.

In a company of people some pairs are enemies. A group of people is called [i]unsociable[/i] if the number of members in the group is odd and at least $3$, and it is possible to arrange all its members around a round table so that every two neighbors are enemies. Given that there are at most $2015$ unsociable groups, prove that it is possible to partition the company into $11$ parts so that no two enemies are in the same part. [i]Proposed by Russia[/i]

In the plane, $10^6$ points are marked, no three of which are collinear. All possible segments between them are drawn. Grisha assigned to each drawn segment a real number with absolute value no greater than $1$. For every group of $6$ marked points, he calculated the sum of the numbers on all $15$ connecting segments. It turned out that the absolute value of each such sum is at least \(C\), and there are both positive and negative such sums. What is the maximum possible value of \(C\)?

Let $m$ and $n$ be positive integers integers such that $2m + 1 < n$, and let $S$ be the set of the $2^n$ subsets of $\{1,2,\ldots,n\}$. Prove that we can place the elements of $S$ on a circle, so that for any two adjacent elements $A$ and $B$, the set $A \Delta B$ has exactly $2m + 1$ elements. [b]Note[/b]: $A \Delta B = (A \cup B) - (A \cap B)$ is the set of elements that are exclusively in $A$ or exclusively in $B$.

Let $F$ be a finite non-empty set of integers and let $n$ be a positive integer. Suppose that $\bullet$ Any $x \in F$ may be written as $x=y+z$ for some $y$, $z \in F$; $\bullet$ If $1 \leq k \leq n$ and $x_1$, ..., $x_k \in F$, then $x_1+\cdots+x_k \neq 0$. Show that $F$ has at least $2n+2$ elements.

Given a finite number of boys and girls, a [i]sociable set of boys[/i] is a set of boys such that every girl knows at least one boy in that set; and a [i]sociable set of girls[/i] is a set of girls such that every boy knows at least one girl in that set. Prove that the number of sociable sets of boys and the number of sociable sets of girls have the same parity. (Acquaintance is assumed to be mutual.) [i](Poland) Marek Cygan[/i]

There are $n>2$ lines given in the plane. No two of the lines are parallel and no three of them intersect at one point. Every point of intersection of these lines is labelled with a natural number between $1$ and $n-1$. Prove that, if and only if $n$ is even, it is possible to assign the labels in such a way that every line has all the numbers from $1$ to $n-1$ at its points of intersection with the other $n-1$ lines.

Find all positive integers $n\geq 3$ such that it is possible to triangulate a convex $n$-gon such that all vertices of the $n$-gon have even degree.