A town-planner has built an isolated city whose road network consists of $2N$ roundabouts, each connecting exactly three roads. A series of tunnels and bridges ensure that all roads in the town meet only at roundabouts. All roads are two-way, and each roundabout is oriented clockwise. Vlad has recently passed his driving test, and is nervous about roundabouts. He starts driving from his house, and always takes the first edit at each roundabout he encounters. It turns out his journey incluldes every road in the town in both directions before he arrives back at the starting point in the starting direction. For what values of $N$ is this possible?

Given a finite set of points in the plane, each with integer coordinates, is it always possible to color the points red or white so that for any straight line $L$ parallel to one of the coordinate axes the difference (in absolute value) between the numbers of white and red points on $L$ is not greater than $1$?

Every street in the city of Hamiltonville connects two squares, and every square may be reached by streets from every other. The governor discovered that if he closed all squares of any route not passing any square more than once, every remained square would be reachable from each other. Prove that there exists a circular route passing every square of the city exactly once. [i]Author: S. Berlov[/i]

Prove that the edges of a complete graph with $3^n$ vertices can be partitioned into disjoint cycles of length $3$.

On sport games there was 1991 participant from which every participant knows at least n other participants(friendship is mutual). Determine the lowest possible n for which we can be sure that there are 6 participants between which any two participants know each other.

$2024$ otters live in the river. Some are friends with each other. Is it possible that, for any collection of $1012$ otters, there is exactly one additional otter that is friends with all $1012$ otters?

For integers $a, b$, call the lattice point with coordinates $(a,b)$ [b]basic[/b] if $gcd(a,b)=1$. A graph takes the basic points as vertices and the edges are drawn in such way: There is an edge between $(a_1,b_1)$ and $(a_2,b_2)$ if and only if $2a_1=2a_2\in \{b_1-b_2, b_2-b_1\}$ or $2b_1=2b_2\in\{a_1-a_2, a_2-a_1\}$. Some of the edges will be erased, such that the remaining graph is a forest. At least how many edges must be erased to obtain this forest? At least how many trees exist in such a forest?

Prove that the for all $n>1000$, we can arrange the number $1,2,\dots, \binom{n}{2}$ on edges of a complete graph with $n$ vertices so that the sum of the numbers assigned to edges of any length three path (possibly closed) is not less than $3n-1000log_2log_2 n$.

Prove that a graph containing a copy of each possible tree on $n$ vertices as a subgraph has at least $n(\ln n - 2)$ edges.

Let $G$ be a bipartite graph in which the greatest degree of a vertex is 2019. Let $m$ be the least natural number for which we can color the edges of $G$ in $m$ colors so that each two edges with a common vertex from $G$ are in different colors. Show that $m$ doesn’t depend on $G$ and find its value.

Consider a graph with $n$ vertices and $\frac{7n}{4}$ edges. (a) Prove that there are two cycles of equal length. (25 points) (b) Can you give a smaller function than $\frac{7n}{4}$ that still fits in part (a)? Prove your claim. We say function $a(n)$ is smaller than $b(n)$ if there exists an $N$ such that for each $n>N$ ,$a(n)<b(n)$ (At most 5 points) [i]Proposed by Afrooz Jabal'ameli[/i]

On every card of a deck of cards a regular 17-gon is displayed with all sides and diagonals, and the vertices are numbered from 1 through 17. On every card all edges (sides and diagonals) are colored with a color 1,2,...,105 such that the following property holds: for every 15 vertices of the 17-gon the 105 edges connecting these vertices are colored with different colors on at least one of the cards. What is the minimum number of cards in the deck?

Let $n\geqslant 2$ be a positive integer. Paul has a $1\times n^2$ rectangular strip consisting of $n^2$ unit squares, where the $i^{\text{th}}$ square is labelled with $i$ for all $1\leqslant i\leqslant n^2$. He wishes to cut the strip into several pieces, where each piece consists of a number of consecutive unit squares, and then [i]translate[/i] (without rotating or flipping) the pieces to obtain an $n\times n$ square satisfying the following property: if the unit square in the $i^{\text{th}}$ row and $j^{\text{th}}$ column is labelled with $a_{ij}$, then $a_{ij}-(i+j-1)$ is divisible by $n$. Determine the smallest number of pieces Paul needs to make in order to accomplish this.

Smallville is populated by unmarried men and women, some of which are acquainted. The two City Matchmakers know who is acquainted with whom. One day, one of the matchmakers claimed: "I can arrange it so that every red haired man will marry a woman with who he is acquainted." The other matchmaker claimed: "I can arrange it so that every blonde woman will marry a man with whom she is acquainted." An amateur mathematician overheard this conversation and said: "Then it can be arranged so that every red haired man will marry a woman with whom he is acquainted and at the same time very blonde woman will marry a man with who she is acquainted." Is the mathematician right?

Given seven points in the plane, some of them are connected by segments such that: [b](i)[/b] among any three of the given points, two are connected by a segment; [b](ii)[/b] the number of segments is minimal. How many segments does a figure satisfying [b](i)[/b] and [b](ii)[/b] have? Give an example of such a figure.

The organizers of a mathematical congress found that if they accomodate any participant in a room the rest can be accomodated in double rooms so that 2 persons living in each room know each other. Prove that every participant can organize a round table on graph theory for himself and an even number of other people so that each participant of the round table knows both his neigbours. [i]Proposed by S. Berlov, S. Ivanov[/i]

For a positive integer $n$ define a sequence of zeros and ones to be [i]balanced[/i] if it contains $n$ zeros and $n$ ones. Two balanced sequences $a$ and $b$ are [i]neighbors[/i] if you can move one of the $2n$ symbols of $a$ to another position to form $b$. For instance, when $n = 4$, the balanced sequences $01101001$ and $00110101$ are neighbors because the third (or fourth) zero in the first sequence can be moved to the first or second position to form the second sequence. Prove that there is a set $S$ of at most $\frac{1}{n+1} \binom{2n}{n}$ balanced sequences such that every balanced sequence is equal to or is a neighbor of at least one sequence in $S$.

Let \(L_3 = \{3\}\), \(L_n = \{3, 4, \ldots, h\}\) (where \(h > 3\)). For any given integer \(n \geq 3\), consider a graph \(G\) with \(n\) vertices that contains a Hamiltonian cycle \(C\) and has more than \(\frac{n^2}{4}\) edges. For which lengths \(l \in L_n\) must the graph \(G\) necessarily contain a cycle of length \(l\)?

We are given a positive integer $ r$ and a rectangular board $ ABCD$ with dimensions $ AB \equal{} 20, BC \equal{} 12$. The rectangle is divided into a grid of $ 20 \times 12$ unit squares. The following moves are permitted on the board: one can move from one square to another only if the distance between the centers of the two squares is $ \sqrt {r}$. The task is to find a sequence of moves leading from the square with $ A$ as a vertex to the square with $ B$ as a vertex. (a) Show that the task cannot be done if $ r$ is divisible by 2 or 3. (b) Prove that the task is possible when $ r \equal{} 73$. (c) Can the task be done when $ r \equal{} 97$?

A directed graph has each vertex with outdegree 2. Prove that it is possible to split the vertices into 3 sets so that for each vertex $v$, $v$ is not simultaneously in the same set with both of the vertices that it points to. [i]David Yang.[/i] [hide="Stronger Version"]See [url=http://www.artofproblemsolving.com/Forum/viewtopic.php?f=42&t=492100]here[/url].[/hide]

Let $G$ be a simple graph on the infinite vertex set $V=\{v_1, v_2, v_3,\ldots\}$. Suppose every subgraph of $G$ on a finite vertex subset is $10$-colorable, Prove that $G$ itself is $10$-colorable.

Let \(Q\) be a set of permutations of \(1,2,...,100\) such that for all \(1\leq a,b \leq 100\), \(a\) can be found to the left of \(b\) and adjacent to \(b\) in at most one permutation in \(Q\). Find the largest possible number of elements in \(Q\).

In ancient times there was a Celtic tribe consisting of several families. Many of these families were at odds with each other, so that their chiefs would not shake hands. At some point at the annual meeting of the chiefs they found it even impossible to assemble four or more of them in a circle with each of them being willing to shake his neighbour's hand. To emphasize the gravity of the situation, the Druid collected three pieces of gold from each family. The Druid then let all those chiefs shake hands who were willing to. For each handshake of two chiefs he paid each of them a piece of gold as a reward. Show that the number of pieces of gold collected by the Druid exceeds the number of pieces paid out by at least three.

Prove that for any finite graph G there is a constant c(G)>0 such that for every n-point graph that does not have an induced subgraph isomorphic to G, there are two disjoint sets of vertices, each with at least $n^{c(G)}$ elements, between which either all edges are connected, or none of the edges are.

Let $n\geq 3$ be a fixed integer. Each side and each diagonal of a regular $n$-gon is labelled with a number from the set $\left\{1;\;2;\;...;\;r\right\}$ in a way such that the following two conditions are fulfilled: [b]1.[/b] Each number from the set $\left\{1;\;2;\;...;\;r\right\}$ occurs at least once as a label. [b]2.[/b] In each triangle formed by three vertices of the $n$-gon, two of the sides are labelled with the same number, and this number is greater than the label of the third side. [b](a)[/b] Find the maximal $r$ for which such a labelling is possible. [b](b)[/b] [i]Harder version (IMO Shortlist 2005):[/i] For this maximal value of $r$, how many such labellings are there? [hide="Easier version (5th German TST 2006) - contains answer to the harder version"] [i]Easier version (5th German TST 2006):[/i] Show that, for this maximal value of $r$, there are exactly $\frac{n!\left(n-1\right)!}{2^{n-1}}$ possible labellings.[/hide] [i]Proposed by Federico Ardila, Colombia[/i]