Given two forests $A$ and $B$ with \(V(A) = V(B)\), that is the graphs are over same vertex set. Suppose $A$ has [b]strictly more[/b] edges than $B$. Prove that there exists an edge of $A$ which if included in the edge set of $B$, then $B$ will still remain a forest. Graphs are undirected

In a group of mathematicians, every mathematician has some friends (the relation of friend is reciprocal). Prove that there exists a mathematician, such that the average of the number of friends of all his friends is no less than the average of the number of friends of all these mathematicians.

There are $m \ge 2$ blue points and $n \ge 2$ red points in three-dimensional space, and no four points are coplanar. Geoff and Nazar take turns, picking one blue point and one red point and connecting the two with a straight-line segment. Assume that Geoff starts first and the one who first makes a cycle wins. Who has the winning strategy?

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.

On some planet, there are $2^N$ countries $(N \geq 4).$ Each country has a flag $N$ units wide and one unit high composed of $N$ fields of size $1 \times 1,$ each field being either yellow or blue. No two countries have the same flag. We say that a set of $N$ flags is diverse if these flags can be arranged into an $N \times N$ square so that all $N$ fields on its main diagonal will have the same color. Determine the smallest positive integer $M$ such that among any $M$ distinct flags, there exist $N$ flags forming a diverse set. [i]Proposed by Tonći Kokan, Croatia[/i]

A country has $100$ cities and $n$ airplane companies which take care of a total of $2018$ two-way direct flights between pairs of cities. There is a pair of cities such that one cannot reach one from the other with just one or two flights. What is the largest possible value of $n$ for which between any two cities there is a route (a sequence of flights) using only one of the airplane companies?

A kingdom has $2021$ towns. All of the towns lie on a circle, and there is a one-way road going from every town to the next $101$ towns in a clockwise order. Each road is colored in one color. Additionally, it is known that for any ordered pair of towns $A$ and $B$ it is possible to find a path from $A$ to $B$ so that no two roads of the path would have the same color. Find the minimal number of road colors in the kingdom.

The plane is divided into regions by $ n \ge 3$ lines, no two of which are parallel, and no three of which are concurrent. Some regions are coloured , in such a way that no two coloured regions share a common segment or half-line of their borders. Prove that the number of coloured regions is at most $ \frac{n(n\plus{}1)}{3}$

Prove that every finite triangle-free graph can be embedded as an induced subgraph in a finite triangle-free graph of diameter 2.

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.

There were 20 participants in a chess tournament. Each of them played with each other twice: once as white and once as black. Let us say that participant $X{}$ is no weaker than participant $Y{}$ if $X{}$ has won at least the same number of games playing white as $Y{}$ and also has won at least the same number of games playing black as $Y{}$ . Do there exist for sure two participants $A{}$ and $B{}$ such that $A{}$ is not weaker than $B{}$? [i]Boris Frenkin[/i]

During a certain lecture, each of five mathematicians fell asleep exactly twice. For each pair of mathematicians, there was some moment when both were asleep simultaneously. Prove that, at some moment, three of them were sleeping simultaneously.

There are $33$ children in a given class. Each child writes a number on the blackboard, which indicates how many other children possess the same forename as oneself. Afterwards, each child does the same thing with their surname. After they've finished, each of the numbers $0,1,2,\dots,10$ appear at least once on the blackboard. Prove that there are at least two children in this class that have the same forename and surname.

There are some cities in both sides of a river and there are some sailing channels between the cities. Each sailing channel connects exactly one city from a side of the river to a city on the other side. Each city has exactly $k$ sailing channels. For every two cities, there's a way which connects them together. Prove that if we remove any (just one) sailing channel, then again for every two cities, there's a way that connect them together. $( k \geq 2)$

From a collection of $n$ persons $q$ distinct two-member teams are selected and ranked $1, \cdots, q$ (no ties). Let $m$ be the least integer larger than or equal to $2q/n$. Show that there are $m$ distinct teams that may be listed so that : [b](i)[/b] each pair of consecutive teams on the list have one member in common and [b](ii)[/b] the chain of teams on the list are in rank order. [i]Alternative formulation.[/i] Given a graph with $n$ vertices and $q$ edges numbered $1, \cdots , q$, show that there exists a chain of $m$ edges, $m \geq \frac{2q}{n}$ , each two consecutive edges having a common vertex, arranged monotonically with respect to the numbering.

The kingdom of Anisotropy consists of $n$ cities. For every two cities there exists exactly one direct one-way road between them. We say that a [i]path from $X$ to $Y$[/i] is a sequence of roads such that one can move from $X$ to $Y$ along this sequence without returning to an already visited city. A collection of paths is called [i]diverse[/i] if no road belongs to two or more paths in the collection. Let $A$ and $B$ be two distinct cities in Anisotropy. Let $N_{AB}$ denote the maximal number of paths in a diverse collection of paths from $A$ to $B$. Similarly, let $N_{BA}$ denote the maximal number of paths in a diverse collection of paths from $B$ to $A$. Prove that the equality $N_{AB} = N_{BA}$ holds if and only if the number of roads going out from $A$ is the same as the number of roads going out from $B$. [i]Proposed by Warut Suksompong, Thailand[/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 $S$.

Consider a finite undirected graph in which each edge belongs to at most three cycles. Prove that its vertices can be colored with three colors so that any two vertices connected by an edge have different colors. [i](A cycle in a graph is a sequence of distinct vertices \( v_1, v_2, \ldots, v_k \), \( k \geq 3 \), such that \( v_i v_{i+1} \) is an edge for each \( i = 1, 2, \ldots, k \); we consider \( v_{k+1} = v_1 \). The edges \( v_i v_{i+1} \) belong to the cycle.)[/i]

There are 25 people. Every two of them are use some language to speak between. They use only one language even if they both know another one. Among every three of them there is one who speaking with two other on the same language. Prove that there exist one who speaking with 10 other on the same language.

This year, some contestants at the Memorial Contest ABC are friends with each other (friendship is always mutual). For each contestant $X$, let $t(X)$ be the total score that this contestant achieved in previous years before this contest. It is known that the following statements are true: $1)$ For any two friends $X'$ and $X''$, we have $t(X') \neq t(X''),$ $2)$ For every contestant $X$, the set $\{ t(Y) : Y \text{ is a friend of } X \}$ consists of consecutive integers. The organizers want to distribute the contestants into contest halls in such a way that no two friends are in the same hall. What is the minimal number of halls they need?

A boy lives in a small island in which there are three roads at every junction. He starts from his home and walks along the roads. At each junction he would choose to turn to the road on his right or left alternatively, i.e., his choices would be . . ., left, right, left,... Prove that he will eventually return to his home.

$1)$. Prove a graph with $2n$ vertices and $n+2$ edges has an independent set of size $n$ (there are $n$ vertices such that no two of them are adjacent ). $2)$.Find the number of graphs with $2n$ vertices and $n+3$ edges , such that among any $n$ vertices there is an edge connecting two of them

Given any positive real number $\varepsilon$, prove that, for all but finitely many positive integers $v$, any graph on $v$ vertices with at least $(1+\varepsilon)v$ edges has two distinct simple cycles of equal lengths. (Recall that the notion of a simple cycle does not allow repetition of vertices in a cycle.) [i]Fedor Petrov, Russia[/i]

Given a finite graph $G$, let $f(G)$ be the number of triangles in graph $G$, $g(G)$ be the number of edges in graph $G$, find the minimum constant $c$, so that for each graph $G$, there is $f^ 2(G)\le c \cdot g^3(G)$.

Consider $9$ points in space, no four of which are coplanar. Each pair of points is joined by an edge (that is, a line segment) and each edge is either colored blue or red or left uncolored. Find the smallest value of $\,n\,$ such that whenever exactly $\,n\,$ edges are colored, the set of colored edges necessarily contains a triangle all of whose edges have the same color.