$2017$ engineers attend a conference. Any two engineers if they converse, converse with each other in either Chinese or English. No two engineers converse with each other more than once. It is known that within any four engineers, there was an even number of conversations and furthermore within this even number of conversations: i) At least one conversation is in Chinese. ii) Either no conversations are in English or the number of English conversations is at least that of Chinese conversations. Show that there exists $673$ engineers such that any two of them conversed with each other in Chinese.

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]

show there exist positive constants $c_1$ and $c_2$ such that for any $n\geq 3$, whenever $T_1$ and $T_2$ are two trees on the set of vertices $X = \{1, 2, ..., n\}$, there exists a function $f : X \to \{-1, +1\}$ for which $$\bigg | \sum_ {x \in P} f (x) \bigg | <c_1 \log n$$ for any path P that is a subgraph of $T_1$ or $T_2$ , but with an upper bound $c_2 \log n / \log \log n$ the statement is no longer true.

Five points are given in space, no four of which are coplanar. Each of the segments connecting two of them is painted in white, green or red, so that all the colors are used and no three segments of the same color form a triangle. Prove that among these five points there is one at which segments of all the three colors meet.

In the City of Islands there are $2021$ islands connected by bridges. For any given pair of islands $A$ and $B$, one can go from island $A$ to island $B$ using the bridges. Moreover, for any four islands $A_1, A_2, A_3$ and $A_4$: if there is a bridge from $A_i$ to $A_{i+1}$ for each $i \in \left \{ 1,2,3 \right \}$, then there is a bridge between $A_{j}$ and $A_{k}$ for some $j,k \in \left \{ 1,2,3,4 \right \}$ with $|j-k|=2$. Show that there is at least one island which is connected to any other island by a bridge.

Some cities in country are connected with oneway road. It is known that every closed cyclic route, that don`t break traffic laws, consists of even roads. Prove that king of city can place military bases in some cities such that there are not roads between these cities, but for every city without base we can go from city with base by no more than $1$ road. [hide=PS]I think it should be one more condition, like there is cycle that connect all cities [/hide]

There are $k$ cities in Belarus and $k$ cities in Armenia, between some cities there are non-directed flights. From any Belarusian city there are exactly $n$ flights to Armenian cities, and for every pair of Armenian cities exactly two Belarusian cities have flights to both of the Armenian cities. a) Prove that from every Armenian city there are exactly $n$ flights to Belarusian cities. b) Prove that there exists a flight route in which every city is visited at most once and that consists of at least $\lfloor \frac{(n+1)^2}{4} \rfloor$ cities in each of the countries. [i]D. Gorovoy[/i]

In the country there're $N$ cities and some pairs of cities are connected by two-way airlines (each pair with no more than one). Every airline belongs to one of $k$ companies. It turns out that it's possible to get to any city from any other, but it fails when we delete all airlines belonging to any one of the companies. What is the maximum possible number of airlines in the country ?

Adam, Benin, Chiang, Deshawn, Esther, and Fiona have internet accounts. Some, but not all, of them are internet friends with each other, and none of them has an internet friend outside this group. Each of them has the same number of internet friends. In how many different ways can this happen? $ \textbf{(A)}\ 60 \qquad\textbf{(B)}\ 170 \qquad\textbf{(C)}\ 290 \qquad\textbf{(D)}\ 320 \qquad\textbf{(E)}\ 660 $

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

Find the maximum number $E$ such that the following holds: there is an edge-colored graph with 60 vertices and $E$ edges, with each edge colored either red or blue, such that in that coloring, there is no monochromatic cycles of length 3 and no monochromatic cycles of length 5.

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 $1$?

There is a graph with 30 vertices. If any of 26 of its vertices with their outgoiing edges are deleted, then the remained graph is a connected graph with 4 vertices. What is the smallest number of the edges in the initial graph with 30 vertices?

Let $A$ be a set of $n$ points in the space. From the family of all segments with endpoints in $A$, $q$ segments have been selected and colored yellow. Suppose that all yellow segments are of different length. Prove that there exists a polygonal line composed of $m$ yellow segments, where $m \geq \frac{2q}{n}$, arranged in order of increasing length.

In the reality show [i]Big Sister Brasil[/i], it is said that there is a [i]treta[/i] if two people are friends with each other and enemies with a third one. For audience purposes, the broadcaster wants a lot of [i]tretas[/i]. If friendship and enmity are reciprocal relationships, given $n$ people, what is the maximum number of [i]tretas[/i]?

A subset of $ n\times n$ table is called even if it contains even elements of each row and each column. Find the minimum $ k$ such that each subset of this table with $ k$ elements contains an even subset

Suppose that $2n$ points of an $n\times n$ grid are marked. Show that for some $k > 1$ one can select $2k$ distinct marked points, say $a_1,...,a_{2k}$, such that $a_{2i-1}$ and $a_{2i}$ are in the same row, $a_{2i}$ and $a_{2i+1}$ are in the same column, $\forall i$, indices taken mod 2n.

In the country of Graphia there are $100$ towns, each numbered from $1$ to $100$. Some pairs of towns may be connected by a (direct) road and we call such pairs of towns [i]adjacent[/i]. No two roads connect the same pair of towns. Peter, a foreign tourist, plans to visit Graphia $100$ times. For each $i$, $i=1,2,\dots, 100$, Peter starts his $i$-th trip by arriving in the town numbered $i$ and then each following day Peter travels from the town he is currently in to an adjacent town with the lowest assigned number, assuming such that a town exists and that he hasn't visited it already on the $i$-th trip. Otherwise, Peter deems his $i$-th trip to be complete and returns home. It turns out that after all $100$ trips, Peter has visited each town in Graphia the same number of times. Find the largest possible number of roads in Graphia.

In a country, some cities are connected by two-way airlines, and one can get from any city to any other city in no more than $n{}$ flights. Prove that all airlines can be distributed among $n{}$ companies so that a route can be built between any two cities in which no more than two flights of each company would meet. [i]From the folklore[/i]

Initially, all edges of the $K_{2024}$ are painted with $13$ different colors. If there exist $k$ colors such that the subgraph constructed by the edges which are colored with these $k$ colors is connected no matter how the initial coloring was, find the minimum value of $k$.

Let $n$ be an integer greater than $1$, and let $p$ be a prime divisor of $n$. A confederation consists of $p$ states, each of which has exactly $n$ airports. There are $p$ air companies operating interstate flights only such that every two airports in different states are joined by a direct (two-way) flight operated by one of these companies. Determine the maximal integer $N$ satisfying the following condition: In every such confederation it is possible to choose one of the $p$ air companies and $N$ of the $np$ airports such that one may travel (not necessarily directly) from any one of the $N$ chosen airports to any other such only by flights operated by the chosen air company.

consider $n\geq 6$ points $x_1,x_2,\dots,x_n$ on the plane such that no three of them are colinear. We call graph with vertices $x_1,x_2,\dots,x_n$ a "road network" if it is connected, each edge is a line segment, and no two edges intersect each other at points other than the vertices. Prove that there are three road networks $G_1,G_2,G_3$ such that $G_i$ and $G_j$ don't have a common edge for $1\leq i,j\leq 3$. Proposed by Morteza Saghafian

This question is both combinatorics and Number Theory : a ) Prove that we can color edges of $K_{p}$ with $p$ colors which is proper, ($p$ is an odd prime) and $K_{p}$ can be partitioned to $\frac{p-1}2$ rainbow Hamiltonian cycles. (A Hamiltonian cycle is a cycle that passes from all of verteces, and a rainbow is a subgraph that all of its edges have different colors.) b) Find all answers of $x^{2}+y^{2}+z^{2}=1$ is $\mathbb Z_{p}$

There are $n\geq 2$ people at a party. Each person has at least one friend inside the party. Show that it is possible to choose a group of no more than $\frac{n}{2}$ people at the party, such that any other person outside the group has a friend inside it.

Let $\mathfrak F$ be the family of all $k$-element subsets of the set $\{1, 2, \ldots, 2k + 1\}$. Prove that there exists a bijective function $f :\mathfrak F \to \mathfrak F$ such that for every $A \in \mathfrak F$, the sets $A$ and $f(A)$ are disjoint.