A crazy physicist discovered a new kind of particle wich he called an imon, after some of them mysteriously appeared in his lab. Some pairs of imons in the lab can be entangled, and each imon can participate in many entanglement relations. The physicist has found a way to perform the following two kinds of operations with these particles, one operation at a time. (i) If some imon is entangled with an odd number of other imons in the lab, then the physicist can destroy it. (ii) At any moment, he may double the whole family of imons in the lab by creating a copy $I'$ of each imon $I$. During this procedure, the two copies $I'$ and $J'$ become entangled if and only if the original imons $I$ and $J$ are entangled, and each copy $I'$ becomes entangled with its original imon $I$; no other entanglements occur or disappear at this moment. Prove that the physicist may apply a sequence of such operations resulting in a family of imons, no two of which are entangled.

Several $good$ points, several $bad$ points and several segments are drawn in the plane. Each segment connects a $good$ point and a $bad$ one; at most $100$ segments begin at each point. We have paint of $200$ colors. One half of each segment is painted with one of these colors, and the other half with another one. Is it always possible to do it so that every two segments with common end are painted with four different colors? [i](M. Qi, X. Zhang)[/i]

There are $n$ students in a class, and some pairs of these students are friends. Among any six students, there are two of them that are not friends, and for any pair of students that are not friends there is a student among the remaining four that is friends with both of them. Find the maximum value of $n$.

At a two-round volleyball tournament participated 99 teams. Each played a match at home and a match away. Each team won exactly half of their home matches and exactly half of their away matches. Prove that one of the teams beat another team twice. [i]Proposed by M. Antipov[/i]

A graph with $10^{100}$ vertices satisfies the following condition: Any simple odd cycle has length > 100. Prove there is an independent set in the graph of size at least $\frac{10^{100}}{102}$

Let $n \geq 2$ be a positive integer. On a spaceship, there are $n$ crewmates. At most one accusation of being an imposter can occur from one crewmate to another crewmate. Multiple accusations are thrown, with the following properties: • Each crewmate made a different number of accusations. • Each crewmate received a different number of accusations. • A crewmate does not accuse themself. Prove that no two crewmates made accusations at each other.

Call a simple graph $G$ [i]quasicolorable[/i] if we can color each edge blue, red, green, or white such that [list] [*] for each vertex v of degree 3 in G, the three edges incident to v are either (1) red, green, and blue, or (2) all white, [*] not all edges are white. [/list] A simple connected graph $G$ has $a$ vertices of degree $4$, $b$ vertices of degree $3$, and no other vertices, where $a$ and $b$ are positive integers. Find the smallest real number $c$ so that the following statement is true: “If $a/b > c$, then $G$ must be quasicolorable.”

In a meeting of $4042$ people, there are $2021$ couples, each consisting of two people. Suppose that $A$ and $B$, in the meeting, are friends when they know each other. For a positive integer $n$, each people chooses an integer from $-n$ to $n$ so that the following conditions hold. (Two or more people may choose the same number). [list] [*] Two or less people chose $0$, and if exactly two people chose $0$, they are coupled. [*] Two people are either coupled or don't know each other if they chose the same number. [*] Two people are either coupled or know each other if they chose two numbers that sum to $0$. [/list] Determine the least possible value of $n$ for which such number selecting is always possible.

How many graphs are there on $6$ vertices with degrees $1,1,2,3,4,5$?

Fix an integer $n \ge 2$, let $Q_n$ be the graph consisting of all vertices and all edges of an $n$-cube, and let $T$ be a spanning tree in $Q_n$. Show that $Q_n$ has an edge whose adjunction to $T$ produces a simple cycle of length at least $2n$.

$ 1650$ students are arranged in $ 22$ rows and $ 75$ columns. It is known that in any two columns, the number of pairs of students in the same row and of the same sex is not greater than $ 11$. Prove that the number of boys is not greater than $ 928$.

In a simple graph, there exist two vertices $A,B$ such that there are exactly $100$ shortest paths from $A$ to $B$. Find the minimum number of edges in the graph. [i]CSJL[/i]

Let S be a set of $n\geq 3$ points in space. We color some line segments having two points in $S$ as their endpoints red, let $r$ be the number of edges colored red (color $r$ edges red). We know no two colored segmentes have the same length. Proof there is a path of red edges increasing in size of length at least $\Bigg\lceil\frac{2r}{n}\Bigg\rceil$.

Let \( G \) be a graph colored using \( k \) colors. We say that a vertex is [b]forced[/b] if it has neighbors in all the other \( k - 1 \) colors. Prove that for any \( 2024 \)-regular graph \( G \) that contains no triangles or quadrilaterals, there exists a coloring using \( 2025 \) colors such that at least \( 1013 \) of the colors have a forced vertex of that color. Note: The graph coloring must be valid, this means no \( 2 \) vertices of the same color may be adjacent.

There are $N$ points on the plane; no three of them belong to the same straight line. Every pair of points is connected by a segment. Some of these segments are colored in red and the rest of them in blue. The red segments form a closed broken line without self-intersections(each red segment having only common endpoints with its two neighbors and no other common points with the other segments), and so do the blue segments. Find all possible values of $N$ for which such a disposition of $N$ points and such a choice of red and blue segments are possible.

At a table tennis tournament, each one of the $ n\ge 2 $ participants play with all the others exactly once. Show that, at the end of the tournament, one and only one of these propositions will be true: $ \text{(1)} $ The players can be labeled with the numbers $ 1,2,...,n, $ such that $ 1 $ won $ 2, 2 $ won $ 3,...,n-1 $ won $ n $ and $ n $ won $ 1. $ $ \text{(2)} $ The players can be partitioned in two nonempty subsets $ A,B, $ such that whichever one from $ A $ won all that are in $ B. $

Find the smallest positive constant $c$ satisfying: For any simple graph $G=G(V,E)$, if $|E|\geq c|V|$, then $G$ contains $2$ cycles with no common vertex, and one of them contains a chord. Note: The cycle of graph $G(V,E)$ is a set of distinct vertices ${v_1,v_2...,v_n}\subseteq V$, $v_iv_{i+1}\in E$ for all $1\leq i\leq n$ $(n\geq 3, v_{n+1}=v_1)$; a cycle containing a chord is the cycle ${v_1,v_2...,v_n}$, such that there exist $i,j, 1< i-j< n-1$, satisfying $v_iv_j\in E$.

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.

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 $\mathbb Z_{p}$

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 $\gcd(x,y)=\gcd(y,z) \neq \gcd(z,x)$.

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.

Let $G = (S, T; E)$ be a finite bipartite graph with a perfect matching. Prove that there exists an injective edge weighting $w: E \to \mathbb{R}$ satisfying the following: 1. If $e_s$ is the edge with the smallest weight among the edges incident to $s$ for all $s \in S$, then $\{ e_s \mid s \in S \}$ forms a perfect matching in $G$. 2. If $e_t$ is the edge with the largest weight among the edges incident to $t$ for all $t \in T$, then $\{ e_t \mid t \in T \}$ forms a perfect matching in $G$.

Integers not divisible by $2012$ are arranged on the arcs of an oriented graph. We call the [i]weight of a vertex [/i]the difference between the sum of the numbers on the arcs coming into it and the sum of the numbers on the arcs going away from it. It is known that the weight of each vertex is divisible by $2012$. Prove that non-zero integers with absolute values not exceeding $2012$ can be arranged on the arcs of this graph, so that the weight of each vertex is zero. [i]Proposed by W. Tutte[/i]

A social network has $2019$ users, some pairs of whom are friends. Whenever user $A$ is friends with user $B$, user $B$ is also friends with user $A$. Events of the following kind may happen repeatedly, one at a time: [list] [*] Three users $A$, $B$, and $C$ such that $A$ is friends with both $B$ and $C$, but $B$ and $C$ are not friends, change their friendship statuses such that $B$ and $C$ are now friends, but $A$ is no longer friends with $B$, and no longer friends with $C$. All other friendship statuses are unchanged. [/list] Initially, $1010$ users have $1009$ friends each, and $1009$ users have $1010$ friends each. Prove that there exists a sequence of such events after which each user is friends with at most one other user. [i]Proposed by Adrian Beker, Croatia[/i]

I've come across a challenging graph theory problem. Roughly translated, it goes something like this: There are n lines drawn on a plane; no two lines are parallel to each other, and no three lines meet at a single point. Those lines would partition the plane down into many 'area's. Suppose we select one point from each area. Also, should two areas share a common side, we connect the two points belonging to the respective areas with a line. A graph consisted of points and lines will have been made. Find all possible 'n' that will make a hamiltonian circuit exist for the given graph