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

The REAL country has $n$ islands, and there are $n-1$ two-way bridges connecting these islands. Any two islands can be reached through a series of bridges. Arctan, the king of the REAL country, found that it is too difficult to manage $n$ islands, so he wants to bomb some islands and their connecting bridges to divide the country into multiple small areas. Arctan wants the number of connected islands in each group is less than $\delta n$ after bombing these islands, and the island he bomb must be a connected area. Besides, Arctan wants the number of islands to be bombed to be as less as possible. Find all real numbers $\delta$ so that for any positive integer $n$ and the layout of the bridge, the method of bombing the islands is the only one. [i]Proposed by chengbilly[/i]

Let $T$ be a tree on $n$ vertices with exactly $k$ leaves. Suppose that there exists a subset of at least $\frac{n+k-1}{2}$ vertices of $T$, no two of which are adjacent. Show that the longest path in $T$ contains an even number of edges. [hide=*]A tree is a connected graph with no cycles. A leaf is a vertex of degree 1[/hide] [i]Vincent Huang[/i]

Let T be a finite tree graph that has more than one vertex. Let s be the largest number of vertices of a subtree $X \subset T$ for which every vertex of X has a neighbor other than X. Let t be the smallest positive integer for which each edge of T is contained in exactly t stars, and each vertex of T is contained in at most 2t - 1 stars. (That is, the stars can be represented by multiplicity.) Prove that s = t. Note: a star of T is a vertex with degree $\geq$ 3 , including its neighouring edges and vertices.

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?

A positive integer $N$ is given. Panda builds a tree on $N$ vertices, and writes a real number on each vertex, so that $1$ plus the number written on each vertex is greater or equal to the average of the numbers written on the neighboring vertices. Let the maximum number written be $M$ and the minimal number written $m$. Mink then gives Panda $M-m$ kilograms of bamboo. What is the maximum amount of bamboo Panda can get?

Captain Jack and his pirate men plundered six chests of treasure $(A_1,A_2,A_3,A_4,A_5,A_6)$. Every chest $A_i$ contains $a_i$ coins of gold, and all $a_i$s are pairwise different $(i=1,2,\cdots ,6)$. They place all chests according to a layout (see the attachment) and start to alternately take out one chest a time between the captain and a pirate who serves as the delegate of the captain’s men. A rule must be complied with during the game: only those chests that are not adjacent to other two or more chests are allowed to be taken out. The captain will win the game if the coins of gold he obtains are not less than those of his men in the end. Let the captain be granted to take chest firstly, is there a certain strategy for him to secure his victory?

Let $T$ be a tree with $n$ vertices; that is, a connected simple graph on $n$ vertices that contains no cycle. For every pair $u$, $v$ of vertices, let $d(u,v)$ denote the distance between $u$ and $v$, that is, the number of edges in the shortest path in $T$ that connects $u$ with $v$. Consider the sums \[W(T)=\sum_{\substack{\{u,v\}\subseteq V(T)\\ u\neq v}}d(u,v) \quad \text{and} \quad H(T)=\sum_{\substack{\{u,v\}\subseteq V(T)\\ u\neq v}}\frac{1}{d(u,v)}\] Prove that \[W(T)\cdot H(T)\geq \frac{(n-1)^3(n+2)}{4}.\]

A set $S$ of rational numbers is ordered in a tree-diagram in such a way that each rational number $\frac{a}{b}$ (where $a$ and $b$ are coprime integers) has exactly two successors: $\frac{a}{a+b}$ and $\frac{b}{a+b}$. How should the initial element be selected such that this tree contains the set of all rationals $r$ with $0 < r < 1$? Give a procedure for determining the level of a rational number $\frac{p}{q}$ in this tree.

Consider $n$ points, some of them connected by segments. These segments do not intersect each other. You can reach every point from any every other one in exactly one way by traveling along the segments. Prove that the total number of segments is $n-1$.

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.

A Magician and a Detective play a game. The Magician lays down cards numbered from $1$ to $52$ face-down on a table. On each move, the Detective can point to two cards and inquire if the numbers on them are consecutive. The Magician replies truthfully. After a finite number of moves, the Detective points to two cards. She wins if the numbers on these two cards are consecutive, and loses otherwise. Prove that the Detective can guarantee a win if and only if she is allowed to ask at least $50$ questions. [i]Proposed by Anant Mudgal[/i]

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