Find the largest positive integer $n$ ($n \ge 3$), so that there is a convex $n$-gon, the tangent of each interior angle is an integer.

If $a_1,a_2,\cdots,a_{2013}\in[-2,2]$ and $a_1+a_2+\cdots+a_{2013}=0$ , find the maximum of $a^3_1+a^3_2+\cdots+a^3_{2013}$.

As shown in figure , $A,B$ are two fixed points of circle $\odot O$, $C$ is the midpoint of the major arc $AB$, $D$ is any point of the minor arc $AB$. Tangent at $D$ intersects tangents at $A,B$ at points $E,F$ respectively. Segments $CE$ and $CF$ intersect chord $AB$ at points $G$ and $H$ respectively. Prove that the length of line segment $GH$ has a fixed value.

For positive integers $n,a,b$, if $n=a^2 +b^2$, and $a$ and $b$ are coprime, then the number pair $(a,b)$ is called a square split of $n$ (the order of $a, b$ does not count). Prove that for any positive $k$, there are only two square splits of the integer $13^k$.

Find all non-integers $x$ such that $x+\frac{13}{x}=[x]+\frac{13}{[x]} . $where$[x]$ mean the greatest integer $n$ , where $n\leq x.$

As shown in figure , it is known that $M$ is the midpoint of side $BC$ of $\vartriangle ABC$. $\odot O$ passes through points $A, C$ and is tangent to $AM$. The extension of the segment $BA$ intersects $\odot O$ at point $D$. The lines $CD$ and $MA$ intersect at the point $P$. Prove that $PO \perp BC$.

Suppose that $\{a_n\}$ is a sequence such that $a_{n+1}=(1+\frac{k}{n})a_{n}+1$ with $a_{1}=1$.Find all positive integers $k$ such that any $a_n$ be integer.

$3n$ ($n \ge 2, n \in N$) people attend a gathering, in which any two acquaintances have exactly $n$ common acquaintances, and any two unknown people have exactly $2n$ common acquaintances. If three people know each other, it is called a Taoyuan Group. (1) Find the number of all Taoyuan groups; (2) Prove that these $3n$ people can be divided into three groups, with $n$ people in each group, and the three people obtained by randomly selecting one person from each group constitute a Taoyuan group. Note: Acquaintance means that two people know each other, otherwise they are not acquaintances. Two people who know each other are called acquaintances.