Let $n$ be a positive integer with $n \geq 2$, and $0<a_{1}, a_{2},...,a_{n}< 1$. Find the maximum value of the sum $\sum_{i=1}^{n}(a_{i}(1-a_{i+1}))^{\frac{1}{6}}$ where $a_{n+1}=a_{1}$

Find the smallest positive real $k$ satisfying the following condition: for any given four DIFFERENT real numbers $a,b,c,d$, which are not less than $k$, there exists a permutation $(p,q,r,s)$ of $(a,b,c,d)$, such that the equation $(x^{2}+px+q)(x^{2}+rx+s)=0$ has four different real roots.

In $\triangle PBC$, $\angle PBC=60^{o}$, the tangent at point $P$ to the circumcircle$g$ of $\triangle PBC$ intersects with line $CB$ at $A$. Points $D$ and $E$ lie on the line segment $PA$ and $g$ respectively, satisfying $\angle DBE=90^{o}$, $PD=PE$. $BE$ and $PC$ meet at $F$. It is known that lines $AF,BP,CD$ are concurrent. a) Prove that $BF$ bisect $\angle PBC$ b) Find $\tan \angle PCB$

Assuming that the positive integer $a$ is not a perfect square, prove that for any positive integer n, the sum ${S_{n}=\sum_{i=1}^{n}\{a^{\frac{1}{2}}\}^{i}}$ is irrational.

Let $S=\{n|n-1,n,n+1$ can be expressed as the sum of the square of two positive integers.$\}$. Prove that if $n$ in $S$, $n^{2}$ is also in $S$.

$AB$ is a diameter of the circle $O$, the point $C$ lies on the line $AB$ produced. A line passing though $C$ intersects with the circle $O$ at the point $D$ and $E$. $OF$ is a diameter of circumcircle $O_{1}$ of $\triangle BOD$. Join $CF$ and produce, cutting the circle $O_{1}$ at $G$. Prove that points $O,A,E,G$ are concyclic.

Let $k$ be a positive integer not less than 3 and $x$ a real number. Prove that if $\cos (k-1)x$ and $\cos kx$ are rational, then there exists a positive integer $n>k$, such that both $\cos (n-1)x$ and $\cos nx$ are rational.

Given a positive integer $ n\geq 2$, let $ B_{1}$, $ B_{2}$, ..., $ B_{n}$ denote $ n$ subsets of a set $ X$ such that each $ B_{i}$ contains exactly two elements. Find the minimum value of $ \left|X\right|$ such that for any such choice of subsets $ B_{1}$, $ B_{2}$, ..., $ B_{n}$, there exists a subset $ Y$ of $ X$ such that: (1) $ \left|Y\right| = n$; (2) $ \left|Y \cap B_{i}\right|\leq 1$ for every $ i\in\left\{1,2,...,n\right\}$.