Let $ABC$ be acute triangle. The circle with diameter $AB$ intersects $CA,\, CB$ at $M,\, N,$ respectively. Draw $CT\perp AB$ and intersects above circle at $T$, where $C$ and $T$ lie on the same side of $AB$. $S$ is a point on $AN$ such that $BT = BS$. Prove that $BS\perp SC$.

Let $a,\, b,\, c$ be side lengths of a triangle and $a+b+c = 3$. Find the minimum of $$a^{2}+b^{2}+c^{2}+\frac{4abc}{3}$$

Sequence $\{a_{n}\}$ is defined by $a_{1}= 2007,\, a_{n+1}=\frac{a_{n}^{2}}{a_{n}+1}$ for $n \ge 1.$ Prove that $[a_{n}] =2007-n$ for $0 \le n \le 1004,$ where $[x]$ denotes the largest integer no larger than $x.$

For every point on the plane, one of $n$ colors are colored to it such that: $(1)$ Every color is used infinitely many times. $(2)$ There exists one line such that all points on this lines are colored exactly by one of two colors. Find the least value of $n$ such that there exist four concyclic points with pairwise distinct colors.

Let $\alpha$, $\beta$ be acute angles. Find the maximum value of $$\frac{\left(1-\sqrt{\tan\alpha\tan\beta}\right)^{2}}{\cot\alpha+\cot\beta}$$

Let $f$ be a function given by $f(x) = \lg(x+1)-\frac{1}{2}\cdot\log_{3}x$. a) Solve the equation $f(x) = 0$. b) Find the number of the subsets of the set $$\{n | f(n^{2}-214n-1998) \geq 0,\ n \in\mathbb{Z}\}.$$

Let $n$ be a positive integer and $[ \ n ] = a.$ Find the largest integer $n$ such that the following two conditions are satisfied: $(1)$ $n$ is not a perfect square; $(2)$ $a^{3}$ divides $n^{2}$.

The inradius of triangle $ABC$ is $1$ and the side lengths of $ABC$ are all integers. Prove that triangle $ABC$ is right-angled.