If $\min \left \{ \frac{ax^2+b}{\sqrt{x^2+1}} \mid x \in \mathbb{R}\right \} = 3$, then (1) Find the range of $b$; (2) for every given $b$, find $a$.

If positive integers, $a,b,c$ are pair-wise co-prime, and, \[\ a^2|(b^3+c^3), b^2|(a^3+c^3), c^2|(a^3+b^3) \] find $a,b,$ and $c$

Find all positive integer $n$ , such that for all 35-element-subsets of $M=(1,2,3,...,50)$ ,there exists at least two different elements $a,b$ , satisfing : $a-b=n$ or $a+b=n$.

Let $O$ be the circumcenter of triangle $ABC$ , a line passes through $O$ intersects sides $AB,AC$ at points $M,N$ , $E$ is the midpoint of $MC$ , $F$ is the midpoint of $NB$ , prove that : $\angle FOE= \angle BAC$

In triangle $ABC$ , $AA_0,BB_0,CC_0$ are the angle bisectors , $A_0,B_0,C_0$are on sides $BC,CA,AB,$ draw $A_0A_1//BB_0,A_0A_2//CC_0$ ,$A_1$ lies on $AC$ ,$A_2$ lies on $AB$ , $A_1A_2$ intersects $BC$ at $A_3$.$B_3$ ,$C_3$ are constructed similarly.Prove that : $A_3,B_3,C_3$ are collinear.

Let $P_i$ $i=1,2,......n$ be $n$ points on the plane , $M$ is a point on segment $AB$ in the same plane , prove : $\sum_{i=1}^{n} |P_iM| \le \max( \sum_{i=1}^{n} |P_iA| , \sum_{i=1}^{n} |P_iB| )$. (Here $|AB|$ means the length of segment $AB$) .

The sequence $(a_n)_{n>=1}$ satisfies that : $a_1=a_2=1$ $a_n=7a_{n-1}-a_{n-2}$ ($n>=3$) , prove that : for all positive integer n , number $a_n+2+a_{n+1}$ is a perfect square .

12 points are located on a clock with the sme distance , numbers $1,2,3 , ... 12$ are marked on each point in clockwise order . Use 4 kinds of colors (red,yellow,blue,green) to colour the the points , each kind of color has 3 points . N ow , use these 12 points as the vertex of convex quadrilateral to construct $n$ convex quadrilaterals . They satisfies the following conditions: (1). the colours of vertex of every convex quadrilateral are different from each other . (2). for every 3 quadrilaterals among them , there exists a colour such that : the numbers on the 3 points painted into this colour are different from each other . Find the maximum $n$ .