Let $a,b$ be real numbers such that the equation $x^3-ax^2+bx-a=0$ has three positive real roots . Find the minimum of $\frac{2a^3-3ab+3a}{b+1}$.

$\triangle ABC$, $AB>AC$. the incircle $I$ of $\triangle ABC$ meet $BC$ at point $D$, $AD$ meet $I$ again at $E$. $EP$ is a tangent of $I$, and $EP$ meet the extension line of $BC$ at $P$. $CF\parallel PE$, $CF\cap AD=F$. the line $BF$ meet $I$ at $M,N$, point $M$ is on the line segment $BF$, the line segment $PM$ meet $I$ again at $Q$. Show that $\angle ENP=\angle ENQ$

A sequence $\{a_n\}$ , $a_1=1,a_2=2,a_{n+1}=\dfrac{a_n^2+(-1)^n}{a_{n-1}}$. Show that $a_m^2+a_{m+1}^2\in\{a_n\},\forall m\in\Bbb N$

There are $12$ acrobats who are assigned a distinct number ($1, 2, \cdots , 12$) respectively. Half of them stand around forming a circle (called circle A); the rest form another circle (called circle B) by standing on the shoulders of every two adjacent acrobats in circle A respectively. Then circle A and circle B make up a formation. We call a formation a “tower” if the number of any acrobat in circle B is equal to the sum of the numbers of the two acrobats whom he stands on. How many heterogeneous towers are there? (Note: two towers are homogeneous if either they are symmetrical or one may become the other one by rotation. We present an example of $8$ acrobats (see attachment). Numbers inside the circle represent the circle A; numbers outside the circle represent the circle B. All these three formations are “towers”, however they are homogeneous towers.)

$f(x)=\sum\limits_{i=1}^{2013}\left[\dfrac{x}{i!}\right]$. A integer $n$ is called good if $f(x)=n$ has real root. How many good numbers are in $\{1,3,5,\dotsc,2013\}$?

$n>1$ is an integer. The first $n$ primes are $p_1=2,p_2=3,\dotsc, p_n$. Set $A=p_1^{p_1}p_2^{p_2}...p_n^{p_n}$. Find all positive integers $x$, such that $\dfrac Ax$ is even, and $\dfrac Ax$ has exactly $x$ divisors

Given a $3\times 3$ grid, we call the remainder of the grid an “angle” when a $2\times 2$ grid is cut out from the grid. Now we place some angles on a $10\times 10$ grid such that the borders of those angles must lie on the grid lines or its borders, moreover there is no overlap among the angles. Determine the maximal value of $k$, such that no matter how we place $k$ angles on the grid, we can always place another angle on the grid.

$n\geq 3$ is a integer. $\alpha,\beta,\gamma \in (0,1)$. For every $a_k,b_k,c_k\geq0(k=1,2,\dotsc,n)$ with $\sum\limits_{k=1}^n(k+\alpha)a_k\leq \alpha, \sum\limits_{k=1}^n(k+\beta)b_k\leq \beta, \sum\limits_{k=1}^n(k+\gamma)c_k\leq \gamma$, we always have $\sum\limits_{k=1}^n(k+\lambda)a_kb_kc_k\leq \lambda$. Find the minimum of $\lambda$