CN-SMO 2013

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 $\frac{2a^3-3ab+3a}{b+1}$ .

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.)

Attachments:

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.