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According to Van der Waerden, the sum of two natural numbers is defined as &x*1=x& &x*y^+= x*y+x& (for every x and every y) The product of two natural numbers is recursively defined as &x*1=x& &x*y^+= x*y+x& (for every x and every y) x^+ ( y^+ ) here means the successor(consequent) of x( y) in the set of natural numbers.

In this paper, we continues to investigate the properties of symmetric inequalities. We study the necessary condition for the extremum to be attained in a semi-algebraic system. From which, we partly solve the problem of the further generalization of Jensen inequality which the author brought up in the previous article. Namely, by using the sign of the (n+1)^th derivative while xing the value of the sum of power of n variables, we get a dimension-descending method. Further, we prove that the necessary and sucient condition for the symmetric inequality of degree m with n variables to hold on R+n is that it holds when the number of nonzero variables does not exceed max&{1, frac{m}{2}}&.

A topic for study derives from an interesting problem on the internet.Some significative conclusions about the problem of number ring transformation are made in the paper. Firstly, an assumption that a given starting number ring could be transformed to the end ring in finite degree is proved. Secondly, transformation degree is proved to be unconcerned with the order of the transformation. Furthermore, a function of computing transformation degree is worked out. Finally, the same end ring is gained with a given starting number ring transformed in different sequence.

In the early 1980s of twenty century, Professor Zhang Zhongfu, an expert in graph theory, raised a question in his research: how many points of intersection of diagonal lines are there inside a regular N polygon, hoping to find out a formula of count. It has been more than 20 years since the question was raised, which has aroused the interest of quite a lot of experts, scholars and those who love mathematics, but still remains unsolved. When N is an odd number, proposition 1 can be derived from formula of counting based on proposition 2, which can also be verified by the programme the author has made. Proposition 1: when N is an odd number, there is no concurrence of 3 or more than 3 diagonal lines of regular N polygon. Proposition 2: when N is an odd number, the number of intersection points of inner diagonal lines of regular N polygon is: &a_n=c^{4}_{n}=frac{1}{24}(n-1)(n-2)(n-3)$ But when N is an even number, it becomes quite complicated. When N are some special even numbers referred to by Chang Jianming, there are some laws: the research is made from the perspective of prime factorization of figures. The writer primarily studies the issue by making use of geometrical drawing and imitation by computer programme, so that the programme is improved and the imitating solutions of the numbers of intersection points of inner diagonal lines of regular N polygon with specific programmes can be solved. The solution is popularizable. The writer has worked out the numbers of intersection points of inner diagonal lines of regular 2N polygon when N 50, if conditions of computers in schools allow. By comparing the counting between drawing and computer imitation, the writer has also found out some interesting laws about the numbers of intersection points of inner diagonal lines of regular N polygon, and has put forward some hypotheses.