A, construction skills:

Its basic form is to construct a new mathematical form with known conditions as raw materials and conclusions as the direction, so that the problem can be solved simply in this form. Common ones are structure diagrams, equations, identities, functions, counterexamples, drawers and algorithms.

Second, the drawing skills:

Its basic form is RMI principle. Let R represent a set of relational structures (or systems) of the original image, which contains the original image to be determined. Let R represent a mapping, through which the original image structure R is mapped into an image relational structure R*, which naturally contains images of unknown original images. If there is a definite method, it is determined by inversion, that is, inverse mapping. This principle is embodied in logarithmic calculation, substitution, introduction of coordinate system, design of mathematical model, construction of generating function and so on. Establishing corresponding relationships to solve problems also belongs to this skill.

Third, recursive skills:

If there is a definite relationship between the former and the latter, then we can recursively get the results at any time from some (several) initial conditions and solve the problem by recursive method, which is related to mathematical induction (but not predicting the conclusion) and infinite descent method. The key is to find out the recursive relationship between the former proposition and the latter proposition.

Fourth, identification skills:

When the "black box of mathematics" is too complicated, it can be divided into several small black boxes to decipher one by one, that is, the parts with the same nature are classified into one category, forming a very distinctive method in mathematics-distinguishing situations or classifying. Without correct classification, it is impossible to master mathematics.

Sometimes, a problem can be arranged into a series of small goals in stages, so that once the previous situation is proved, it can be used to prove the later situation. This is the so-called climbing procedure. For example, solving Cauchy function equation is to turn integers into natural numbers, then rational numbers into integers, and finally real numbers into rational numbers.

Distinguishing the situation not only distinguishes the difficulty of the problem, but also attaches a known condition to the classification standard itself, so the difficulty of solving each sub-problem is greatly reduced.

Five, dyeing skills:

Dyeing is an intuitive expression of classification, and there are a large number of questions in mathematical competitions, which are characterized by few knowledge points, strong logic and strong skills; At the same time, dyeing, as a means of solving problems, is also widely used in mathematics competitions. Here are some well-known results.

1. (Point) There are two homochromatic points with a distance of 1 or 2 on the dichroism line;

2. (Point) There are three homochromatic points of arithmetic progression on the dichroism line;

3. There is a monochromatic regular triangle (triangle with three vertices of the same color) with a side length of 1 or on the (vertex) dichromatic plane;

4. Let T 1, T2 be two triangles, the side length of T 1 is 1, and the side length of T2 is. If the plane is (point) dichroic, you can always find a monochromatic triangle that is all equal to T 1 or T2.

5. On the tricolor plane, there must be two points with the same color, and the distance is1;

6. (Point) There must be a right triangle with the hypotenuse of 1 on the trichromatic plane, and its three vertices are all of the same color or different colors;

7. There must be a single triangle (three sides of the same color) in the (edge) colored sixth-order complete graph;

8. There are at least two monochromatic triangles in the (edge) colored sixth-order complete graph.

Six, extreme skills:

In some mathematical problems, the status of each element is unbalanced, and an extreme element or the extreme state of an element often has the special property of taking precedence over other elements, which just provides a breakthrough for solving problems. Starting from the extreme elements, we can simply solve the problem, which is commonly known as the "extreme principle".

Seven, symmetry skills:

Symmetry analysis is to combine the symmetrical beauty of mathematics with the conditions or conclusions of the topic, and then rely on knowledge, experience and aesthetic intuition to determine the overall thinking or direction of solving the problem. Its essence is that the revelation of beauty and the pursuit of nothingness become the macroscopic guiding force in the process of solving problems. Yang Zhenning, a famous physicist, once spoke highly of the symmetry method: "When we silently think about the beauty and completeness of the mathematical reasoning it contains and compare its complex and in-depth physical achievements, we cannot but deeply admire the power of the symmetry law."

Eight, collocation skills:

There are various forms of pairing, such as rounding or yoke pairing of numbers, symmetrical pairing or whole pairing of analytical expressions, pairing of subsets and their complements, image pairing between sets and original images, etc. All these reflect the pursuit and strength of mathematics for harmonious beauty. Little Gauss and (1+2+…+99+ 100) started the pairing.

Nine, professional skills:

Specialization embodies the idea of retreating to advance: from the general to the special, from the complex to the simple, from the abstract to the concrete, from the whole to the part, from the strong conclusion to the weak conclusion, from the high dimension to the low dimension, to the simplest situation of maintaining characteristics, to the smallest independent and complete system, first solve the particularity, and then induce, associate and discover the generality. Mr. Hua said that when solving a problem, we should retreat to the place where we can easily see the problem, understand it thoroughly, drill deep, and then go up. Specialization is not only a way to find solutions, but also a way to solve problems directly.

Ten, general skills:

Advancing to the general is to transform low-dimensional or weakly abstract related problems into high-dimensional and strongly abstract problems, and solve the problems by considering the overall nature or essential relationship. Discrete problems can be summarized by continuous means, and finite problems can be summarized by mathematical induction, because special cases often involve some irrelevant details, so they cover up the problems. The key is that the general case expresses the essence of the problem more clearly. Paulia said: "This seems to be contradictory, but when we transition from one question to another, we often see that new ambitious questions are easier to master than the original ones, more questions may be easier to answer than just one question, more complicated theorems may be easier to prove, and more common problems may be easier to solve." Hilbert also said: If we don't succeed in solving a mathematical problem, it is often because we don't realize the more general view that we need to solve more than one link in a series of related problems now.

Eleven, digital skills:

The advantages of digitalization are: transforming practical problems into mathematical problems, and transforming abstract reasoning into concrete calculations.

Twelve, orderly skills:

When there are many parameters and elements (numbers, letters, points, angles, line segments, etc. In a topic, if you rearrange it according to certain rules (such as the size of numbers and the order of points, etc.). ), sorting itself adds a known condition (effective addition) to the topic, thus greatly reducing the difficulty of the problem. Especially when dealing with inequality, this is an effective skill.

Thirteen, the skills of invariants:

In a changing mathematical process, there are often individual invariants or special invariants, which show relatively stable and good properties. It is a good idea to choose these invariants as the breakthrough of solving problems.