abstract

I. Calculation

1. Elementary arithmetic complex fraction

(1) Operation sequence

⑵ mixed operation skills of fractions and decimals

Generally speaking:

① In addition and subtraction operations, the number of units that can be converted into finite decimals is in decimal form;

② In multiplication and division, it is unified in fractional form.

⑶ Interoperability of Band Score and False Score.

⑷ Simplification of complex fractions

2. Simple calculation

(1) brainstorm

(2) the concept of benchmark number

(3) Cracking and splitting

(4) extracting common factors

5] Quotient invariant property

[6] Change the operation sequence.

(1) the comprehensive application of algorithm.

② the nature of continuous reduction

③ the essence of persistent division

④ The nature of transposition in the operation at the same level.

⑤ The nature of increasing or decreasing brackets

⑥ Variation extraction of common factors

Shape image:

estimate

Finding the integer part of the formula: expansion method

Comparative size

① Comprehensive score

A. Common denominator

B. Universal molecules

② Compared with "intermediary"

③ Take advantage of reciprocity.

If so, c>b> answers. The shape is like:, then.

5. Define a new operation

6. Sum of special series

Use the relevant formula:

①

②

③

④

⑤

⑥

⑦ 1+2+3+4…(n- 1)+n+(n- 1)+…4+3+2+ 1 = n

Second, number theory.

1. parity problem

Odd = even odd × odd = odd

Odd even = odd odd × even = even

Even number = even number x even number = even number

2. Bit value principle

Shape: =100a+10b+c.

3. The divisible characteristics of numbers:

Characteristics of divisible numbers

2 ends with 0, 2, 4, 6 and 8.

The sum of each number is a multiple of 3.

5 ends with 0 or 5.

The sum of the digits of 9 is a multiple of 9.

1 1 The difference between the sum of odd numbers and the sum of even numbers is a multiple of 1 1.

The last two digits of 4 and 25 are multiples of 4 (or 25)

8 and the last three digits of 125 are multiples of 8 (or 125).

7. The difference between the last three digits and the first three digits of 1 1 and 13 is a multiple of 7 (or113).

4. Separability

(1) If c|a and c|b, then c|(a b).

② If bc|a, then b|a, C | A. ..

③ If b|a, c|a, and (b, c)= 1, then BC | a.

(4) if c|b, b|a, then c | a.

⑤ In a continuous natural number, exactly one number must be divisible by A. ..

5. Division with remainder

Generally speaking, if A is an integer and B is an integer (b≠0), then there must be two other integers Q and R, 0 ≤ R < B, so that A = B× Q+R.

When r=0, we say that A is divisible by B.

When r≠0, we say that A is not divisible by B, R is the remainder of A divided by B, and Q is the incomplete quotient of A divided by B. Division with remainder can also be expressed as a ÷ b = q...r, 0 ≤ r < b a = b× q+r.

6. Unique decomposition theorem

Any natural number n greater than 1 can be written as the continued product of prime numbers, i.e.

N= p 1 × p2 ×...× primary key.

7. Factors and Theorem of Sum of Factors

Let the prime factorization formula of natural number n be n= p 1 × p2 ×...×pk, then:

The divisor of n: d (n) = (a1+1) (a2+1) ...

Sum of all divisors of n: (1+p1+… p1) (1+p2+… p2) ……… (1+PK+… PK).

8. congruence theorem

① Definition of congruence: If two integers A and B are divided by the natural number M, and the remainder is the same, they are said to be congruences of module M, and expressed as a≡b(mod m) by formula.

② If two numbers A and B are evenly divided by the same number C to get the same remainder, then the difference between A and B will be evenly divided by C..

③ The sum of two numbers divided by m equals the sum of two numbers divided by m respectively.

(4) The remainder of the difference between two numbers divided by m is equal to the difference between two numbers divided by m respectively.

⑤ The product of two numbers divided by the remainder of m is equal to the product of these two numbers divided by the remainder of m respectively.

9. Properties of Complete Square Numbers

① Square difference: A -B =(A+B)(A -B), and the parity of A+B and A-B should also be noted.

2 divisor: the divisor of odd numbers is a complete square.

Divider 3 is the square of the prime number.

(3) prime factor decomposition: decompose a number so that its product is a square number.

④ Sum of squares.

10. Sun Tzu's theorem (China's remainder theorem)

1 1. Reversing department

12. Common methods of solving problems by number theory:

Enumeration, induction, disproof, construction, pairing and estimation

Third, geometry.

1. Plane graphics

The sum of interior angles of (1) polygon

The sum of internal angles of n polygons = (n-2) × 180.

(2) Equal area deformation (displacement, cutting and repair)

(1) Triangle with equal base and equal height.

(2) Triangle with equal base and equal height on parallel lines.

③ Transitivity of public parts.

④ Extreme value principle (change and invariability)

(3) The triangle area is proportional to the base.

s 1∶S2 = a∶b； S 1∶S2=S4∶S3 or S 1×S3=S2×S4.

(4) Nature of similar triangles (number of shares, proportion)

① ; S 1∶S2=a2∶A2

②s 1∶S3∶S2∶S4 = a2∶B2∶ab∶ab； S=(a+b)2

5] Dovetail theorem

S△ABG:S△AGC = S△BGE:S△GEC = BE:EC；

S△BGA:S△BGC = S△AGF:S△GFC = AF:FC；

S△AGC:S△BCG = S△ADG:S△DGB = AD:DB；

[6] Principle of difference invariance

Knowing that 5-2=3, the number of points is more than 3.

(7) Equivalent substitution of implicit conditions.

For example, the relationship between the long side and the short side in a chord diagram.

Being thinking method of combined graphics

(1) is broken into parts.

② Make up before you go.

③ Positive and negative combination

2. Stereo graphics

(1) Formulas of surface area and volume of regular solid figure.

⑵ Surface area of irregular three-dimensional graphics

Holistic observation method

(3) Equal volume deformation

① The object is immersed in water: V liter of water =V object.

② Measure the volume of beer bottle: V=V air +V water.

(4) Three views and expanded drawings

The shortest line and shape of expanded graph

5] dyeing problem

The relationship between the number of blocks dyed on several faces and the number of "cores", side length, vertices and faces.

Fourth, typical application problems

1. Planting trees

① Open and closed.

② Relationship between spacing and number of plants

2. Square matrix problem

Outside length -2= inside length.

(outer length-1)×4= outer circumference.

Outside length 2- hollow side length 2= actual area.

3. The train crosses the bridge

(1) traverse+bridge length = speed × time.

② Captain A+ Captain B = speed and × meeting time.

(3) conductor A+ conductor B = speed difference × catch-up time.

Encounters and problems between a train and people, cyclists or drivers on another train.

Captain = speed and × meeting time

Captain = speed difference × catch-up time

4. Age problem

Principle of difference invariance

The chicken and the rabbit are in the same cage.

Problem-solving thought of hypothesis method

6. Cattle eat grass

Raw grass quantity = (grazing speed of cattle-grass growth speed) × time

7. General problems

8. Profit and loss issues

Analysis of difference relation

9. Sum and difference problem

10. Sum times problem

1 1. Differential time problem

12. Inverse problem

Reduction method, starting from the result

13. Replace

List exclusion method

Equivalent conditional substitution

Verb (abbreviation of verb) travel problem

1. Encountered a problem

Sum of distances = speed and x meeting time.

Step 2 track down the problem

Distance difference = speed difference × catch-up time

Sail by water

Downstream speed = ship speed+current speed

Current speed = ship speed-water speed

Ship speed = (downstream speed+upstream speed) ÷2

Water velocity = (downstream velocity-upstream velocity) ÷2

Step 4 meet many times

Linear distance: the total distance between line A and line B * * * = number of encounters ×2- 1.

Circular distance: the total distance between line A and line B * * * = the number of encounters.

Where the distance of * * * line = the distance traveled in a complete trip * * * the number of complete trips.

5. Circular runway

6. The application of positive and negative proportional relationship in travel problems.

The distance is fixed, and the speed is inversely proportional to the time.

The speed is constant and the distance is proportional to time.

Time is constant, and distance is proportional to speed.

7. Catch-up problem on the clock face.

① The hour hand and minute hand are in a straight line;

② The hour hand and the minute hand are at right angles.

8. Combine some types of scores, engineering sum and difference problems.

9. Travel problems often use the thinking methods of "going back in time" and "assuming".

Sixth, the counting problem.

1. addition principle: classified enumeration.

2. Multiplication principle: permutation and combination

3. The principle of inclusion and exclusion:

① Total amount =A+B+C-(AB+AC+BC)+ABC

② Commonly used: Total amount =A+B-AB

4. Dove cage principle:

At most, it is a problem.

shake?hands;?handshake?(n.)

It is widely used in graphic counting.

(1) Angle, line segment, triangle,

② Rectangular, trapezoidal and parallelogram

③ Square

Seven, the score problem

1. Quantity ratio corresponds.

2. Take the invariant as "1"

3. Profit problem

4. Concentration problem

Inverted triangle principle

Example:

5. Engineering problems

① Cooperation problem

(2) The problem of water entering and leaving the pool.

6. Proportional distribution

Eight, equation solving

1. equivalence relation

(1) associated quantity representation.

For example: A+B = 100 A-B =3.

x 100-x 3x

② Skills of solving equations

Same deformation

2. Solve the binary linear equation

Substitution and elimination methods

3. Analysis and solution of indefinite equation

Take the coefficient as the angle of trial value.

4. Analysis and solution of inequality equation

Nine, find the law

(1) Periodic Problem

(1) Year, month and day.

② Application of remainder

(2) Sequence problem

(1) arithmetic progression.

The general formula an = a1+(n-1) d.

Find the number of items: n=

Sum: S=

② Geometric series.

Sum: S=

③ Peibonahi sequence

(3) Strategic issues

(1) Grab the newspaper 30

2 put coins.

(4) the maximum problem

① Shortest route

A. line-by-line reading of character array groups

B. The shortest number of walks on the grid route

② Optimization problem

A. integrated approach

B. Pancake problem

X. Formula puzzle

1. filled type

2. Substitution type

3. Fill in the operation symbol

4. Horizontal to vertical

5. Combining the knowledge points of number theory

XI。 Digital array problem

1. Equation and Sum Problem

2. Series grouping

(1) Know the number of rows and columns and find a certain number.

⑵ Know a certain number and find the number of rows and columns.

3. Magic Square

(1) Magic Square Problem:

Yang Huifa-Roberts method

⑵ Even order magic square problem:

Even order: symmetric exchange method

Single even order: concentric square array method

Twelve. binary system

1. Binary notation

(1) binary bit value principle

② Conversion between binary numbers and decimal numbers.

③ Binary operation

2. Other Hexadecimal (Hexadecimal)

XIII. Stroke

1. One-hit theorem:

(1) There can only be 0 or 2 singularities in a stroke graph;

(2) Two singularities must enter from one singularity and exit from the other;

2. Hamilton cycle and Hamilton chain

3. Multi-stroke theorem

Number of strokes =

Fourteen, logical reasoning

Conversion of 1. Equivalence Condition

2. List method

Step 3 match the chart

The competition problem involves the common sense of sports competition.

Fifteen, the match stick problem

1. Move the matchstick to change the number of numbers.

2. Move the matchstick to change the formula to keep it unchanged.

Sixteen, intelligence problems

1. Break through the mindset

2. Some special cases.

Seventeen, problem solving methods

(Combined with miscellaneous questions)

1. substitution method

2. Exclusion method

3. Backward push method

4. Hypothetical method

Step 5 reduce to absurdity

6. Extreme value method

7. Set number method

8. Overall analysis

9. Drawing method

10. List method

1 1. exclusion method

12. Dyeing method

13. Construction method

14. Contrast method

15. Column equation

⑴ equation

⑵ Indefinite equation

⑶ Inequality equation