summary

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 unity that can be converted into finite decimals is in decimal form;

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

⑶ Mutualization of Band Score and False Score

⑷ Simplification of complex fraction

2. Simple calculation

(1) gather together ideas

⑵ the idea of benchmark number

(3) Cracking and splitting

(4) extracting common factor

5] Quotient invariant property

[6] Change the operation order.

① Comprehensive application of operation law

② the nature of continuous reduction

③ the nature of continued division

④ The nature of transposition in the same level operation.

⑤ The nature of increasing or decreasing brackets

⑥ Variant extraction of common factor

Shaped like:

Step 3 estimate

Finding the integer part of a formula: expansion and contraction method

Step 4 compare sizes

① General score

A. Common denominator

B. universal molecule

② Compared with "intermediary"

③ Using reciprocal property

If so, c> b> a.。 Shaped 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 even = even even × even = even

2. Bit value principle

Shape: = 100a+ 10b+c

3. The divisible characteristics of numbers:

Characteristics of divisible number

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

The sum of the numbers on each digit 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 sum of numbers in odd digits and the sum of numbers in even digits, and the difference between them is a multiple of 1 1

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

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

7. The difference between the last three digits and the first digits of 1 1 3 is a multiple of 7 (or113).

4. Divisible nature

(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 There must be exactly one number in a continuous natural number that can 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 (also called quotient for short). 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 ×...×pk

7. The number of divisors and the theorem of the sum of divisors

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

Number of divisors of n: d (n) = (a1+1) (a2+1) ... (AK+1)

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

8. congruence theorem

① definition of congruence: if two integers a and b are divided by natural number m and have the same remainder, then they are said to be congruence of modulo m, and expressed as a≡b(mod m) by the formula.

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

③ The sum of two numbers divided by m is equal to the sum of the two numbers divided by m respectively.

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

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

9. Properties of Complete Square Numbers

① squared difference: A -B =(A+B)(A-B), among which we should also pay attention to the parity of A+B and A-B.

2 divisors: odd numbers of divisors are complete squares.

The divisor number 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)

11.

12. Common methods of solving problems by number theory:

Enumeration, induction, disproof, construction, pairing and estimation

Third, geometric figures

1.Plane graphics

(1) sum of interior angles of polygons

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

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

(1) triangle with equal base and equal height in the triangle.

(2) triangle with equal base and equal height in parallel lines.

③ Transitivity of the public part

④ Extreme value principle (change and invariability)

(3) the triangle area is proportional to the bottom.

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

(4) Nature of similar triangles (number of copies, 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 dots is more than 3.

(7) Equivalent substitution of implicit conditions.

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

⑻ Thinking method of combined graphics

(1) break the whole into parts

② Make up first and then go.

③ Positive and negative combination

2. Stereo graphics

(1) Formula of surface area and volume of regular three-dimensional figure.

⑵ Surface area of irregular three-dimensional figure

Holistic observation method

(3) Equal volume deformation

① immersion of objects in water: v liters of water =V objects.

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

(4) Three views and development drawings

Shortest Line and Shape of Unfolded Diagram

5] dyeing problem

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

Fourth, typical application problems

1.Tree planting problem

① Open type and closed type.

② Relationship between interval and number of plants

2. Square matrix problem

Outer side length -2= inner side length.

(outer side length-1)×4= outer perimeter.

Outer side length 2- hollow side length 2= real area.

3. Train crossing the bridge

(1) conductor+bridge length = speed x time.

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

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

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

Captain = speed and × meeting time

Captain = speed difference × catch-up time

4. Age problem

Difference invariance principle

5. Chickens and rabbits in the same cage

Problem-solving thought of hypothesis method

6. Cattle eat grass

Original grass quantity = (cattle eating speed-grass growing speed) × time

7. The average problem

8. profit and loss problem

Analysis of difference relation

9. Sum and difference problem

10. Sum times problem

11.

12. Inverse problem

Reduction method, starting with the result

13. substitution problem

List elimination method

Equivalent conditional substitution

V. Travel issues

1.Encounter problem

Distance sum = speed and x meeting time

2. Follow up the problem

Distance difference = speed difference × catch-up time

Step 3 sail with water

Downstream speed = ship speed+water speed

Current speed = ship speed-water speed

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

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

4. Meet many times

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

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

Where the distance of a * * * line = the distance traveled in a single whole journey * * * the number of whole journeys.

5. Circular runway

6. The application of positive and negative proportional relation in the travel problem.

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

The speed is constant, and the distance is proportional to the 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 fractional, engineering and sum-difference problems.

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

Sixth, the counting problem

1.addition principle: classified enumeration.

2. Multiplication principle: permutation and combination

3. The principle of inclusion and exclusion:

① Total quantity =A+B+C-(AB+AC+BC)+ABC

② Commonly used: total quantity =A+B-AB

4. pigeonhole principle:

At most, at least the problem

Step 5 shake hands

It is widely used in graphic counting.

(1) Angle, line segment, triangle,

② Rectangular, trapezoidal and parallelogram

③ Square

Seven, the score problem

1.Volume-rate correspondence

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 in and out of the pool.

6. Proportional distribution

Eight, equation solving

1.Equivalence relation

(1) the associated quantity representation.

Example: A+B = 100 A-B =3

x 100-x 3x x

② skill of solving equations

identical deformation

2. Solving binary linear equations

Substitution method and elimination method

3. Analysis and solution of indefinite equation

Take the coefficient as the trial value angle.

4. Analysis and solution of inequality equation

Nine, find the law

(1) Periodic problem

(1) year, month and day of the week.

② Application of remainder

⑵ sequence problem

① arithmetic progression.

General formula an=a 1+(n- 1)d

Find the number of items: n=

Sum: S=

② geometric progression.

Sum: S=

③ Peibonachi sequence

(3) Strategic issues

(1) grab the newspaper 30

② Put coins.

(4) Maximum problem

① Shortest route

A. line-dividing reading of a character array group

B. the shortest walking number on the grid route

② Optimization problem

A. Integrated approach

B. Pancake problem

Ten, 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. Number array problem

1.Equality and Value 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) odd magic square problem:

Yang Hui-Fa-Rober method

⑵ Even order magic square problem:

Bieven order: symmetric exchange method

Single even order: concentric square array method

XII. Binary system

1.Binary notation

(1) the principle of binary bit value

② Mutual conversion between binary numbers and decimal numbers.

③ binary operation

2. Other hexadecimal (hexadecimal)

Thirteen, a stroke

1.One stroke 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 of1.Equivalence Condition

2. List method

3. Match diagram

Competition problems, involving common sense of sports competitions.

Fifteen, the match stick problem

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

2. Move the matchstick to change the formula and make it hold.

Sixteen, intelligence problems

1.Break through the mindset

2. Some special situation problems

Seventeen, problem solving methods

(Combined with the treatment of miscellaneous questions)

1.substitution method

2. Elimination method

3. Backward push method

4. Hypothesis method

5. reduction to absurdity

6. Extreme value method

7. Set number method

8. Holistic approach

9. Drawing method

10. List method

11.exclusion method

12. Dyeing method

13. Construction method

14. Pairing method

15. Column equation

⑴ equation

⑵ Indefinite equation

⑶ inequality equation