First, multiple choice questions

1. (Neijiang City, Sichuan Province, 2009) As shown in the figure, Chen Xiao starts from point O, goes forward for 5 meters and then turns right for 20O.

Go five meters further, then turn right 20 degrees, ..., and so on.

When he first returned to the starting point o, he left ().

A.60 B. 100

C.90 D. 120m。

Answer C.

2. (In 2009, Qiandongnan Prefecture, Guizhou Province) When Mr. Li, a biology teacher of a school, was doing experiments in the biology laboratory, she divided rice seeds into groups for germination experiments. 1 group takes 3 grains, the second group takes 5 grains, and the third group takes 7 grains ... that is, each group takes 2 grains more than the previous group. According to this rule, then please guess that there should be () grains in group N.

A, B, C, D,

Keywords exploration law

Answer a

3. (Jiangsu Province in 2009) The following is a list of figures arranged according to certain rules:

No.1:;

Second place:;

Third place:;

……

Number:.

Then, among the numbers 10, 1 1, 12 and 13, the largest number is ().

A. number10 b. number11C. number12 d. number 13.

Answer a

4.(Xiaogan, 2009) For each nonzero natural number n, the parabola intersects with the X axis at two points an and Bn to indicate the distance between these two points, and the value is

A.B. C. D。

Answer d

5. (Chongqing, 2009) Observe the following figures. The number of triangles in the first figure is ().

A.B. C. D。

Answer D.

6. (Hebei 2009) The famous Pythagorean school in ancient Greece called the numbers 1, 3, 6, 10 … as "trigonometric numbers" and the numbers 1, 4, 9, 16 … as "square numbers". As can be seen from fig. 7,

The square number of can be regarded as the sum of two adjacent triangular numbers. In the following equation, which conforms to this law is ().

a . 13 = 3+ 10 b . 25 = 9+ 16

c . 36 = 15+2 1d . 49 = 18+3 1

Answer c

Second, fill in the blanks

1. (Neijiang City, Sichuan Province, 2009) Cut a piece of paper into 4 pieces, then take some from the obtained paper, cut each piece into 4 pieces, and so on until it is cut. Then _ _ _ _ _ _ in the four numbers of 2007, 2008 and 20 10 may be the number of pieces of paper cut out.

Answer 2008

2.(2009 Xiantao) As shown in the figure, the straight line Y = X+ 1 intersects with the Y axis at point A 1, and a square with an edge of OA 1B 1C 1 is taken as the first square; Then extend the intersection of C 1B 1 and straight line Y = x+ 1 at point A2, and then make a square with C 1A2 as the edge, and record it as the second square; Similarly, the extension line C2B2 intersects with the straight line Y = x+ 1 at point A3, and then a square C2A3B3C3 is made with C2A3 as the edge, which is recorded as the third square; … and so on, the side length of the nth square is _ _ _ _ _ _ _ _ _ _ _.

answer

3. (Luzhou, 2009) As shown in figure 1, it is known that in Rt△ABC, AC=3, BC= 4, the vertex C passing through the right angle is CA 1⊥AB, the vertical foot is A 1, and then A1c6. Let A2 be A2C2⊥BC, the vertical foot C2, …, and so on, a group of line segments CA 1, A 1C 1, …, then CA 1=,

The answer is,

4. (Guilin and Baise in 2009) As shown in the figure, in △ABC, ∠ A =. ∠A=。 ∠ABC and ∠ACD.

The bisector passes through point A 1 and gets ∠ a1; The bisector phase of ∠A 1BC and ∠A 1CD

At point A2, you get ∠ A2; ……; The equation between ∠A2008BC and ∠A2008CD.

The branch line intersects at point A2009 and gets ∠ A2009. Then ∠ A2009 =

answer

5. (Wuhan, 2009) 14. Arrange some small circles with the same radius as shown in the figure: 1 graph has 6 small circles, the second graph has 10 small circles, the third graph has 16 small circles, and the fourth graph has 24 small circles.

Answer 46

6.(2009 Chongqing Qijiang) Observe the following equation:

;

;

;

…………

Then the (positive integer) th equation is _ _ _ _ _.

answer

7. (Chengdu, 2009) The expression = _ _ _ _ _ _ inferred by calculation is known and recorded.

(expressed by algebraic expression with n)

answer

8. (Zibo, 2009) As shown in the figure, every quadrilateral in the grid is a diamond. If the area of grid triangle ABC is s, the area of grid triangle A 1B 1C 1 is 19S, and the area of grid triangle A3B3C3 is .37s..

Answer 37S

9. (Loudi, 2009) Wang Jing arranged the following three patterns in the shape of Chinese characters with matchsticks. According to this rule, the pattern in the shape of the nth Chinese character needs a matchstick.

Answer 6n+3 or 9+6(n- 1)

10 (Lishui City, 2009) As shown in the figure, Figure ① is a regular triangular cardboard with a side length of 1 and a perimeter of P 1. After cutting a regular triangular cardboard with a side length of, figure ② is obtained, and then cutting a smaller regular triangular cardboard along the same bottom (that is, its side length is the side length of the previous regular triangular cardboard).

answer

1 1 (Enshi City, 2009) Observation Table

According to the arrangement of the numbers in the table, the number represented by letters is _ _ _ _ _ _ _ _.

Answer-10

12. (Nanning, Guangxi, 2009) Positive integers are arranged according to the rule in Figure 8. Please write the number in the 20th row and column 2 1.

Answer 420

13. (Mudanjiang City, 2009) has the column number …, so the seventh number is.

answer

14. (Guangzhou, 2009) As shown in Figure 7-①, Figure 7-②, Figure 7-③, Figure 7-④, …, it is a "wide" character line with chess pieces arranged according to certain rules. According to this rule, the number of pieces in the fifth word "Guang" is _ _ _ _ _.

answer

15. (Yiyang City, 2009) Figure 6 is a set of regular patterns. 1 mode consists of four basic modes, the second mode consists of seven basic modes, ..., and (n is a positive integer) mode consists of four basic modes.

-

Answer 3n+ 1

16. (Jining City, 2009) Observing the arrangement of white triangles in each big triangle in the picture, there are three white triangles in the fifth big triangle.

Answer 12 1

17. (Yibin, 2009) As shown in the figure, the diagonal length of the rhombic ABCD is: take the midpoint of each side of the rhombic ABCD as the vertex, make a rectangle A 1b 1d 1, and then take the rectangle A1b/kloc-. In this way, the area of quadrilateral A2009B2009C2009D2009 is represented by the contained algebraic expression.

The answer.

18. (Sunshine in 2009) Square A1b1o, AB2C2C1,A3B3C3C2, … are arranged as shown in the figure. Points A 1, A2, A3, … and C65438.

Then the coordinate of Bn is _ _ _ _ _ _ _.

Answer (,).

19. (Qinzhou, Guangxi, 2009) A set of formulas arranged according to certain rules:-,-,…, (a≠0) Then the nth formula is _▲_(n is a positive integer).

answer

20. (Wuzhou, Guangxi, 2009) Figure (3) is a square made of matchsticks with side lengths of 1, 2 and 3 matchsticks respectively. When the side length is n matchsticks, let the number of matchsticks used in a square be =★. (expressed by algebraic expression of n).

answer

2 1. (Zhaoqing, 2009) 15. Observe the following kinds:,,, …, according to the observation calculation: =. (n is a positive integer)

answer

22. (Huzhou, 2009) As shown in the figure, it is known as the midpoint of the hypotenuse, which is crossed by a link; Work, linking to; If for …, and so on, we can get the points of …, and … in turn, and the areas of … are …, then = _ _ _ _ _ (expressed by the contained algebraic expression).

answer

23. (Xianning City, 2009) In the operation program shown in the figure, if the initial input value is 48, we find that the output result of 1 time is 24, the output result of the second time is 12, and the output result of the 2009th time is _ _ _ _ _ _.

Answer 3

24. (Jingzhou, Hubei, 2009) 13. Put four playing cards with the same pattern face up in two piles. If you can read the top one of a pile at a time (don't put it back) and read it all, the flop will be different.

answer

25. (Guangdong Province, 2009) As shown in the following figure, if the ground is paved with black and white square bricks of the same specification, will there be black brick in the third figure? _ _ _ _ _ _ _ _ _ _ _ block, the first figure needs a black tile _ _ _ _ _ _ _ _ block (represented by inclusion algebra).

Answer 10,

26. (Shanxi Province, 2009) The following pattern is a part of the pane of Shanxi Merchants' Courtyard, in which "○" stands for paper-cut pasted on window paper, and the number of paper-cut pasted in the first picture is.

answer

27.(2009 Daxing 'anling, Heilongjiang) As shown in the figure, in the rhombus with the side length of 1, the diagonal line is connected to make a second rhombus, so that; Connect, and then make a third diamond with the edge, like this; ....., the side length of the first diamond made according to this rule is.

Keywords: the nature and judgment of diamonds

answer

28. (Benxi, 2009) 16. As shown in the figure, it is known that points, and are equilateral triangles in turn, so that one side is on the axis and the other vertex is on the side, and the equilateral triangles made are 1, 2, 3, … respectively, then the side length of the first equilateral triangle is equal to.

answer

29. Observe the table below and answer the questions:

Serial number 1 2 3 …

chart

…

The number of △ in the first picture is five times that of ○.

Answer 20

30. (Mianyang City, 2009) If positive integers are arranged in four columns according to the following table, according to the arrangement rules in the table, the position where the number 2009 should be arranged is the third column of the 670th row.

Column 65438 +0 Column 2 Column 3 Column 4

Line 1 1 2 3

Line 2 6 5 4

Line 3 7 8 9

The fourth line:12110.

……

Answer 670, 3

3 1. (Tieling City, 2009) As shown in the figure, if black pieces with the same size are placed on the edge of a regular polygon, the number of black pieces required for the first figure is.

Answer or or

32. (Qinghai, 2009) Observe the following list of monomials:,,, … According to the law you found, the seventh monomial is; The first monomial is

Answer;

33. (Longyan, 2009) Observe the following set of numbers:,,, ... They are arranged according to certain rules. Then the k-th number of this set of numbers is.

answer

34. (Fushun City, 2009) Observe the following figures (the smallest triangle in each figure is congruent). Please write down the number of the smallest triangles in the first figure.

35. (Meizhou City, 2009) As shown in Figure 5, there are several diamonds of different sizes in each picture. 1 The picture has 1, the second picture has 3, and the third picture has 5, so there is one in the fourth picture and one in the nth picture.

Third, answer questions.

1. (Xiantao, 2009) As shown in the figure, in △ABC, D and E are points on AB and AC respectively, and DE‖BC, as shown in Figure ①. Then rotate △ADE clockwise around point A for a certain angle to get Figure ②, and then extend BD and CE to M and N respectively, so that DM= BD and EN = CE.

(1) If AB = AC, please explore the following quantitative relationship:

① In Figure ②, the quantitative relationship between BD and CE is _ _ _ _ _ _ _ _ _ _ _ _;

② In Figure ③, guess the quantitative relationship between AM and AN, and the quantitative relationship between ∠MAN and ∠BAC to prove your guess;

(2) if ab = k? Ac (k > 1), according to the above operation method, figure ④ is obtained. Please continue to explore the quANtitative relationship between AM and an, and the quantitative relationship between ∠MAN and ∠BAC. Write your guess directly, without proof.

Answer: (1) ① BD = ce; ②AM=AN，∠MAN=∠BAC。

(2)AM= AN，∠MAN=∠BAC。

2. (Taizhou City, 2009) Arrange positive integers 1, 2, 3, … from small to large according to the following rules. If the number in the second column of row 4 is 32, then

① ; ② The numbers in rows and columns are (indicated by).

Column, column, column ... column

Line 1 …

line ...

line ...

… … … … … …

The answer is 10, (2 points in the first box and 3 points in the second box; Answer 3 points, answer 2 points)

3. (Hang, 2009) As shown in the figure, in the isosceles trapezoid ABCD, ∠ C = 60, AD‖BC, AD=DC, E and F are on the extension lines of AD and DC respectively, and DE=CF, AF and BE intersect at point P.

(1) verification: af = be

Please guess the degree of ∠BPF and prove your conclusion.

The answer (1) be = af;

(2) Guess ∠ BPF = 120.

4. (Enshi City, 2009) The rectangle with the ratio of length to width is called the golden rectangle, which is pleasing to the eye and gives us a harmonious and symmetrical aesthetic feeling, as shown in Figure 9. If you draw a square in a golden rectangle, is the rectangle on the left still a golden rectangle? Please prove your conclusion.

Answer: The left rectangle CDFE is a golden rectangle.

5. (Baiyin City, 2009) No.29. 19 test paper is entitled: if, try to compare the size of a and b, without fractions and decimals. Observe the characteristics of numbers A and B in this question and the process of size comparison, and write the general conclusion you found directly.

Answer 29. Solution: Students may write different levels of generalization conclusions and get different points from different levels of generalization.

If m and n are arbitrary positive integers and m > n, then.

If m and n are arbitrary positive real numbers and m > n, then.

If m, n and r are arbitrary positive integers, and m > n；; Or m and n are any positive integers, r is any positive real number, and m > n, then.

If m and n are any positive real numbers, r is any positive integer, and m > n；; Or m, n and r are any positive real numbers, and m > n, then.

6. (Quzhou, 2009) As shown in the figure, AD is ⊙ o in diameter.

(1) As shown in Figure ①, if two chords B 1C 1 perpendicular to AD and B2C2 divide the circumference into four equal parts, then the degree of ∠B 1 is, and the degree of ∠B2 is;

(2) As shown in Figure ②, three chords perpendicular to AD B 1C 1, B2C2, B3C3 divide the circumference into six equal parts, and find ∠B 1, ∠B2,

The degree of B3;

(3) As shown in Figure ③, n chords B 1C 1 are perpendicular to the bisected circumference 2n of AD, B2C2, B3 C3, …, BnCn. Please use an algebraic expression containing n to express the number of ∠Bn (just write the answer directly).

Answer: (1) 22.5, 67.5.

(2) 45 , 75 .

(3). (or)

7. (Anhui, 2009) 19. The decorative part of the guardrail along the school botanical garden is designed into a number of congruent diamond patterns, and the length of the decorative pattern is increased with each diamond pattern, as shown in the figure. It is known that the side length of each diamond pattern is cm, and one of the inner angles is 60.

(1) If d = 26, the decoration needs 23 1 diamond pattern, and find the length l of the decoration;

(2) When d = 20, how many such diamond patterns are needed to keep the length of (1) unchanged?

The answer (1)60 10 cm(2) requires 300 such diamond patterns.