Attached are 2 test questions in 2007.

In 2007, Ningbo Donghai Cup Grade Two Mathematics Competition Examination Paper II

1. Multiple choice questions: (6 points for each question, ***30 points. Of the four options given in each question, only one is correct)

1. Make the line segment AB with the known length of l0cm into a semicircle with the diameter of AB upward, and mark the circumference of the semicircle as c1; Divide AB into two halves, make a semicircle upward with half the diameter of its line segment, and write down the sum of the perimeters of the two semicircles as C2; Then divide AB into three parts, make a semicircle upward with one third of its line segment as the diameter, and sum the perimeters of two semicircles as C3; Continue like this, remember that the sum of the perimeters of each semicircle is Ck when K is equal, then with the increase of the equal score K, the sum of the perimeters of each semicircle is Ck ().

A. it's getting bigger and bigger. B. It's getting smaller and smaller. C. unchanged. D. impossible to judge

There are six playing cards on the table, all facing down. You have been told that there are two and only two are old K, but you don't know where old K is. As long as you open two, the following two things will happen:

(1) At least one of the two cards is an old K; (2) Neither card is an old K. Comparing the possibilities of these two situations, we can see that ()

A.( 1) is very likely. B is very likely. C.they are the same. D. they can't be compared.

3. There are two vertical mirrors that can rotate around the vertical axis. I stand at a point in front of these two mirrors, which is located on the bisector of the included angle between the mirrors. If the angle between two mirrors is 5o, I will see that my mirror number is ().

A. 10

As shown in the figure, each vertex of the parallelogram is connected with the midpoint of two opposite sides by a straight line. The graphic area enclosed by these lines is () times that of the original parallelogram.

A. one quarter b one sixth c one eighth d one tenth

5. A writer was troubled by a strange difficulty. The more he writes to the end, the slower he writes. When he starts writing a work, the number he finishes every day is in direct proportion to the number of pages left to write. Take a book for example, it took him 10 days to write the first page, but it took him 50 days to write the last page. The number of pages in this book and the number of days he finished writing are ()

(Whenever the remaining pages to be written and the number of days used are not integers, it is always normalized to an integer greater than it and closest to it. )

7 125 c . 6 120d . 5 1 15

Fill in the blanks (6 points for each question, 30 points for * * *)

6. As shown in the figure, the rectangle is divided into several blocks by some line segments whose lengths are known. If these small pieces can be put together into a square, then the circumference of the square is. (This question is wrong)

(Question 6) (Question 8)

7. The number 3 can be expressed in four ways as the ordered sum of 1 or several positive integers, and 3, 1+2, 2+ 1, l+l+ 1, so for a general positive integer n, the number of such expressions is.

8. As shown in the figure, two circles with radii r and r intersect (R≥r), so the difference between the areas of the non-overlapping parts of the two circles is,

9. Products are sold in both liquid and powder forms. A market survey shows that:

The consumers surveyed don't use powder;

Consumers surveyed do not use liquid drugs;

427 consumers surveyed used both liquid and powder;

The consumers surveyed don't use this product at all.

The number of consumers surveyed is.

10. Zhang Bin sells cloth. His own retail price is 40% higher than the wholesale price. But he found that because the rice scale he used was inaccurate, he only earned 39%. There are many more rulers1"m" used to sell cloth in Zhang Bin than the standard1m. ..

Three. Problem solving (65438+ 05 points for each question, ***60 points)

1 1. Xiao Chen started from Paris by car and kept moving forward on the highway. Soon, he passed a "milestone" (of course, it should actually be called "kilometer monument") with two digits on it. An hour later, he passed another milestone. The two numbers were the same as before, but in the opposite order. After another hour, he passed the third milestone, which is a three-digit number with a zero (sequential or reverse order) between the two numbers. What's the speed in Xiao Chen?

12. If point P is the intersection of two known circles, try to make a straight line L without male chord (male chord refers to the connecting line segment between the intersection of two circles) so that it can be cut into two equal line segments by two circles.

13. Wang Ming, Li Hong and Zhao Liang take the same series of tests. In each test, the scores of the three people are different positive integers x, y and z. The total score of each person is as follows: 20 points for Wang Ming, 9 points for Li Hongluo and 9 points for Zhao Liang. If Li Hong is the first in algebra, who is the second in geometry?

14. There is a balance, a 2g weight and a 7g weight. It is required that a package of salt weighing140g can be divided into 90g and 50g only by using this balance for three times. In addition, in order to reduce the error, the specified weight is an integer kilogram every time the salt is divided. Please design as many schemes as possible and explain the basic reasons.

Reference answers and grading opinions

First, multiple-choice questions (6 points for each question, ***30 points)

1.C

No matter what the fraction is, the sum of the perimeters of each semicircle is always 5π.

2.A

Let's number the six cards with the numbers 1 to 6, assuming that the numbers 5 and 6 are two old K. Now, we list all the different combinations of two of the six cards, and there are 15 such combinations:

1—2, 2—3, 3—4, 4—5, 5—6

1—3, 2—4, 3—5, 4—6

1—4, 2—5, 3—6

1—5, 2—6

1—6

Note that nine of the 15 decks contain the old K (numbers 5 and 6). Since the possibility of each deck is the same, this means that in the long run, you will turn an old K at least 9 times for every 15 attempts. In other words, the probability of digging out at least one old k is 9/ 15, and this score can be reduced to 3/5. This is certainly better than 1/2, so the answer to this question is: you are more likely to dig up at least one old K than an old K.

3.C

Let the included angle of mirror surface be α. Object A has initial mirrors A 1 and A 1' in each mirror. They will have mirror images in the opposite mirror, the mirror image of A 1' is A2, and the mirror image of A 1 is A2' ……………………………………………………………………………………………………………………………………………………………………………………………………………………. The other mirror symmetrically reflects the mirror images A 1', A2', A3', …, An'. These mirror images lie in planes with angles α, 2α, 3α, …, nα. Where nα is the maximum included angle less than or equal to. If nα is exactly equal to 180, the two images of An And an' will overlap, so the total * * * will produce 2n- 1 images. If α is between and, then you will always get 2n images. Because, so always * * will get 6 images.

4.B

As shown in the figure, connect the midpoints of the opposite sides of the parallelogram and divide the parallelogram into four parallelograms. Pay attention to the parallelogram AEPH in the upper left corner, and the quadrilateral PQRS is the part of the desired figure in AEPH.

Note that R is the intersection of two midlines of △ADB, so A, R and P are three lines, and AP=3RP.

So there are S△APS=3S△RPS, S△AQP=3S△RQP, so SPQRS= SPQAS= SAEPH.

Similar reasoning can be used for the other three sub-parallelograms, and the final required conclusion is one sixth.

5.D

Because he writes 1/50 pages every day when there is only one page left to finish, generally speaking, his daily writing volume is always 1/50 pages left. He started writing at the rate of110 pages a day, which means that this work has five pages. It took him 10 days to write the first page; He finished the second page at the speed of 4/50 pages a day, so it took 12 days; Page 3 was written at the rate of 3/50 pages per day, and it took 16 days; Page 4 was written at the rate of 2/50 pages per day, which took 25 days; It took 50 days to write page five. So it takes 1 14 days in total, and the rounding is 1 15 days.

Fill in the blanks (6 points for each question, 30 points for * * *)

6.48.

It is easy to draw that the size of this rectangle is 9× 16, so the area of the square is 144, the side length is 12 and the circumference is 48.

7.2n- 1 An ordered sum A 1+A2+...+AK = n, ai≥ 1 can be expressed by a line of 1 separated by a slash. Namely111…1…111…/kloc-0 In order to get all these expressions (for all 1≤k≤n), n L's can be arranged in a row, and in every n-1 spaces generated between two adjacent1,either a slash is put or not put, so that 2N- 1 different expressions can be generated.

8.π(R2-r2)

If the overlapping area of two circles is S0, the non-overlapping areas of two circles are S 1=πr2-S0 and S2=πR2-S0, respectively. Then the difference between the areas of non-overlapping parts is S=S2-S 1=π(R2-r2). All respondents can be divided into four parts:

A: This product is not used at all; B: only use liquid medicine;

C: both liquid and powder are used; D: just use powder.

8.735

Assuming that the number of consumers surveyed is X, then

Judging from the meaning of the question, the equations can be listed.

The solution is x=735.

10.7.2mm

Suppose Zhang Bin buys cloth at the price of per meter 1 yuan, and let L be the actual length of 1 "meter" on his ruler.

Zhang Bin thought he was selling L meters of cloth, but he was actually selling L meters. The purchase price of this L rice cloth is Shangyuan, and the selling price is 1.40 yuan, so the profit he earns is (1.40-L), so the following equation can be listed:

39/ 100=( 1.40-L)/L

So l =1140/139 =1.0072 is solved. So in fact, his L "meter" is 7.2 mm longer.

Three. Problem solving (65438+ 05 points for each question, ***60 points)

1 1. Solution: The kilometers on the first milestone can be written as10a+b; The number of kilometers on the second milestone can be written as10b+a. As for the number of kilometers on the third milestone, it must be one of the following two: 100A+B or100b+a. Because Xiao Chen is driving at a constant speed, the distance between the first and second milestones is equal to the distance between the second and third milestones. Therefore, the maximum hundred digits on the third milestone can only be 1(4 points). This is because 9L+(91-19) =163. Moreover, because A is the ten digits on the first milestone, it must be less than the ten digits on the second milestone, so it can only be 100A= 100. Therefore, the equation can be listed: (10b+1)-(10+b) = (100+b)-(10b+1) and B=6 to solve. (4 points) So the figures on these three milestones are: 16, 6 1, 106. (Each score is 1, and ***3) The speed in Xiao Chen is 45 kilometers per hour. (score 4 points)

12. As shown in the figure, observing the required results, notice that point Ql can be regarded as point Q2 rotated around p 180, so point Q 1 is not only on the circle O 1, but also on the graph obtained by rotating l80 around the point O2. The graph has the same size as the circle O2 and is tangent to the point P. ..

13. Assume that x>y & gtZ≥ 1, and n is the number of test items. According to the meaning of the question (series test), N> 1, while (x+y+z)N=20+ 10+9=39. Because x+y+z≥3+2+ 1=6 (score 2 points), we know that N≤6 (score 2 points), and because n can be divisible by 3 9, so N=3 (score 2 points), x+y+z =13 (.

(x，y，z)=( 1 0，2， 1):(9，3， 1)，(8，4， 1)，(8，3，2)，(7，5， 1)，(7，4，2)，(6，5，2)，(6，4，3)

Based on Wang Ming's total score of 20, only (8,4, 1) is possible. (Score 2) Delete other situations. So Li Hong's algebra test score is 8,4, 1, and the maximum value is 8. In this way, the problem is transformed into the filling of the following table, so that each row is 8,4, 1, and the sum of the three columns is 20,10,9 respectively.

Wang Mingli Liang

Algebra 8

geometry

Other subjects

20 10 9

Easy to find the only solution to the problem (3 points)

Wang Mingli Liang

Algebra 4 8 1

Geometry 8 1 4

Other subjects 8 1 4

20 10 9

Therefore, Zhao Liang ranked second in the geometry exam. (2 points)

14. There are many answers to this question. Considering the existing weight and their different placement, the specified weight can be divided into two parts, and their weight difference (grams) is limited to: 0, 2, 5, 7 and 9. Therefore, the following mathematical model can be set:

(Scoring: 2 points for the first type and 32 rounds for the second type)

Therefore, the following five solutions can be obtained.

(Rating: 2 points for each scheme, *** 10)

However, if the weighed salt is considered as a new weight, other solutions can be obtained (omitted