The 2nd1The 2nd "Hope Cup" National Mathematics Invitational Tournament

Try 2

201April 1 1 9:00 am to 1 1:00 am.

1. Multiple choice questions (4 points for each question, ***40 points. Only one of the following four options is correct. Please write the English letters indicating the correct answers in brackets after each question.

1. If the inverse of a-b is 2b-a, then b= ()

(1)-1. (B)0。 (C) 1。 (D)2。

2. The output value of a factory in March increased by 10% compared with that in February, and the output value in April decreased by 10% compared with that in March, so ()

(a) The output value in April is equal to that in February. (b) The output value in April is higher than that in February.

(c) The output value in April was lower than that in February. (d) The output value in April was lower than that in February.

3. As shown in figure 1, in △ABC, the outer angles of ∠A, ∠B and ∠C are denoted as α, β and γ respectively. If α: β: γ =3:4:5,

Then ∠ A: ∠ B: ∠ C = ()

(A)3:2: 1。 1:2:3。 3:4:5。 5:4:3。

4. If m=, then m is ()

(1) Odd numbers and complete squares. (b) Even numbers and complete squares.

(c) Odd numbers, but not perfect squares. (d) Even numbers, but not perfect squares.

5. There are two prime numbers with two digits, their difference is equal to 6, and their squares are the same.

The group number of this two-digit prime number is ()

1。 (B)2。 (C)3。 (D)4。

6. As shown in Figure 2, the area of square ABCD is l69cm2, and

Thombus BCPQ is 156cm2. Then the area of the shaded part is ()

23 square centimeters. 33 square centimeters. 43 square centimeters. 53 square centimeters.

(English-Chinese dictionary: square; Thombus diamond)

7. To change 40kg of salt water with a concentration of 16% into salt water with a concentration of 20%, it is necessary to evaporate the water ().

Eight kilos. Seven kilos. Six kilos. Five kilos.

8. As shown in Figure 3, the waist length of the isosceles right angle △ABC is 2cm. Rotate △ ABC 90 counterclockwise around point C.

Then the area swept by the line segment AB is ()

9. If a two-digit number is exactly four times the sum of its digits, it is called a "smart number". Then the number of two digits that are not "smart numbers" is ().

(A)82。 (B)84。 86。 (D)88。

10. If you write a positive integer on each face of the cube and then write a number on each vertex, this number is equal to the product of the numbers on three faces passing through this vertex, then when the sum of the numbers on each vertex of the cube is 290, the sum of the numbers on each face is equal to ().

(A)34。 35 years old. (C)36。 (D)37。

Fill in the blanks (4 points for each small question, ***40 points. )

1 1.A and B drive from A to B, A leaves 6 hours later than B, and the speed ratio of A to B is 4: 3. Six hours after A left, the speed increased.

Higher than 1 times. If two cars A and B arrive at B at the same time, it will take several hours for A to walk from A to B..

12. If the rational numbers x, y and year satisfy the equation, then

13. Figure 4 is a hexagonal star, in which

14. To process a workpiece, three processes must be carried out in turn, and the workload ratio is 2: 1: 4. A needs t to complete the first two processes of 1 workpiece and the second workpiece. Given that the treatment efficiency ratio of A and B is 6:7, T _ _ _ is needed.

15. The three views and related data of a regular quadrangular prism are shown in Figure 5, and its top view is rhombic.

So the side area of this right-angled quadrangular prism is

16. There is an algorithm to measure whether the weight is normal: the standard weight of a boy (unit:

Kg) is equal to its height (in cm) minus 1 10. When the actual weight reaches the standard weight,

When it is between 90% and 1 10% (excluding the boundary), it is considered that the weight of boys is normal and known.

Boy A's height 16 1 cm and actual weight are 55 kg. According to the above algorithm,

A's weight is normal weight (fill in "Yes" or "No").

17. If a2-a+ 1 and az +a -3 are numbers opposite to each other,

The reciprocal of a is less than the inverse of a,

Then =

(English-Chinese dictionary: countdown; Opposite)

18. Starting from the line segment with the length of 1, the first operation divides it in two and deletes the middle line segment.

Segment; The second operation divides the remaining line segments into two and deletes the middle line segment.

Every subsequent operation is carried out according to this law. Fig. 6 is a schematic diagram of the first few operations, when

When the sixth operation is completed, the sum of the lengths of all remaining line segments is

19. It is known that m and n are both positive integers. If the maximum value of is a and the minimum value is 6, then a+b=

20. If the sum of n consecutive prime numbers (n is a natural number greater than 1) is a complete square number, then when n is the smallest,

Third, write the calculation process for each question.

2 1. (The full mark of this question is 10)

Let a= and prove that a is a multiple of 37.

22. (The full mark of this question is 15)

(1) There are four straight lines A, B, C and D in the known plane. Lines a, b and c intersect at one point. Lines b, c and d also intersect at one point. How many intersections do these four straight lines have? And explain your reasons.

(2) Let the fifth straight line E be parallel to the straight line D in (1). Explanation: How many line segments are there that end at the intersection of these five lines?

23. (The full mark of this question is 15)

The track AB is16.8m long, and there is a station every 2.4m from the starting station A to the terminal station B. Two robots A and B start from the station A at the same time, arrive in bilibili, then return and move repeatedly between the two stations. The speed of A and B is 0.8 meters per second. A needs to rest 1 second every time he arrives at a station, while B never rests.