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The 17th "Hope Cup" National Mathematics Invitational Tournament, the second test of Senior One.

April 2006, 10: 30 to 10: 30.

Class _ _ _ _ _ _ _ Student ID _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _

First, multiple-choice questions (4 points for each small question, 40 points for * * *)

1.a and b are rational numbers satisfying ab≠0. There are four propositions: ① the reciprocal is; ② The inverse number of A-B is the difference between the inverse number of A and the inverse number of B; ③ The inverse of AB is the product of the inverse of A and the inverse of B; ④ The reciprocal of AB is the product of the reciprocal of A and the reciprocal of B, and the true proposition is ()

1 (B)2 (C)3 (D)4。

2. In the figure below, it is () that is not the flat expansion diagram of the cube.

(A) (B) (C) (D)

3. In the algebraic expression xy2, if the values of x and y decrease by 25% respectively, the values of the algebraic expression decrease ().

50% (B)75% (C) (D)

4. if a < b < 0 < c < a, the following conclusion is correct ().

(a) A+B+C+D must be a positive number; (b) D+C-A-B may be negative.

D-C-B-A must be a positive number.

5. In figure 1, da = db = dc, then the value of x is ().

10 20 30 40

6. It is known that a, b and c are integers, and m = | A+B |+| B-C |+| A-C |, then ()

(A)m must be an odd number; (B)m must be an even number.

(c) Only when A, B and C are even numbers, the parity of M is uncertain.

7. The lengths a, b and c of three sides of a triangle are integers, [a, b, c] = 60, (a, B) = 4 and (b, C) = 3. (Note: [a, b, C] stands for the least common multiple of a, b, c, (a, b).

30 (B)3 1 (C)32 (D)33

8. As shown in Figure 2, the rectangular ABCD consists of 3×4 small squares. In this diagram, rectangles that are not squares have ().

40 (B)38 (C)36 (D)34。

9. Let [a] be a rational number, and use [a] to represent the largest integer not exceeding a, such as [1.7] = 1, [- 1] =- 1, [0] = 0, [-650].

(a) [a]+[-a] = 0 (b) [a]+[-a] equals 0 or-1.

(c) [a]+[-a] ≠ 0 (d) [a]+[-a] equals 0 or 1.

10. There are two points A and B on the number axis corresponding to numbers 7 and B respectively, and the distance between A and B is less than 10. Let m=5-2b, then the value range of m is ()

(A)- 1 < m < 39(B)-39 < m < 1(C)-29 < m < 1 1(D)- 1 1 < m < 29

(English-Chinese dictionary: number axis；; Little by little; The response corresponds to ...; Respectively respectfully; Distance distance; Less than; Value value; Range)

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

1 1. 1-2+3-4+5-6+7-8+9=_______.

12. If m+n-p = 0, the value of is equal to _ _ _ _.

13. Figure 3 is a street map of the residential area. A, b, c, …X, y, z are 17 intersections where roads cross. Standing at any intersection, you can see that all the streets at this intersection are in a straight line. Now, in order for the sentry to see all the streets in the residential area, at least _ _ _ _ _ _ _ _ _ _ _ _

14. If m-=-3, then m-=-3-= _ _ _ _ _ _.

15.=__________.

16. After the table tennis match, give the winner some table tennis. Give half to the first place. Take the remaining half and add half, and send it to the second place; Take the remaining half and add half to the third place; Take the remaining half and add half to the fourth place; Finally, take the remaining half and add half to the fifth place, and all the table tennis will be served. There are _ _ _ _ _ table tennis.

17.A, b, c and d, the average age of every three people plus the age of the remaining one is 29, 23, 2 1 and 17 years old respectively, so the difference between the maximum age and the minimum age of these four people is _ _ _ _ _ _ _ _.

18. Students in Class Two, Grade One stand in a row. They count from left to right from "1" and then from right to left from "1". It is found that there are exactly 65438 students (including these two students) between the two students who reported "20".

The last digit of 19.2m+2006+2m (m is a positive integer) is _ _ _ _ _ _ _.

20. suppose that a, b, c and d are all integers, and the four equations (a-2b)x= 1, (b-3c)y= 1, (c-4d)z= 1, w+100.

(English-Chinese dictionary: hypothesis; Integer；; Equation equation; Solution (of the equation); Positive positive; Respectively respectfully; Minimum value)

Three. Solution (this big question ***3 small questions, 2 1 question 10, questions 22 and 23 15 ***40) requires that the calculation process be written.

2 1.( 1) proves that if the square of an odd number is divided by 8, the remainder is 1.

Please further prove that 2006 cannot be expressed as the sum of squares of odd numbers 10.

22. As shown in Figure 4, the area of triangle ABC is 1, e is the midpoint of AC, and o is the midpoint of BE. Connect AO and extend the intersection BC to d, Connect co and extend the intersection AB to f, and find the area of quadrilateral BDOF.

23. The teacher took two students to visit the museum 33 kilometers away from the school. The teacher rides a motorcycle at a speed of 25 kilometers per hour. This motorcycle can ride a student in the back seat at a speed of 20 kilometers per hour. The student walks at a speed of 5 kilometers per hour. Please design a plan so that three teachers and students can arrive at the museum within 3 hours after they leave at the same time.

The 17th "Hope Cup" National Mathematics Invitational Tournament

Try to refer to the answer in the second exam of Grade One.

First, multiple choice questions

1, c, prompt: ① ② ④ Correct, ③ Wrong.

2, C, Tip: A vertex in the flat expansion diagram of a cube can connect four squares.

3, c, prompt:

4. C. Prompt: (a) Not sure, A is wrong;

(b), b is wrong;

(c), c right;

(d) Not sure, D is wrong.

5. A. Prompt: As shown in the figure,

,

,

.

6, b, hint: because if there are any, they all appear in even multiples of them.

7, b, hint: then, then, then, then, then.

.

8, a, hint: * * has 60 rectangles and * * * has 20 squares.

9. D. Hint: At that time, at that time,

10, c, prompt:,, that is.

Second, fill in the blanks

1 1、,

12、,

13,4, prompt: as shown in Figure 4: D, N, Y, F.

14, prompt:

15、4026042; Tip:

16、3 1; Tip: If there are 10 table tennis balls, give the first place: 10;

Give second place: one,

To the third place: one, to the fourth place: one, to the fifth place: one.

Then,.

17、 18 ; Tip: Let the ages of A, B, C and D be

①

②

③

④

①+②+③+④ get ⑤, and substitute ⑤ into ①, ②, ③ and ④ respectively.

,。

18,53, prompt:

19,0, prompt: the last digit of is 2, the last digit of is 4, and the last digit of is 5, so it is 0.

20,2433, hint: it's an integer again,

,

3.2 1, (1) Proof: Let an odd number be, then;

(i) When it is an odd number, it can be divisible by 8, so dividing by 8 is1;

(2) If it is an even number, it can be divisible by 8, so it can be divisible by 8 to 1.

So the square of the odd number divided by 8 is equal to 1.

(2) It is proved that the sum of squares of odd numbers 10 is:

Therefore, 2006 cannot be expressed as the sum of squares of odd numbers 10.

22. solution: as shown in figure, is the midpoint, is the midpoint,

,,

Settings,

, that is,,.

,, that is to say,

.

23. Solution: Let a classmate go first. After the teacher took classmate B to ride a motorcycle for an hour, he asked classmate B to walk to the museum. The teacher came back to pick up a classmate and took him to the museum.

At that time,,,

,, yes,

So, let classmate A go first, and the teacher took classmate B by motorcycle for 1.2 hours, that is, after 24 kilometers, let B walk to the museum, and the teacher came back to pick up classmate A. In this way, three people arrived at the museum at the same time after three hours.