/Article _ D/2007-0 1/887725649973753 . htm
The 17th "Hope Cup" National Mathematics Invitational Tournament, the second test of Senior One.
1. Multiple choice questions (4 points for each question, ***40 points. Only one of the four options in each question below is correct. Please fill in the English letters indicating the correct answer in the brackets after each question.
1.a and b are rational numbers satisfying ab≠0, and there are four propositions:
The inverse of ① 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;
The reciprocal of AB is the product of the reciprocal of A and the reciprocal of B 。
Among them, the real proposition is ()
1。 (b) 2。 (c) 3。 (d) 4。
[answer] c
[Analysis] In ③, the inverse of ab is -ab, the inverse of A is -a, the inverse of B is -a, and the inverse of their product is ab.
[Test site] This question examines the flexible use of the definitions of reciprocal and reciprocal.
2. In the figure below, it is () that is not the flat expansion diagram of the cube.
[answer] c
[Resolution] Restores the graph expanded in the topic, and only the answer B cannot be restored to a cube.
[test site] the characteristics of the cube expansion diagram investigated in this question.
3. In the algebraic expression, if the values of x and y decrease by 25% respectively, the value of the algebraic expression decreases by ().
50%。 75% (C) (D)。
[answer] c
[Analysis] Let the algebraic expression after reduction be m, then there is m= =.
[Test site] This topic examines the operation and flexible application of algebraic expression multiplication.
4. if a
(A)a+b+c+d must be a positive number. (b) D+C-A-B may be negative.
D-c-b-a must be positive. (d) C-D-B-A must be positive.
[answer] c
[Analysis] This question applies the special value exclusion method. For A, if a=-2, b=- 1, c= 1 and d=2, then a+b+c+d=0 is not positive; For b, d+c > 0，-a & gt； -b & gt； 0, so d+c-a-b must be greater than zero; For d, let a=-2, b=- 1, c= 1, d=5, then c-d-b-a=- 1.
The operation of rational numbers.
5. In figure 1, DA=DB=DC, then the value of x is ().
(1) 10. (B)20。 (C)30。 Forty years old.
[Answer] A.
[Analysis] According to the sum of the internal angles of a triangle, x=
[Test site] Investigate the calculation of triangle angle.
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) M is even only if A, B and C are odd or even. (d) The parity of m is uncertain.
[answer] b
[Analysis] Using the special value method, set a specific number and substitute it into the algebraic formula to discharge options A, C and D. ..
The operation of rational numbers.
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).
(A)30。 (B)3 1。 (C)32。 (D)33。
[answer] b
[Analysis] Starting from the least common multiple, it is known from the meaning of the question that there must be 15 and 4 in the three numbers, and then further inference is made according to the known conditions that the greatest common divisors are 4 and 3 respectively.
Maximum common divisor, minimum common multiple and triangle edge.
8. As shown in Figure 2, the rectangular ABCD consists of 3×4 small squares. In this diagram, rectangles that are not squares have ().
(A)40。 (b) 38。 (c) 36。 (d) 34。
[Answer] A.
[Analysis] This problem can be considered from two aspects. Starting from the front, count the number of rectangles with one side, two sides and three sides respectively, and then sum them up. Second, from the opposite side, first find the total number of squares and rectangles, and then find the number of squares, total-number of squares = number of rectangles.
[test site] to investigate the understanding of graphics.
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, [-/kloc-0.
=-2, then in the following four conclusions, the correct one is ()
[answer] D.
[Analysis] Using the special value method, let a=0, then; Let a=- 1.2, then there is
Flexible use of rational numbers.
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 range of m is ().
(English-Chinese dictionary: number axis；; Little by little; The response corresponds to ...; Respectively respectfully; Distance distance; 1ess than is less than; Value, numerical value; Range)
[answer] c
[Analysis] First, the inequality group is listed according to the meaning of the question, from which the range of b is found, and then the relationship between m and b is found according to m=5-2b, that is, the range of m is obtained by solving the inequality.
[Test site] The flexible application of group solution of linear inequality and infinitive in one variable.
Fill in the blanks (4 points for each small question, ***40 points. )
[answer]
[Analysis] Change the original text into = = =
[Test center] A simple algorithm for testing scores in this question.
[Answer] -3
[Analysis] From the known situation, the original formula = = is further deformed.
[test site] This question examines the operation of algebraic expressions.
13. Figure 3 is a street map of the residential area. A, B, C, …, X, Y and Z are 17 intersections where roads cross. When you stand 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, you should at least set _ _ _ _ _.
[Answer] 4
[Resolution] Find points that meet the requirements of the problem, and they at least meet the requirements.
[Test site] This topic examines the recognition and understanding of graphics.
[Answer] -36
[Analysis] According to the meaning of the question, the original formula = =-36.
[Test site] This question examines the flexible application of the cubic difference formula.
=_________.
[Answer] 4026042
[Resolution] The numerator and denominator of the original formula are operated separately. The numerator is 2007 and the denominator is 2006.
[Test site] The idea of simple operation in fractional operation was investigated.
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.
[Answer] 3 1
[Analysis] The key to solving this problem is to display the number of table tennis balls given to each winner separately and find a general rule.
[Detailed explanation] Solution: Assuming there are X table tennis balls, the number of balls that get the first place from the meaning of the question is: the second place:; Third place:
And so on, fifth place:. So there is: the solution is 3 1.
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 highest age and the lowest age of these four people is _ _ _ _.
[Answer] 18
[Analysis] Set the age of four people, and according to the meaning of the question, respectively represent the sum of the average age of three people and the age of another person.
Suppose four people are all of the same age, and then make the difference between four formulas according to the meaning of the question:,,,.
So the difference between the maximum age and the minimum age is 18.
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".
[Answer] 53 or 25
[Analysis] This question diverges and is considered in two situations.
[Detailed explanation] Solution: Suppose there are X people in the class. According to the meaning of the question, there are two situations: first, when counting from right to left, the students who reported 20 did not reach the position of the students who reported 20 for the first time: Second, when counting from right to left, the students who reported 20 exceeded the position of the students who reported 20 for the first time.
The last digit of is _ _ _ _ _ _.
[Answer] 0
[Analysis] The original formula is deformed and makes full use of the special value method.
[Detailed explanation] The original formula =, Ling, the original formula =, because the last digit of the power of 2 is a cycle of 4 numbers, which are 2, 4, 8 and 6 respectively, so the last digit of the original formula is 8, so the last digit of the original formula is 0.
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=d always has positive solutions x, y, z, w, then the minimum value of a is _ _ _ _ _ _ _ _ _ _.
(English-Chinese dictionary: hypothesis; Integer；; Equation equation; Solution (of the equation); Positive positive; Respectively respectfully; Minimum value)
[Answer] 2433
[Test site] This question examines the discussion ideas of indefinite equations.
Third, the solution (this big question is ***3 small questions, ***40 points. ) Requirements: Write out the calculation process.
2 1. (Full score for this small question 10)
(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.
(1) 【 Resolution 】 Set the general formula of odd numbers.
Proof: Set any odd number as = = according to the meaning of the question.
The product of two consecutive integers must be even, so 4k(k+ 1) can be divisible by 8.
So I have to prove it.
(2) Suppose that 2006 can be expressed as the sum of squares of odd numbers of 10, that is
Where,,,,, are all odd numbers.
Divide the left side of the equation by 8, and divide 2006 by 8. 6. contradiction!
Therefore, 2006 cannot be expressed as the sum of squares of odd numbers 10.
22. (The full score of this short question is 15)
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, extend the intersection of BC to d, Connect co, extend the intersection of AB to f, and find the area of quadrilateral BDOF.
set up
Because e is the midpoint of AC and 0 is the midpoint of BE,
therefore
rule
allow
that is
and
allow
That's right, so
23. (The full score of this short question is 15)
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. A student can sit in the back seat of this motorcycle 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.
[Analysis] The key to solving this problem is that the analysis teacher takes a student to a certain position and then returns to pick up another student. And make clear the distance traveled in each time period.
[Detailed explanation] Solution: Suppose the teacher took a student for x meters, and then put the student down to pick up another student. According to the suggestion, the whole process is divided into three time periods: the time when the teacher takes the first student away, the time when the teacher comes back to pick up the second student, and the time when the teacher takes the second student to the museum.
The solution is =24, so the teacher takes one student out of 24 meters and goes back to take another student, which can ensure that after three people leave at the same time, the time to arrive at the museum will not exceed 3 hours.