Who has a prize for the math junior high school competition question bank! ! Four questions! ! Four questions! !

Examination questions of mathematics competition in senior high schools in Hebei Province in 2008

First, multiple-choice questions (this big question ***6 small questions, 6 points for each small question, out of 36 points)

1. If the image of the function passes (-1, 3), then the image of the function must pass () relative to the axisymmetric graph.

A ( 1，-3) B (- 1，3) C (-3，-3) D (-3，3)

2. If 2008 is expressed as the square difference of two integers, there are () different representations.

A 4 B 6 C 8 D 16

3. If the function has a minimum value, the value range of a is ().

A B C D

4. The minimum value of the known rule is ().

A B C 2 D 1

5. If known, the value range of is ().

A B C D

6. The function is a monotonically increasing function on. When, and, the value of is equal to ().

A 1 B 2 C 3 D 4

2. Fill in the blanks (this big question is ***6 small questions, 9 points for each small question, out of 54 points)

7. Let a set be a subset of S and satisfy:, then the number of subsets satisfying the condition is.

8. If the known sequence satisfies, then = _ _ _.

9. It is known that three points on the coordinate plane are points on the coordinate plane, and the trajectory equation of the points is.

10. In the triangular pyramid, the maximum volume of the triangular pyramid is.

1 1. Choose two men from m boys and n girls () as the team leader. Suppose event A means that two men are of the same sex, and event B means that two men are of the opposite sex. If the probability of a and the probability of b are equal, the possible values of (m, n) are.

12. is three points on the plane, vector, let p be a line segment AB, any point on the perpendicular of the vector. If, the value is _ _ _ _ _.

Third, solve the problem (this big question ***5 small questions, the answer to each small question needs a reasoning process, 13 small questions 10, 17 small questions 14, the rest of each small question 12, out of 60 points)

13. are two unequal positive numbers, and satisfy, find all possible integers c, and so on.

14. As shown in the figure, all sides of the oblique triangular prism are, respectively, the side and the bottom.

(1) Find the distance between straight lines on different planes;

(2) Find the degree of dihedral angle formed by side surface and bottom surface.

15. Let the vector be the unit vector in the positive direction of X axis and Y axis in the rectangular coordinate plane. If the vectors are,, and.

(1) Find the trajectory equation of the point satisfying the above conditions;

(2) Suppose, ask whether there is a constant, which makes the constant hold? Prove your conclusion.

16. In series, is a given non-zero integer.

(1) If, ask;

(2) It is proved that infinite terms can choose to form two different constant series.

17. Let the function defined on [0,2] meet the following conditions:

(1) for, there is always, and,;

② For，if，then。

Proof: (1) (); (2),.

Reference Answers and Grading Criteria of Mathematics Competition in Senior High Schools in Hebei Province in 2008

(Time: 8: 30am ~11:30am on May 8th)

First, multiple-choice questions (this big question ***6 small questions, 6 points for each small question, out of 36 points)

1. If the image of the function passes (-1, 3), then the image of the function must pass () relative to the axisymmetric graph.

A ( 1，-3) B (- 1，3) C (-3，-3) D (-3，3)

Answer: B.

2. If 2008 is expressed as the square difference of two integers, there are () different representations.

A 4 B 6 C 8 D 16

Answer: C.

Solution: Suppose .2008 has eight positive factors, namely 1, 2, 4, 8, 25 1, 502, 1004, 2008, and the sum can only be even, then the corresponding equation is

Therefore, * * * has eight different sets of values:; .

3. If the function has a minimum value, the value range of a is ().

A B C D

Answer: C.

Solution: It was a decreasing function at that time. Because there is no maximum and there is no minimum. At that time, having a minimum value is equivalent to having a minimum value greater than 0. Therefore, this is equivalent to.

4. The minimum value of the known rule is ().

A B C 2 D 1

Answer: a.

Solution: remember, then, (if and only if you take the equal sign). So choose a.

5. If known, the value range of is ().

A B C D

Answer: D.

Solution: Assuming that it is easy to get, that is, because, therefore, it is solved.

6. The function is a monotonically increasing function on. When, and, the value of is equal to ().

A 1 B 2 C 3 D 4

Answer: b

Solution: Order, and you will get it.

If, then, and contradiction;

If, then, it is contradictory to "monotonic increase in the world";

If, then, it also contradicts the "monotonous increase in the world"

So choose B.

2. Fill in the blanks (this big question is ***6 small questions, 9 points for each small question, out of 54 points)

7. Let a set be a subset of S and satisfy:, then the number of subsets satisfying the condition is.

Answer: 37 1.

Solution: At that time, there were six choices, so * * * had a choice; At that time, once it was decided, there were two options, so there were two options.

To sum up, there are 37 1 subset * * eligible.

8. If the known sequence satisfies, then = _ _ _.

Answer:.

Solution: from what is known.

So, that is, {} is the first arithmetic progression, and the tolerance is 1, so =n, that is, there is.

9. It is known that three points on the coordinate plane are points on the coordinate plane, and the trajectory equation of the points is.

Answer:.

Solution: As shown in the figure, as a regular triangle, it can be proved because it is also a regular triangle.

Here we go again, so just click * * *.

, so the point p is on the circumscribed circle, because the trajectory equation is

10. In the triangular pyramid, the maximum volume of the triangular pyramid is.

Answer:.

Solution: Suppose, according to the cosine theorem,

Therefore, since the height of the pyramid does not exceed the length of its side, it can be verified that the known conditions are met by actually summing, and the volume of the pyramid can reach the maximum.

Answer: (10,6).

Solution: Because, therefore, it is a complete square number, therefore.

Solve (unqualified).

So ...

12. is three points on the plane, vector, let p be a line segment AB, any point on the perpendicular of the vector. If, the value is _ _ _ _ _.

Answer: 8.

Solution: As shown in the figure, it is the middle vertical line of line segment AB.

13. are two unequal positive numbers, and satisfy, find all possible integers c, and so on.

Solution: Yes, so,

This led to.

Because the same, therefore. ...........................................................................................................................................................................

Because the order is ................... 6 points.

At that time, t was monotonically increasing, so,

Therefore, we can take 1, 2, 3. .............................................................................................................................................................

14. As shown in the figure, all sides of the oblique triangular prism are, respectively, the side and the bottom.

(1) Find the distance between straight lines on different planes;

(2) Find the degree of dihedral angle formed by side surface and bottom surface.

Solution: (1) As shown in the figure, take the midpoint d and connect the lines.

∴.

4 points after ................

A plane.

Therefore, the distance between straight lines on different planes is equal to .................................................................................................................................................................

(2) As shown in the figure,

Eight points.

...................... 12.

15. Let the vector be the unit vector in the positive direction of X axis and Y axis in the rectangular coordinate plane. If the vectors are,, and.

(1) Find the trajectory equation of the point satisfying the above conditions;

(2) Suppose, ask whether there is a constant, which makes the constant hold? Prove your conclusion.

Solution: (1) According to the condition:.

Defined by hyperbola, we get the trajectory equation of point P: ................................................ 4 points.

(2) In the first quadrant, ................................................ scored 6 points.

When PF is not perpendicular to the X axis and P is in the first quadrant, the following proof holds.

By, by.

Substitute the above formula and simplify it to ... 10.

According to symmetry, the same holds true when p is at the fourth pixel limit.

Therefore, there is a constant, which makes the constant hold ............................................................................................................... 12.

16. In series, is a given non-zero integer.

(1) If, ask;

(2) It is proved that infinite terms can choose to form two different constant series.

Solution: (1)∵,,,,,, ...

∴ From the 22nd item, every three adjacent items are given the values 1, 1, 0 periodically, so = 1...4 points.

(2) First, it is proved that there must be zero term after the finite term. Assuming that there is no zero term,

Because, so ............................... 6 points.

At that time, ();

At that time, (),

That is to say, the value of is either at least less than 1 or at least less than 1...................8 points.

Let's order then.

Because it is a definite positive integer, if it goes on like this, there must be a term, which is contradictory, so there must be a zero term in it. ........................................................................................................................................

If the first zero item is, remember, then every three adjacent items are assigned periodically from the first item, that is,

Therefore, infinite items can be selected from the series to form two different constant series ... 12 points.

17. Let the function defined on [0,2] meet the following conditions:

(1) for, there is always, and,;

② For，if，then。

Proof: (1) (); (2),.

Proof: If we know that the function image is symmetrical about a straight line, then we can know from ②: For, if, then.

Settings, and then.

∵

∴ on [0, 1], which is an irreducible function ..................................................... 4.

( 1)∵,

∴