In urgent need of the 2003 junior high school mathematics competition in Taiyuan! ! ! ! ! ! !

2003 Junior Middle School Mathematics Competition in Taiyuan City

1. Multiple choice questions (***5 small questions, 6 points for each small question, out of 30 points. Each of the following questions gives four conclusions in English code, one and only one of which is correct. Please put the code of the correct conclusion in parentheses after the question. Don't fill in, fill in too much or fill in the wrong, get zero)

1. If 4x-3y-6z = 0 and x+2y-7z = 0 (XYZ ≠ 0), the value of is equal to ().

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

2. For sending ordinary letters at this port, the postage is 0.80 yuan when the quality of each letter does not exceed 20g, 1.60 yuan when it exceeds 20g and does not exceed 40g, and so on. For each 20g increase, the postage is 0.80 yuan (the letter quality is within 100g). If the quality of a letter sent is 72.5g, the postage will be paid ().

(1) 2.4 yuan (2) 2.8 yuan (3) 3 yuan (4) 3.2 yuan.

3. As shown in the figure below, ≈A+≈b+≈C+≈D+≈E+≈F+≈G = ()

360 (B) 450 (C) 540 (D) 720

4. The lengths of the four line segments are 9, 5, x, 1 respectively (where x is a positive real number), and they are used to form two right-angled triangles, and AB and CD are two of them (as shown above), so the number of acceptable values of x is ().

2 (B)3 (C)4 (D) 6。

5. Students and teachers of two graduating classes in Grade Three of a school *** 100 people took photos of graduation photo on the steps. Photographers should be arranged in a trapezoidal queue (number of rows ≥3), and the number of people in each row must be a continuous natural number, so that everyone in the back row can stand in the gap between two people in the front row. Then, the arrangement scheme that meets the above requirements is ().

(A) 1 species (B)2 species (C)4 species (D) 0 species.

Fill in the blanks (***5 small questions, 6 points for each small question, out of 30 points)

6. Then you will know.

7. If the real numbers x, y and z satisfy,, then the value of xyz is.

8. Observe the following figure:

① ② ③ ④

According to the laws in Figures ①, ② and ③, the number of triangles in Figure ④ is.

9. As shown in the figure, it can be seen that the telephone pole AB stands upright on the ground, and its shadow just shines.

On the slope surface CD and the ground BC of the soil slope, if CD is 45? ，∠A=60？ ,

CD=4m, BC= m, then the length of telephone pole AB is _ _ _ _ _ m. 。

10. It is known that the image of quadratic function (where a is a positive integer) passes through point A (-1, 4) and point B (2, 1) and has two different intersections with the X axis, so the maximum value of b+c is.

Iii. Answer questions (***4 questions, 0/5 point for each minor question, out of 60 points)

1 1. As shown in the figure, it is known that AB is the diameter of ⊙O, BC is the tangent of ⊙O, OC is parallel to the chord AD, passes through point D, and makes DE⊥AB at point E to connect AC, and intersects with DE at point P. Are EP and PD equal? Prove your conclusion.

Solution:

12. When someone rents a car from City A to City B, the possible cities along the way and the time (unit: hours) required to pass between the two cities are shown in the figure. If the average speed of a car is 80 km/h, the average cost of a car per 1 km is 1.2 yuan. Try to point out this person's shortest route from City A to City B (there must be a reasoning process).

Solution:

13B .. As shown in the figure, in △ABC, ∠ ACB = 90.

(1) When point D is within the hypotenuse AB, it is proved that.

(2) When point D coincides with point A, does the equation in item (1) exist? Please explain the reason.

(3) When point D is on the extension line of BA, does the equation in item (1) exist? Please explain the reason.

14B ... It is known that real numbers A, B and C satisfy: a+b+c=2, abc=4.

(1) Find the maximum and minimum among A, B and C;

(2) Find the minimum value.

Note: 13B and 14B are easier questions than the following 13A and 14A. 13B and 14B and the previous questions 12 make up the test paper. The next two pages are 13A and 14A.

13A ... As shown in the figure, the length of the diameter ⊙O is the largest integer root of the quadratic equation about x (k is an integer). P is a point outside ⊙O, and the tangent PA and secant PBC of ⊙ o pass through P, where A is the tangent point, and point B and point C are the intersection points of straight lines PBC and ⊙ o.

Solution:

14A .. There are some numbers on the circumference. If there are four numbers a, b, c and d connected in turn, inequality >; 0, then the positions of b and c can be interchanged, which is called an operation.

(1) If there are numbers 1, 2, 3, 4, 5 and 6 on the circumference in turn, ask: Can all four numbers A, B, C and D connected in turn on the circumference be ≤0 after a limited number of operations? Please explain the reason.

(2) If there are 2003 positive integers 1, 2, …, 2003 on the circumference from small to large clockwise, may I ask: Are the four numbers A, B, C and D connected in turn on the circumference ≤0 after a limited number of operations? Please explain the reason.

Solution: (1)

(2)

In 2003, "Really? Reference Answers and Grading Criteria of "Xinli Cup" National Junior Middle School Mathematics Competition

First, multiple-choice questions (6 points for each small question, out of 30 points)

1.D

You can get it by substituting it into the solution.

2.D

Because 20× 3

3.C

As shown in the figure, ∠ B+∠ BMN+∠ E+∠ G = 360, ∠ FNM+∠ F+∠ A+∠ C = 360,

And ∠ BMN+∠ FNM = ∠ D+ 180, so

∠A+∠B+∠C+∠D+∠E+∠F+∠G=540。

4.D

Obviously AB is the longest of the four line segments, so AB=9 or AB = X. 。

(1) If AB=9, when CD=x,,;

When CD=5,,;

When CD= 1, …

(2) If AB=x, when CD=9, …,;

When CD=5,,;

When CD= 1, …

Therefore, the number of x values is 6.

5.B

Let there be k people in the last row and n rows in * *, then the number of people in each row from back to front is k, k+ 1, k+2, …, k+(n- 1).

Because k and n are positive integers and n≥3, n

6.。

= .

7. 1.

Because,

Therefore, a solution is obtained.

Therefore,.

So ...

8. 16 1.

According to the laws of ①, ② and ③ in the figure, the number of triangles in Figure ④ is

1+4+3× 4+=1+4+12+36+108 =161(pieces).

9.。

As shown in the figure, extend the ground plane of AD to E, and cross D to make DF⊥CE to F. 。

Because ∠ DCF = 45, ∠ A = 60, CD=4m,

So CF=DF= m, ef = df tan 60 = (m).

Because, so (m).

10.-4.

Because the image of quadratic function passes through point A (-1, 4) and point B (2 1), it is.

solve

Because the quadratic function image has two different intersections with the X axis,

That is, because a is a positive integer,

So ≥2。 And because b+c =-3a+2 ≤-4, and when a=2, B =-3, C =- 1, the meaning of the question is satisfied, so the maximum value of b+c is -4.

Iii. Answer questions (***4 questions, 0/5 point for each minor question, out of 60 points)

1 1. As shown in the figure, it is known that AB is the diameter of ⊙O, BC is the tangent of ⊙O, OC is parallel to the chord AD, passes through point D, and makes DE⊥AB at point E to connect AC, and intersects with DE at point P. Are EP and PD equal? Prove your conclusion.

Solution: DP=PE. The evidence is as follows:

Because AB is the diameter ⊙O and BC is the tangent,

So AB⊥BC.

From Rt△AEP∽Rt△ABC, get ①...(6 points)

AD‖OC，so ∠DAE=∠COB，so Rt△AED∽Rt△OBC。

So ②...( 12 points)

ED=2EP is obtained from ① and ②.

So DP = PE...( 15 points)

12. Someone rented a car from city A to city B, and the possible cities along the way, and

The time (unit: hours) required for traffic between two cities is shown in the figure. If the car is running,

The average car speed is 80 km/h, and the average car cost is 1 km.

For 1.2 yuan. Try to point out the shortest route for this person from City A to City B (there must be reasoning).

Cheng), and find out how much is the lowest cost?

Solution: The route from city A to city B is divided into the following two categories:

(1) Depart from City A to City B and pass through City O, because from City A to City O,

The shortest time required for a city is 26 hours, and the shortest time required for a city to travel from O to B.

22 hours. Therefore, the shortest time required for such routes is 26+22=48 (hours) ... (5 minutes).

(2) Starting from city A to city B, without going through city O, at this time, from city A to city B, it must go through cities C, D, E or cities F, G and H, and it takes at least 49 hours ... (10)

To sum up, the shortest time from city A to city B is 48 hours, and the route is as follows:

A → F → O → E → B...( 12 points)

The minimum cost required is:

80×48× 1.2=4608 (yuan) ... (14 points)

Answer: The shortest route for this person from City A to City B is A→F→O→E→B, and the minimum cost is 4608 yuan …( 15 points).

13B .. As shown in the figure, in △ABC, ∠ ACB = 90.

(1) When point D is within the hypotenuse AB, it is proved that:

(2) When point D coincides with point A, does the equation in item (1) exist? Please explain the reason.

(3) When point D is on the extension line of BA, does the equation in item (1) exist? Please explain the reason.

Solution: (1) is DE⊥BC, and the vertical foot is e, which is obtained by Pythagorean theorem.

So ...

Because of DE‖AC, so.

Therefore ... (10)

(2) When point D coincides with point A, the equation in (1) still holds. At this time, there are AD=0, CD=AC and BD=AB.

So ...

Therefore, the equation in the sub-item (1) holds ... (13 points).

(3) When point D is on the extension line of BA, the equation in (1) is not valid.

For DE⊥BC, the extension line of BC is at point E, then

Besides,

Therefore ... (15 points)

[Description] As long as the answer to sub-question (3) is not valid (no points will be deducted if the reason is not clear).

14B ... It is known that real numbers A, B and C satisfy: a+b+c=2, abc=4.

(1) Find the maximum and minimum among A, B and C;

(2) Find the minimum value.

Solution: (1) Let A be the largest of A, B and C, that is, a≥b and a≥c, from which we can see that a>0,

And b+c=2-a,.

So b and c are two real roots of a quadratic equation, ≥0,

≥0, ≥0. So a ≥ 4...(8 points)

When a=4 and b=c=- 1, the meaning of the question is satisfied.

Therefore, the minimum value of the maximum value in A, B and C is 4...( 10).

(2) Because of abc>0, A, B and C are all greater than 0 or one plus two minus.

1) If A, B and C are all greater than 0, then (1) shows that the largest of A, B and C is not less than 4, which contradicts a+b+c=2.

2) If any one of A, B and C is positive and negative, let A >；; 0, b<0, c< So, 0

,

According to (1), a≥4, so 2a-2≥6. When a=4 and b=c=- 1, the problem setting conditions are met and the inequality equals sign is established. Therefore, the minimum value is 6...( 15 points).

13A ... As shown in the figure, the length of the diameter ⊙O is the largest integer root of the quadratic equation about x (k is an integer). P is a point outside ⊙O, and the tangent PA and secant PBC of ⊙ o pass through P, where A is the tangent point, and point B and point C are the intersection points of straight lines PBC and ⊙ o.

Solution: Let the two roots of the equation be,, ≤ It is derived from the relationship between roots and coefficients.

- ①, - ②

From the title and ①, we know that it is an integer. From ①, ②, eliminate K, we get,

.

As can be seen from the above formula, when k=0, the maximum integer root is 4.

So the diameter of ⊙O is 4, so BC≤4.

Because BC = PC-Pb is a positive integer, BC= 1, 2, 3 or 4...(6 points)

Connect AB and AC, because ∠PAB=∠PCA, PAB∽△PCA,

Therefore, ③...( 10)

(1) When BC= 1, it is derived from ③, so it is contradictory!

(2) When BC=2, it is obtained from ③, so it is contradictory!

(3) When BC=3, it is obtained from ③, so,

Because PB is not a complex number, it is only possible.

, ,

Solve.

At this time.

(4) When BC=4, it is derived from ③, so it is contradictory.

To sum up ... (15 points)

14A .. There are some numbers on the circumference. If there are four numbers a, b, c and d connected in turn, inequality >; 0, then the positions of b and c can be interchanged, which is called an operation.

(1) If there are numbers 1, 2, 3, 4, 5 and 6 on the circumference in turn, ask: Can all four numbers A, B, C and D connected in turn on the circumference be ≤0 after a limited number of operations? Please explain the reason.

(2) If there are 2003 positive integers 1, 2, …, 2003 on the circumference from small to large clockwise, may I ask: Are the four numbers A, B, C and D connected in turn on the circumference ≤0 after a limited number of operations? Please explain the reason.

Solution: (1) The answer is yes. The specific operation is as follows:

..... (5 points)

(2) The answer is yes. Suppose that the sum of the products of two adjacent numbers of this 2003 number is P ... (7 points).

At the beginning, =1× 2+2× 3+3× 4+…+2002× 2003+2003×1. After k(k≥0) operations, the sum of the products of two adjacent numbers of these 2003 numbers is. At this time, if the four numbers A and B are connected in turn on the circumference, it is 0, that is, AB+CD >; Ac+bd, after exchanging the positions of b and c, the sum of the products of two adjacent numbers of these 2003 numbers is, and.

Therefore, every operation, the sum of the products of two adjacent numbers will be reduced by at least 1. Because the product of two adjacent numbers is always greater than 0, after a finite number of operations, there must be ≤ 0...( 15 minutes) for any four numbers A, B, C and D connected in turn.