(Jiangxi Division on April 24th, 2004 at 8: 30 am-11:00 am supplementary question)

1. Multiple-choice questions (the full score of this question is 42 points, and each small question is 7 points)

1. The hypotenuse length of a right-angled triangle is an integer, and the two right-angled sides are the two roots of the equation 9X2-3 (k+ 1) x+k = 0, so the value of k2 is …………………………………………………………………………………………………………………………………………………………….

2 (B)4 (C)8 (D)9

2. The value of (8+3) 9+is ................................ ().

(a) odd (b) even (c) rational numbers, not integer (d) irrational numbers.

3. Three cubes with side lengths of 2, 5 and 7 are bonded together. Among these cubes bonded together in various ways, the cube with the smallest surface area is ...................... ().

(A)4 10(B)4 16(C)394(D)402

x+yz= 1

4. If three real numbers x, y, z satisfy: y+zz= 1, then the solution group (x, y, z) suitable for the condition has ().

z+xy= 1

(A)3 groups (B) 5 groups (C)7 groups (D)9 groups

5.8a≥ 1, the value of is ()

(A) 1 (B) 2 (C)8a (D) cannot be determined.

6. The integer solution of the equation is ()

(A) 1 group (B)3 groups (C)6 groups (d) infinite groups.

2. Fill in the blanks (the full score of this question is 28 points, and each small question is 7 points)

1. The minimum value of the function y = x2-2 (2k- 1) x+3k2-2k+6 is m. Then when m reaches the maximum value, x =

2. For 1, 2, 3,. . 9 is the product of every two different numbers, and the sum of all these products is

3. As shown in the figure, AB, CD is the diameter of circle O, AB⊥CD, P is a point on the extension line of CD, PE tangent circle O is E, BE and CD intersect at F, AB=6cm, PE=4cm, then the length of EF =

4. Cover the 3x4 grid table completely with 6 rectangular sheets of 1x2, so there are different covering methods.

Three. Comprehensive problem

1。 There are two groups of numbers: group a 1, 2,. . , 100 B group 12, 22, 32,. . 1002 If for X in group A, there is a number Y in group B, so that X+Y is also a number in group B, then X is called a correlation number, and the number of correlation numbers in A is found.

2. quadratic function y = ax2+bx+c (a >; 0) has only one intersection with X axis and Y axis, namely A and B..

AB=3, b+2ac=0, and the image of the linear function y=x+m passes through point A and intersects with the image of the quadratic function at another point D, so as to find the area of △DAB.

3. In the equilateral triangle ABC, D is the point on the side of BC, BD=2CD, and P is the point on AD.

∠CPD=∠ABC, verification: BP⊥AD.

Answer: a CBDBAB

Two, 1. 1 2。 870 3。 4。 1 1

Three, 1. 73 2。 9 3。 (omitted)

2005 national junior high school mathematics league preliminary examination paper

2: 30-4: 30 pm on March 25th or 9: 00- 1 1: 30 am on March 26th.

School _ _ _ _ _ _ Candidate's name _ _ _ _ _ _ _ _

Title number 12345

Take the lead

commentator

chessman

1. Multiple choice questions: (7 points for each small question, * * * 42 points)

1. If both a and b are real numbers, the correct one of the following propositions is ().

(A)A > b a2 > B2； (B)a≠B a2≠B2； (C)| a | > b a2 > B2； (D)a>|b| a2>b2

2. Given that A+B+C = 3 and A2+B2+C2 = 3, the value of a2005+b2005+c2005 is ().

0 (B) 3 (C) 22005 (D)3？ 22005

There is a football made of several pieces of black and white cowhide. Black leather is a regular pentagon, and white leather is a regular hexagon (as shown in the figure). If there are 12 pieces of sewn football black leather, there are () pieces of white leather.

(A) 16 (B) 18 (C)

4. In RT △ ABC, the hypotenuse AB=5, and the lengths BC and AC of right angles are two roots of the unary quadratic equation X2-(2m-1) x+4 (m-1) = 0, then the value of m is ().

(a) 4 (b)- 1 (c) 4 or-1 (d)-4 or 1.

5. In the rectangular coordinate system, the abscissa is an integer point called the whole point, and let k be an integer. When the intersection of the straight line y = x-3 and y=kx+k is an integer, the value of k can be ().

2 (B)4 (C)6 (D)8

6. As shown in the figure, if the straight line x= 1 is the symmetry axis of the image of the quadratic function y=ax2+bx+c, there is ().

(A)A+B+C = 0(B)B > A+C(C)C > 2b(D)ABC < 0

Fill in the blanks: (7 points for each small question, * * * 28 points)

1. It is known that x is a non-zero real number and = a, then = _ _ _ _ _ _ _ _ _ _.

2. It is known that A is a real number, and the quadratic equation x2+a2x+a = 0 about X has a real root, so the maximum value that can be obtained by the root X of this equation is _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _.

3.p is a point on the extension line of the diameter AB of ⊙o, PC is tangent to ⊙o at point C, and the bisector of ∠APC intersects with AC at point Q, then ∠ PQC = _ _ _ _ _ _ _ _

4. For a natural number n, if natural numbers A and B can be found and n=a+b+ab, then N is called a "good number", for example, 3 =1+1,then 3 is a "good number".

3. Let A and B be points on the parabola Y = 2x2+4x-2, and the origin is located at the midpoint of the line segment AB. Try to find the coordinates of a and B.

As shown in the figure, AB is the diameter of ⊙o, and AB = D. Let A be the tangent of ⊙o, and take a point C on it, so that AC=AB, the point D is called ⊙o, and the extension line connecting OC intersects AC at point E to find the length of AE.

5. (The full mark of this question is 25) Let X = A+B-C, Y = A+C-B and Z = B+C-A, where A, B and C are prime numbers to be found. If x2=y, =2, try to find all possible values of the product abc.

Reference solution and grading standard

First, multiple-choice questions (7 points for each small question, * * * 42 points)

1、D 2、B 3、C 4、A 5、C 6、C

Fill in the blanks (7 points for each small question, 28 points for * * *)

1、a2-2 2、3、45 4、 12

Third, the solution: ∵ The origin is the midpoint of the line segment AB, and the points A and B are symmetrical about the origin.

If the coordinates of point A are (a, b), the coordinates of point B are (-a, -b)...5 points.

A and b are points on the parabola, and their coordinates are substituted into the parabolic analytical formula respectively, so:

........................ 10.

Solution: a = 1, b = 4 or a =- 1, b =-4. .............................................................................................................................

Therefore, a is (1, 4), b is (-1, 4) or a is (-1, 4), and b is (1, 4)...20 points.

4. Solution: If AD is connected as shown in the figure, then ∠ 1=∠2=∠3=∠4.

∴δcde∽δcad

∴ (1) ................................... 5 points.

∵δade?bda

∴ ② 10.

From ①, ② and AB=AC, AE = CD ................, we can get 15 points.

It can also be obtained by δ CDE ∽ δ CAD, that is, AE2=CD2=CE? 20 points for calcium ..................

Let AE=x, then CE = D-X, then X2 = D (D-X).

AE = x = (negative value discarded) ... 25 points.

Verb (abbreviation of verb) solution: ∫A+B-C = X, A+C-B = Y, B+C-A = Z,

∴ A =, B =, C = ............................. 5 points.

∫y = x2，

Therefore, a =-(1);

b= - (2)

c= - (3)

∴x= - (4)

∫x is an integer, which gives 1+8a=T2, where t is a positive odd number. ................. 10 point

So 2a=, where a is a prime number, so there is = 2, = a.

∴ t = 5, a = 3.................. 15.

Substitute a=3 into x=2 or -3 in (4).

When x=2 and y=x2=4,

So-2 = 2，z= 16，

Substituting (2) and (3) can get b=9 and c= 10.

The contradiction with b and c is a prime number and should be abandoned. 20 points.

When x =-3 and y = 9. -3 = 2,

∴z=25

Substituting (2) and (3) can give b= 1 1 and c= 17.

∴ ABC = 3×1/kloc-0 /×17 = 56125 points.

2006 National Junior Middle School Mathematics League

First attempt

First, multiple-choice questions (7 points for each small question, ***42 points)

1. It is known that the quadrilateral ABCD is an arbitrary convex quadrilateral, where E, F, G and H are the midpoints of AB, BC, CD and DA respectively, and S and P are used to represent the area and perimeter of the quadrilateral ABCD respectively; S 1 and p 1 respectively represent the area and perimeter of the quadrilateral EFGH. Assume that the following statement is correct ().

(a) Both are constant values; (b) All are constant values, but not constant values.

(c) Not a constant, but a constant; (d) Both are not constants.

2. It is known that it is a real number, which is related to two equations. Then the value of is ().

(A) (B) (C) (D) 1

3. The equation about has only two different real roots. Then the range of real numbers is ().

(A)a>0 (B)a≥4 (C)2