200 1 preliminary examination of national junior high school mathematics league

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

1, a, b and c are rational numbers, and the equation a+b√2+c√3 = √(5+2√6) holds, so the value of 2a+999b+ 100 1c is ().

(a)1999 (b) 2000 (c) 2001(d) cannot be determined.

2. if ab≠ 1, 5a2+200 1a+9 = 0, 9b2+200 1b+5 = 0, then the value of a/b is ().

(A)9/5(B)5/9(C)-2006/5438+0/5(D)-2006/5438+0/9

3. It is known that in △ABC, ∠ACB=900, ∠ABC= 150 and BC= 1, then the length of AC is ().

(A)2+√3(B)2-√3(C)3/ 10(D)√3-√2

4. In △ABC, d is a point on the side of AC. In the following four cases, the case that △ABD∽△ACB is not necessarily true is ().

(A)AD BC = AB BD(B)AB2 = AD AC(C)∠ABD =∠ACB(D)AB BC = AC BD

5.① In the real number range, the root of the unary quadratic equation ax2+bx+c = 0 is x =-b/2a √ (B2-4ac)/2a; ② In △ABC, if ac2+bc2 > ab2, then △ABC is an acute triangle; ③ In △ABC and △A'B'C, A, B and C are three sides of △ABC respectively. If a > a', b > b' and c > c', the area s of △ABC is greater than that of △A'B'C'. In the above three propositions, the number of false propositions is ()

(A)0(B) 1(C)2(D)3

6. A shopping mall gives customers a discount, which stipulates that: ① shopping does not exceed 200 yuan at one time, and there is no discount; (2) If you are dissatisfied with 500 yuan after shopping in 200 yuan at one time, you will get a 10% discount on the marked price; (3) If the shopping exceeds 500 yuan at one time, 500 yuan will be given a discount according to Article (2), and the part exceeding 500 yuan will be 20% off. Someone went shopping twice and paid 168 yuan and 423 yuan respectively; If he only shops for the same goods once, the amount payable is ()

(A)522.8 yuan (B)5 10.4 yuan (C)560.4 yuan (D)472.8 yuan.

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

1, it is known that the coordinate of point P in rectangular coordinate system is (0, 1), O is the coordinate origin, ∠QPO= 1500, and the distance from P to Q is 2, so the coordinate of Q is _ _ _ _ _ _.

2. Given that two circles with radii of 1 and 2 are tangent to point P, the distance from point P to the tangent of these two circles is _ _ _ _ _.

3. It is known that x and y are positive integers, and xy+x+y=23, then x2+y2 = _ _ _ _ _ (Not the original title)

4. Positive integers. If you add 100 and 168 respectively, you can get two complete squares. This positive integer is _ _ _ _ _.

2008 National Junior Middle School Mathematics League

April 2008 13 8:30-9:30 am

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

1, let a 2+1= 3a, b 2+1= 3b, a ≠ b, then the value of algebraic+is ().

5 (B)7 (C)9 (D) 1 1

2. As shown in the figure, let AD, BE and CF be the three heights of △ABC. If AB = 6, BC = 5, EF = 3, the length of the line segment BE is ().

(A) (B)4 (C) (D)

3. If two cards are randomly selected from five cards with the numbers 1, 2, 3, 4 and 5 written on them, and the number on the first card is taken as ten digits, and the number on the second card is taken as single digits to form a two-digit number, the probability that the number formed is a multiple of 3 is ().

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

4. In △ABC, ∠ ABC = 12, ∠ ACB = 132, BM and CN are bisectors of these two angles respectively, and points M and N are on straight lines AC and AB respectively, then ().

(A)BM & gt； CN(B)BM = CN(C)BM & lt； The relationship between CN (D)BM and CN is uncertain.

From today, the prices of five different commodities with the same price will be reduced by 10% or 20% respectively. After a few days, the prices of these five commodities will be different. Let the ratio of the highest price to the lowest price be r, then the minimum value of r is ().

(A)3(B)4(C)5(D)

6. It is known that real numbers x and y satisfy (x–) (y–) = 2008.

The value of 3x2–2y2+3x–3y–2007 is ().

(A)-2008 (B)-2008 (C)- 1(D) 1

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

1, let a =, then =.

2. As shown in the figure, the side length of square ABCD is 1, m and n are two points on the straight line where BD is located, AM =, ∠ man = 135, then the area of quadrilateral AMCN is.

3. It is known that the abscissas of the two intersections of the image with quadratic function y = x 2+ax+b and the X axis are m and n, respectively, | m |+| n | ≤ 1. Let the maximum and minimum values of b satisfying the above requirements be p and q respectively, then | p |+| q | =.

4. Square the positive integer 1, 2,3, … in a string:1491625364964810012165438 …, ranking first.

Answer: b, d, c, b, b, d; – 2、 、 、 1。

2003 National Junior Middle School Mathematics League

First, multiple-choice questions (the full score of this question is 42 points, and each small question is 7 points)

1.2 √ (3-2 √ 2)+√ (17-12 √ 2) is equal to

a . 5-4√2 b . 4√2- 1 c . 5d . 1

2. Among all internal angles of convex 10 polygon, the number of acute angles is the largest.

A.0 B. 1 C.3 D.5

3. if the image of function y = kx (k > 0) and function y = 1/x intersect at point a and point c, and AB is perpendicular to the x axis in b, then the area of △ABC is

A. 1 B.2 C.k D.k2

4. The number of positive integer pairs satisfying the equation x √ y+y √ x-√ (2003xy) = 2003 is

A. 1

5. Let the area of △ABC be 1, D be a point on the side of AB, and AD/AB = 1/3. If we take a point E on the AC side and make the area of the quadrilateral 3/4 of DECB, the value of CE/EA is

A. 1/2b 1/3c 1/4d 1/5

6. As shown in the figure, in the parallelogram ABCD, the intersection of circles passing through points A, B and C is AD, which is tangent to CD. If AB=4 and BE=5, the length of DE is

A.3 B.4 C. 15/4 D. 16/5

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

1. parabola y = ax2 +bx +c intersects with X axis at points A and B, and intersects with Y axis at point C. If △ABC is a right triangle, AC = _ _ _ _ _ _ _

2. Let m be an integer, and both equations 3x2+mx-2 = 0 are greater than -9/5 and less than 3/7, then m = _ _ _ _ _ _ _ _ _

3. As shown in figures AA' and BB', they are the bisectors of ∠EAB and ∠DBC respectively. If AA' = BB' = AB, the degree of ∠BAC is _ _ _ _ _ _ _.

4. It is known that the difference between positive integers A and B is 120, and their least common multiple is 105 times of their greatest common divisor, so the larger number of A and B is _ _ _ _ _ _ _ _.

2007 National Junior Middle School Mathematics League

First attempt

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

1. The known value is ().

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

2. When the values are 2, …, 2006, and 2007 respectively, calculate the values of the algebraic expression, add the obtained results, and the sum is equal to ().

In 2007

3.Let is the length of three sides, and the quadratic function takes the minimum value. Then △ABC is ().

(a) isosceles triangle (b) acute triangle

(c) obtuse triangle (d) right triangle

4. It is known that the distance from vertex A of acute angle △ABC to vertical center H is equal to the radius of its circumscribed circle. Then the degree of ∠A is ().

30 (B)45 (C)60 (D)75

5. Let k be any point in △ABC, and the centers of gravity of △KAB, △KBC and △KCA are D, E and F respectively, then the value of S △ DEF: S △ ABC is ().

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

There are five red balls, six black balls and seven white balls in the schoolbag. Now 15 balls are drawn from the bag, and the probability that there are exactly three red balls in the drawn balls is ().

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

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

1. Let be a decimal part, and a decimal part.

2. For all natural numbers not less than 2, two roots of a quadratic equation are written. then

= .

3. It is known that the four sides of the right-angled trapezoid ABCD are AB=2, BC=CD= 10, and AD=6. If two points pass through B and D to make a circle, intersect with the extension line of BA at point E, and intersect with the extension line of CB at point F, then the value of BE-BF is.

4. If the sum is a four-digit number and a complete square number, the value of the integer is.

A second attempt

volume one

One, (20 points) is set as a positive integer, and if it is all real numbers, it is a quadratic function.

The distance between the two intersections of the image and the axis is not less than the value of.

(25min) As shown in Figure L, the quadrilateral ABCD is trapezoidal, point E is a point on the upper bottom AD, the extension line of CE intersects with the extension line of BA at point F, the parallel line of BA and E intersects with the extension line of CD at point M, and BM intersects with AD at point N. Proof: ∠ AFN = ∠ DME.

(25 points) Known as a positive integer. If the roots of the equation are all integers, then the value of sum and the integer roots of the equation.

Volume II

1.(20 points) is a positive integer, the distance between the image of quadratic function and the two intersections of axis is, and the distance between the image of quadratic function and the two intersections of axis is. If true for all real numbers, it is the required value.

2.(25 points) Same as the second question in Volume A. 。

3.(25 points) Let it be a positive integer and the quadratic function be an inverse proportional function. If the intersection of the images of two functions is an integer point (both the abscissa and the ordinate are integer points), then.

Volume c

I (20 points) is the same as the first question in volume B.

2.(25 points) Same as the second question in Volume A. 。

3. Let (25) be a positive integer. If the images of quadratic function and inverse proportional function have integer points (points whose abscissa and ordinate are integers), evaluate and the corresponding integer 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