(9: 00 am on March 24th, 2007 ~ 1 1 p.m)

First attempt

First, multiple-choice questions (this big question is * * * six small questions, each with 7 points and ***42 points)

The last digit of 1 and 20072007 is ().

a、 1 B、3 C、D、

2. The result of simplification is ()

A, B, C, D,

3. If it is a positive number, it is known that a quadratic equation with one variable has two equal real roots, then the root of the equation is ().

A, there is no real root b, there are two equal real roots c, there are two unequal real roots d, and the situation of roots is uncertain.

4. If all three vertices of a right triangle are taken from the vertices of a regular dodecagon, then the number of such right triangles is ().

a、36 B、60 C、96 D、 120

5, for a given unit square, if two diagonal lines and every two sides.

Connect the center lines to get the right picture, in which the triangle pairs are similar to each other.

Have ()

a、44 B、552 C、946 D、 1892

6. If the triangle with three height lines of X, Y and Z is expressed as (X, Y, Z), then the following four triangles (6,8, 10), (8, 15, 17), (12).

a， 1 b，2 c，3 d，4

2. Fill in the blanks (this big question is ***4 small questions, with 7 points for each small question and 28 points for * * *).

7. All equations are satisfied.

The sum of real numbers x is

8. The number of triangles with an integer side length and a circumference of 20 is

9. In a square ABCD with a side length of 1, A, B, C and D are respectively

The center of the circle is an arc with a radius of 1, and the square is divided into nine small pieces in the figure.

The area of the central block is

10, with the numbers 1, 2, 3, 4 arranged into a four-digit number, so that this number is a multiple of 1 1, there is a four-digit number * *.

A second attempt

Iii. Answer: (This topic is entitled ***3 small questions, ***70 points, 1 1 20 points, 12, 13 small questions.

25 points each)

1 1, find all positive integers, and make the unary quadratic equation about

Both are integers.

12, the diagonal AC and BD of quadrilateral ABCD intersect at p, the intersection point P is a straight line, and the intersection point of AD and BC is F. If PE=PF, AP+AE=CP+CF, it is proved that quadrilateral ABCD is a parallelogram.

13. If a number can be expressed as the sum of squares of two natural numbers (the same is allowed), it is called a "good number". How many "good numbers" are there in the first 200 positive integers 1, 2, …, 200?

Answers to the preliminary contest of junior high school mathematics competition in Jiangxi Province in 2007

First, multiple-choice questions:

1.b； 2.b； 3.d； 4.b； 5.c； 6.A。

Second, fill in the blanks:

7.； 8.8; 9.； 10.8.

Third, answer questions:

1 1. Settings. b∈N

By using prime factorization,

From integer to integer

Another way of writing: let (5a2-26a-8)+4 (a2-4a+9) = B2B ∈ n.

Simplification: (3a-7-b)(3a-7+b)=2 1.

Using prime factorization:

3a-7-b 1 3-2 1-7

3a-7+b 2 1 7- 1-3

A 64-

Empirical calculation: a=6

12. extend AC, take n above c and m below a, so that AM=AE and CN=CF, and it is easy to prove △ PAE.

△PCF, PA=PC, and then prove △ ped △ pfb. Pb = PD∴ABCD is a parallelogram.

13. A number that can be expressed as *** 14; 1 ~ 100, the number that can be expressed is10; One can be expressed as; 10 1~200, there are 8 numbers that can be expressed as,

There are seven, five, two, and 9 1 above * *.

But in numbers less than 40, 5, 10, 13, 17, 20, 25, 26, 29, 34, 37, 40 and the product of these numbers multiplied by 5 can be expressed as the sum of the squares of two numbers in two ways, or in two ways, so

There are 91-12 = 79 numbers that meet the requirements.