Many math application problems in grade one are in urgent need.

You have to pass the preliminaries and get the right to participate in the semi-finals, and then you can go to the provincial capital city to participate in the semi-finals.

The first, second and third prizes (of course, there are also those without prizes) will be distributed in proportion to the rematch, that is, the provincial prize.

You must win the first prize (that is, the top few in the province) before you can enter the provincial team and participate in the winter camp of mathematics competition. The winter camp is the national finals, including the selection of the national team. Only the winter camp has a national prize, and everyone has it. The first, second and third prizes correspond to gold, silver and bronze medals.

Some national gold medals will stay for national team training, and then the national team will be selected to participate in the International Mathematical Olympiad (IMO), which is the highest level of mathematical competition.

Examination Paper of 2006 National Junior Middle School Mathematics Joint Competition (Liaoning)

First attempt

(April 9, 8:30-9:30 am)

First, multiple-choice questions (the full score of this question is 42 points, and each small question is 7 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. Let K = SS 1 and K 1 = PP 1, then the following statement about k and K 1 is correct ().

A.k and K 1 are both constant values. B.K is a constant and K 1 is not a constant.

C.k is not a constant, K 1 is a constant, and D.K. and K 1 are not constants.

2. It is known that m is a real number, and sinα and cosα are two of the equations 3x2 -mx+ 1 = 0 about x, so the value of sin4α+ cos4α is ().

3. The equation about x | x2x–1| = a has only two different real roots, so the value range of real number a is ().

A.a & gt0 b . a≥4 c . 2 & lt； a & lt4d . 0 & lt； A< four

4. Set b>0, a2 -2ab+c2 = 0, bc & gtA2, then the size relation of real numbers A, B and C is ().

A.b>c> the first century BC. a & gtb C.a & gtb & gtc D.b & gta & gtc

5. Let A and B be rational numbers and satisfy the equation a+b3 =6? 6? 6 1+4+23, then the value of a+b is

( ).

A.2 B.4 C.6 D.8

6. Arrange the condition that "at least one number 0 is a positive integer multiple of 4" in a column from small to large: 20, 40, 60, 80, 100, 104, ..., then the number of 158 in this column is ().

2000-2004

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

1. The sum of the abscissas of the intersection point between the image with the function y = x2 -2006|x|+ 2008 and the X axis is equal to _ _ _ _ _ _ _.

2. In the isosceles Rt△ABC, AC = BC = 1, m is the midpoint of BC, CE⊥AM intersects AB at E and F, then S ⊿ MBF = _ _ _ _ _ _ _ _

3. Let the value of the real number X with the minimum value of x2+4+(8-x)2+ 16 be _ _ _ _ _ _ _.

4. In the plane rectangular coordinate system, the vertex coordinates of the square OABC are O (0,0), A (100,0), B( 100, 100) and C (0 0,0100) respectively. If the OABC (boundary and 6? 6 S⊿PBC = S⊿PAB？ 6? 6S⊿POC, that is, the grid point P is called "good point", and the number of "good points" in the square OABC is _ _ _ _ _ _ _.

(Note: The so-called "grid point" refers to the point in the plane rectangular coordinate system whose abscissa and ordinate are integers. )

Examination Paper of 2006 National Junior Middle School Mathematics Joint Competition (Liaoning)

A second attempt

(April 9 10:00- 165438+ 0:30 am)

1. (The full mark of this question is 20 points)

Given that the unary quadratic equation x2 +2(a+2b+3)x+(a2+4b2+99)= 0 has no different real roots, how many ordered positive integer groups (a, b) are there?

Second, (the full score of this question is 25 points)

As shown in the figure, D is the midpoint of the bottom BC of isosceles △ABC, and E and F are AC and its extension lines respectively.

We also know that ∠ EDF = 90, ED = DF = 1, AD = 5 ... Find the length of BC line.

Iii. (Full score for this question is 25 points)

In the parallelogram ABCD, the bisector of ∠A is the extension line of BC and DC respectively.

The straight line intersects at point E and point F, and point O and point O 1 are the outer centers of △CEF and △ABE respectively. (1) Verification:

O, e, O 1 three-point * * * line; (2) Verification: If ∠ ABC = 70, find ∠OBD.

Degree.

Reference answer:

Multiple choice question: BCDABC

Fill in the blank:1.02.1123.384.197.

Solution:1.162.1073. (1) proves that the corresponding angles of similar triangles are equal; (2)35 .

① 1/3x=-4

x=- 12

②6x-a=0

x=a/6

The solution of ① is 5 larger than that of ②.

So-12-a/6=5.

a/6=- 17

a=- 102

③x/a-2/5 1=0

x/(- 102)-2/5 1=0

x/ 102=-2/5 1

x=-4

1. Multiple choice questions (4 points for each question, 40 points for * * *) Only one of the four options below is correct. Please write the English letters indicating the correct answers in the table below.

The title is 1234556789 10 * *.

answer

1. Among the four rational numbers,, 18, the negative number * * * has ().

1 (B)2 (C)3 (D)4。

Xiaoming drew four corners in his exercise book, and their degrees are as shown in figure 1. The obtuse angle in these angles is ().

1 (B)2 (C)3 (D)4。

3. If the nth prime number is 47, then n is ().

12(B) 13(C) 14(D) 15

(English-Chinese dictionary: prime number n, prime number n)

4. The positions of the corresponding points of rational numbers A, B and C on the number axis are shown in Figure 2, and the following four propositions are given:

(A)abc<。 0 (B)

(C)(a-b)(b-c)(c-a)>0 (D)

The correct proposition is ()

(A)4 (B)3 (C)2 (D) 1。

5. As shown in Figure 3, among the four artistic terms of "People's Olympics", the axisymmetric figure is ().

1 (B)2 (C)3 (D)4。

6. It is known that p, q, r and s are mutually different positive integers, and they satisfy, then ()

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

7. Teacher Han specially made four identical cubes and placed them as shown in Figure 4(a) and Figure 4(b) respectively. Then the sum of the points of the four bottom squares in Figure 4(b) is ().

(A) 1 1(B) 13(C) 14(D) 16

8. As shown in Figure 5, if AB//CD, the relationship between ∠B, ∠C and ∠E is ().

(A)∠B+∠C+∠E= 180？ 0? 2 (B)∠B+∠E-∠C= 180？ 0? 2

(C)∠B+∠C-∠E= 180？ 0? 2 (D)∠C+∠E-∠B= 180？ 0? 2

9. Equation 2007x+2007a+2008a with X as unknown number = 0 (A, B is rational number, B >；; 0) has a positive integer solution, then ab is ()

(a) Negative numbers (b) Non-negative numbers (c) Positive numbers (d) Zero

10. Define a new operation for any four rational numbers A, B, C and D: =ad-bc, and if it is known = 18, then x= ().

(A)- 1 (B)2 (C)3 (D)4

Two. Group A fills in the blanks (4 points for each small question, ***40 points)

1 1. Xiaoming played 20 games and 95% won. If he doesn't lose the next game, the winning rate is 96%. Xiaoming needs to play another game.

12. As shown in Figure 6, point D is on BC, at right angles to Rt△ABC, BD=2, DC=3. If AB=m and AD=n, then

= 。

13. the average of p, q and r is 4, and the average of p, q, r and x is 5, so x = 1.

(English dictionary: average)

14. Calculation: =

15. If and are opposite numbers, then =.

16. As shown in Figure 7, the square ABCD has an area of 25 square centimeters, with point E on AB, BE= 1.5AE, point F on BC, and BE=4CF, so the distance from point D to EF is square centimeters.

17. If three rational numbers A, B and C satisfy A: B: C = 2: 3: 5, then a+b+c=.

18. A male and a female athlete practice long-distance running on the circular track. Male athletes are faster than female athletes. If they start from the same starting point in the opposite direction at the same time, they meet every 25 minutes. Now they are starting from the same starting point and heading in the same direction at the same time. The male athlete caught up with the female athlete after 15 minutes, running more laps than the female athlete 16.

19. Given that m, n and p are integers, then =.

20. If known, then =

Three. Fill in the blanks in group B (8 points for each small question, 49 points for * * *, two spaces for each question, 4 points for each space)

2 1. At present, there is brine 100 kg, containing brine 15%. To increase the salt content of this brine by 5%, it is necessary to add 10 kg of pure salt; To reduce the salt content of this brine by 5%, it is necessary to add kilograms of water.

22. After Liu Xiang, a famous track and field athlete in China, set a new world record of 1 10 meter hurdle, the expert group input the images and data of Liu Xiang's previous competitions and trainings into the computer and analyzed them, showing that each "hurdle period" (the time taken to cross two adjacent hurdles) when he crossed 10 hurdle (the distance between two adjacent hurdles was equal) was as follows. The distance from the last column to the finish line is 14.02 meters, and Liu Xiang's best performance in this section is 1.4 seconds. According to the above data, the distance between two adjacent columns is seconds. Theoretically, the best performance of Liu Xiang 1 100 meter hurdles can reach seconds.

23. A poet praised the water of Lijiang River in this way: how deep love is, how beautiful dreams are. /If love is like a dream, the water of Lijiang River. In the translated and published poems, the English translation of this passage is: "Love is deep and dreams are sweet/Lijiang River once flowed". Please count the number of English letters that appear in this English poem (26). The English letters that appear the least are; The English letter with the highest frequency is.

24. If, then = =.

25. Fold a thin iron with a length of 25 cm into a triangle with prime sides (unit: cm). If the three sides of such a triangle are a, b, c and a≤b≤c, then (a, b, c) has a set of solutions, and all triangles are triangles.

Answers to the 18th "Hope Cup" National Mathematics Invitational Tournament (Grade One)

First, multiple-choice questions:

The title is 1 23455 6789 10.

Answer B A D B C C D B A C

Hint: 2,90

3. If the nth prime number is 47, then n = _ _ _ .2, 3, 5, 7, 1 1, 13, 17, 19, 23, 29, 3655.

7. From (a):1-5,2-4,3-6, so 1+3+6+6= 16.

8. If e is taken as EG//AB, we can get: ∠ b+∠ e-∠ c = 180.

9. To solve the equation, x= is a positive integer, so-2007a-2008b >; 0, because b>0, so A.

Two. Group a fills in the blanks

Prompt: 1 1, assuming that the x field is still needed, you get 20×95%+X =(20+X)×96%:X = 5.

Pythagorean theorem: m2 = bc2+ac2 = 52+ac2n2 = dc2+ac2 = 32+ac2 Available: m2-n2 = 16.

13, the average value of p, q and r is 4, the average value of p, q, r and x is 5, and x=?

P+q+r=4×3= 12, p+q+r+x=5×4=20, so x=8.

14, original formula = = =

15、- 1

16, even number DE, DF, from known AB=BC=CD=DA=5, AE=2, BE=3, BF=4, CF= 1, EF=5, s △ def =11.

17, let a = 2k, b = 3k and c = 5k substitute to get k=, so a+b+c= 10k=

18, assuming that the female athlete ran x laps, then the male athlete ran x+ 16 laps.

Then:

Solution: x= 10

19, from the meaning of the question: m=n+ 1, p=m or m=n, p=m+ 1, when m=n+ 1, p=m, the original formula = 3; When m = n and p = m+ 1, the original formula =3. So the original formula =3.

20. The original formula = 3a6+12a4-(a3+2a)+12a2-4.

=3a6+ 12a4+ 12a2-2

=3a3(a3+2a+2a)+ 12a2-2

=3(-2a-2)(-2+2a)+ 12a2-2

= 12- 12 a2+ 12 a2-2

= 10

Third, fill in the blanks in group B.

Tip:

Interpretation of 2 1 and 6.25 50

22、( 1 10- 13.72- 14.02)÷( 10- 1)=9. 14

2.5+0.96×9+ 1.4= 12.54

23、8;