Guangdong ++= 10

In the above formula, Guangdong, Hui, Zhou, Hua, Luo, Geng, Jin, Bei and Sai represent 9 different numbers: 1 ~ 9. Please give a filling method to make the equation hold.

2.

When skipping rope, you can think that the midpoint of the rope moves on the same circle. If Xiaoguang skips the "single shake" in 0.5 seconds and the "double shake" in 0.6 seconds, what is the ratio of the speed of the midpoint of the rope when jumping the "single shake" to the speed of the midpoint of the rope when jumping the "double shake"?

(Note: "Single shake" means that the foot leaves the ground once and the rope turns once; "Double shake" means that the foot leaves the ground once and the rope turns twice. )

3.

As shown in the figure, the vertex of the shadow square is the midpoint of each side of the large square EFGH, and half the diameter of each side of the large square is used as a semicircle, and then the diameter of each side of the shadow square is used as a semicircle to form eight "crescent shapes". The total area of these eight "crescent shapes" is 5 square centimeters. What is the area of this big square, EFGH?

4.

The least common multiple of two natural numbers A and B is equal to 50. How many possible values does A+B have?

5.

Can you fill in a 3×3 grid table (as shown in the figure) with 9 different natural numbers (each grid can only have one number) so that the product of three numbers in each row, column and diagonal is equal to 2005? If yes, please give an example, if not, please explain the reason.

7.

As we all know, the length of a rectangle is 8 and the width is 4. Fold the rectangle diagonally and flatten it as shown. Find the area of the overlapping part (gray triangle).

8.

There are three numbers at the beginning: 1, 1, 1. Every operation converts one number into the sum of two other numbers. What is the maximum possible value of the largest number among the three numbers obtained after 10 operation?

9.

The ratio of potassium nitrate, sulfur and charcoal used to prepare "black powder" in ancient China was15: 2: 3. There are 50 kilograms of charcoal today. How many kilograms of charcoal does it take to make "black powder" 1000 kilograms?

10.

The area of the large square ABCD in the figure is18cm2, the side MN of the gray square MNPQ is on the diagonal BD, the vertex p is on the side BC, and the q is on the side CD. How many square centimeters is the area of the gray square MNPQ?

1 1.

Stack 25 cube building blocks with a side length of 1 into a geometric body, as shown in the figure. See who has the smallest surface area? What is the minimum surface area? Note: This is a practical problem. Four members of each team should use their hands and brains, and have a good spirit of cooperation. If no team obtains the stacking mode with the minimum surface area of 54, the team with the smallest stacking surface area wins. So this question is based on "whose surface area is the smallest in stacking geometry?" What is the minimum surface area? "ask questions)

13.

A parallelogram paper WXYZ with an area of 7.17cm2 is placed on another parallelogram paper EFGH, as shown on the right of the previous page, and four intersections of A, C, B and D, AB‖EF, CD‖WX, are obtained. How many square centimeters is the area of paper EFGH? Explain why.

14;

How many simplest fractions * * * are less than 10, and the denominator is 36?

16.

Can you arrange two natural numbers with a card with numbers written on it, so that one number is twice that of the other? If yes, please give examples, if not, please explain the reasons.

17.

Starting from the equilateral triangle shown in Figure A below, divide each side of the triangle into three equal parts, and then make a new equilateral triangle with the middle line segment as the edge, as shown in Figure B, and get a "snowflake hexagon". Then divide the 12 sides of the "snowflake hexagon" into three equal parts, and make a new equilateral triangle with the middle line segment as the side, as shown in Figure C, and get a new "snowflake shape". Q: What is the ratio of the area of Figure C to that of Figure A?

18.

Form a natural number. All the numbers of are different from each other, and the product of these numbers is equal to 360. Find the maximum value of n.

19.

There are many swans in the West Lake in Goose City, two geese in one lake and three geese in one lake. How many swans are there?

Description: Huizhou is also called "Goose City". West Lake in Hangzhou is a famous scenic spot, which consists of five lakes: Phoenix Lake, Crocodile Lake, Pinghu Lake, Linghu Lake and Nanhu Lake. The title means: a pair of swans landed in the lake area with two more swans and three fewer swans. How many swans are there? )

20. Nine children numbered 1 ~ 9 have a Chinese character pinned to their chests: Hui, Zhou, Hu, Feng, Crocodile, Ping, Ling and Nan. As shown in the figure, the sum of the number of children in each circle is 13. Please point out that they are marked with "Feng".

2 1.

13 students participated in a disaster relief donation, and the amount donated by each student was an integer. Ma Xiaohu quickly calculated that their average donation was 64.96 yuan, but the percentage figure was wrong. Q: What is the total donation of this 13 student?

22.

The two sliders A and B in the picture on the right are connected by connecting rods and can slide on vertical and horizontal slideways respectively. At the beginning, slider A is 20 cm away from point O, and slider B is 15 cm away from point O.: When slider A slides down to point O, how many centimeters does slider B slide?

23.

On the abacus, two decimal numbers are listed on the left and right respectively. The one on the left is 7 digits, and the one on the right is 4 digits. What is the quotient of the number on the left divided by the number on the right?

24.

As shown in the figure, ten points on the circumference divide the circumference into ten equal parts, and connect the equal parts separated by two points to get ten chords of the circle, which cross each other to form various geometric figures. Please answer: How many parallelograms are there in the picture?

25.

100 points on the circle divide the circle into 100 equal arcs, and randomly dye some of them as red dots. To ensure that at least four red dots are the four vertices of a square, ask: How many points do you have to dye red at least?

26.

Fill in a 6×6 grid table with the numbers 1, 2, 3, 4, 5 and 6, as shown in the right figure, and only one number is filled in each small grid. The sum of four numbers in each 2x2 square is called the "marked number" of this 2x2 square. Can you give a filling method so that any two "scales" are different? If yes, please give examples; If not, please explain why.

27.

Archaeological discovery of a regular polygon fragment, as shown in the figure: only a pair of student triangles and a pencil are used as tools. Please judge the number of sides of this regular polygon.

(Note: ∠ EAB = ∠ ZFBA = ∠ 165 in a given regular polygon fragment needs to be measured by the player. )

28.

The following two banners:

Friendship and discussion of China Junior Cup.

Huizhou Hongzhi, descendants of the Yellow Emperor, rejuvenating China.

Each word represents a non-zero natural number less than 25, different words represent different numbers, and the same word represents the same number. It is known that the average value of the 34 numbers represented by these words is 12. What is the maximum sum of natural numbers represented by the word "China"?

Answer:

1. If the answer is not unique, just write one. For example:

9+ + = 10,9+ + = 10,9+ + = 10,

9+ + = 10,7+ + = 10, 6+ + = 10,

++6 = 10 are all answers.

2. Answer: 3: 5

Analysis: If the radius of the circle in which the midpoint of the rope moves is r, then the distance of the midpoint of the rope moves is 2πr when the rope rotates once, and the speeds of "single shake" and "double shake" are sum respectively, then the speed ratio is.

: = : = : =3:5

3. Answer: 10

Analysis: As shown in the figure, connecting AB and CD intersect at O, and the area of triangle and ACH can be easily obtained from Pythagorean theorem and semicircle area formula, that is, the area of triangle AOC is equal to the sum of the areas of two "crescent shapes" on AH and HC. Therefore, the total area of these eight crescent shapes is equal to the area of the square ACBD.

Since the total area of these eight "crescent shapes" is 5 square centimeters, and the area of EFGH is twice that of ACBD, the area of EFGH is equal to 10 square centimeter.

4. Answer: 8

Analysis: Because: 50=2×5, A and B are both divisors of 50, and only 1, 2,5,10,25,50 can be taken. Let a≥b, and when a = 50, b = 1, 2,5,10,25,50; When a = 25, b = 2 10.

Therefore, a+b * * * has eight possible different values.

The least common multiple of two natural numbers A and B is equal to 50. When a≥b, the different values of A+B can be listed as follows:

5. Answer: 59

Analysis: connect AY, CX and BZ, as shown in the figure, the area of triangle XYZ is equal to 24, yz = 2zc, and the area of triangle XZC is equal to 12.

While zx = 3xa, the area of triangle XZC is equal to 12, so the area of triangle AXC is equal to 4. The area of triangle AYX is equal to 8. Note that xy = 4yb, and the area of triangle ABY is equal to 2. The area of triangle ZBY is equal to 6, and the area of triangle CBZ is equal to 3.

So the area of the triangle ABC = 24+ 12+4+8+2+6+3 = 59.

6. answer; cannot

Analysis: If it can be filled in, nine different natural numbers in 2005 are nine different divisors, but there are only four positive divisors in 2005, namely 1, 5,401,and 2005, so it cannot be filled in.

7. Answer: 10

Analysis: As shown in the figure, because EBD = EDB, obviously

BE=DE，AE=CE

Then let be = de = x

AE=CE=8-x

Derived from Pythagorean theorem

(8-z)+4 = x

The result is x = 5.

So, s =? Is it? CD= ×5×4= 10

8. Answer: 144

Analysis: Sort three numbers from small to large at a time, and then replace the smallest number in front with the sum of the last two numbers. The results are {1, 1, 1}→{ 1, 1, 2}→{ 1, 2,3.

It is observed that the largest number constitutes a Fibonacci sequence, and the first two numbers are 1, 2. Starting from the third item, each number is the sum of the first two numbers. Therefore,

1,2,3,5,8, 13,2 1,34,55,89, 144

After the lO operation, the number 1 1 in the sequence is 144, that is, the maximum possible value of the maximum number is 144.

9. Answer: 100

Analysis: According to the ratio of potassium nitrate, sulfur and charcoal 15: 2: 3, the ratio of charcoal is 0. Therefore, the preparation of 1000 kg of "black powder" requires 1000× = 150 (kg) of charcoal, and now there are 50 kg of charcoal, so it is still needed.

10. Answer: 4

Analysis: connect AC alternating BD to O, make a square EFGH and connect a big square ABCD. As shown in the figure, the area of the square EFGH is 36 square centimeters. Therefore, db = AC = 6 cm.

It is easy to know that DM = MQ = Mn = Nb = 2cm.

So the area of the gray square is 4 square centimeters.

1 1. Answer: 54

Analysis: Twenty-five cubic building blocks with a side length of 1 are stacked into a geometric body, and the surface area of small building blocks is the smallest when the self-overlapping surfaces are the most. Imagine 27 cubes with a side length of 1 and a surface area of 54 (Figure A).

Now it is necessary to split two small building blocks into 25 blocks, and its total surface area will not be reduced. In order to minimize the total surface area, it is found that when two adjacent small building blocks are removed from one corner (Figure B) or one small building block is removed from two corners (Figure C), the total surface area remains unchanged, which is equal to the surface area of a cube with a side length of 3, that is, 3× 3× 6 = 54. So the minimum surface area of stacking 25 small building blocks is 54.

12. Answer: 127

Analysis: This is a quick calculation problem to find the law.

The line number of 1 is1; The sum of 2 numbers in line 2 is 2; The sum of the three numbers in the third row is 4; The sum of the four numbers in line 4 is 8; The sum of five numbers in line 5 is16; The sum of 6 numbers in line 6 is 32; The sum of the seven numbers in the seventh row is 64. Total:1+2+4+8+16+32+64 =127.

13. Answer: 7. 17

Analysis: Connect AC, CB, BD and DA, as shown in the figure. Because the area of AB‖EF‖GH, ABC is half that of parallelogram AEFB, and the area of△△△ Abd is half that of parallelogram AHGB, the area of quadrilateral ACBD is half that of parallelogram EFGH.

Similarly, the area of the quadrangle ACBD is half that of the parallelogram WXYZ. Therefore,

Parallelogram area EFGH = parallelogram area WXYZ = 7. 17 cm2.

14. Answer: 120.

Analysis: Let the number satisfying the problem condition be x, then X =, where 0≤n≤9, and r is a natural number 1, 5,7,1,13, 17,1.

Therefore, the simplest fraction * * with a denominator of 36 and less than 10 has LOX12 =120 (pieces).

15. Answer: 32.5

Analysis: As shown in the figure, draw parallel lines passing through M, N, P and Q on each side of the rectangle ABCD. It is easy to know that the side length of the shadow square crossing in the middle is 3 cm and the area is equal to 9 cm 2.

Let the sum of the areas of △MQD, △NAM, △PBN and △QCP be s, and the area of quadrilateral MNPQ be equal to x, then

Solving the above equation gives 2x=65, so x=32.5 square centimeters.

Answer: No.

Analysis: Let a number be A and another number be B. According to the meaning of the question, A = 2B, then S = A+B = 2B+B = 3B, so 3 | S. But the remainder of S divided by 3 is equal to the remainder of A+B divided by 3, which is equal to the sum of A divided by 3 and B divided by 3, which is equal to 2+3+4+5+6+7. So you can't arrange two natural numbers with a card with numbers written on it, so that one natural number is twice as large as the other.

17. Answer: 40: 27

Analysis: Let the area of the equilateral triangle in Figure A be L, and the added area of each side of the equilateral triangle in Figure B be. * * * Three equilateral triangles have been added, so the ratio of the area of Figure B to that of Figure A.. Similarly, in Figure C, the area of small equilateral triangles added to the outer edge is =, * * * plus 12 small equilateral triangles, so the area of Figure C is+12× =+=.

Therefore, the area ratio of Figure C to Figure A is 40: 27.

18. Answer: 9542 1

Analysis: 360 = 2× 3× 5 = 1× 2× 4× 5× 9, so the maximum value of a is 9542 1.

19. Answer: 12.

Analysis:

Method 1: (Arithmetic method) Know that the number of swans is a multiple of 2 from "one lake and two geese"; From "one lake with three fewer", we know that the number of swans is a multiple of three. And (2,3) =1,we know that the number of swans is a multiple of 6: 6, 12,18,24, ... * * There are 12 swans.

Method 2: (Algebraic method) Let the number of swans be x and fall in y lakes. Then 2y+2 = 3y-3,

If y = 5, then x = 2y+2 = 2× 5+2 = 12, that is, * * * 12 swan.

20. Answer: 8

Analysis: one branch

Hui+Zhou+West+Lake+Feng+Crocodile+Ping+Ling+South = 45 ①

Hui +2× Zhou+West +2× Lake+Feng +2× Crocodile+Ping +2× Ling+South = 13× 5 = 65②.

②-① Obtain:

State+Lake+Crocodile+Ling = 20

You Hu+Feng+Crocodile = 13

If "Feng" = 9, only

Lake+crocodile = 1+3

At this time, it is obtained by ③.

Status+Ling = 20- 1-3 = 16

But the status and diamond are less than 9, so

State+diamond ≤ 8+7 = 15

Contradiction! So "Feng" is not equal to 9, and the maximum possible value of "Feng" is equal to 8. In fact, Hui = 4, Zhou = 9, = 1, Hu = 3, Feng = 8, Crocodile = 2, Ping = 5, Ling = 6 and Nan = 7, which meet the requirements. So the maximum value of "wind" is equal to 8.

2 1. Answer: 844

Analysis: Let the total donation of these 13 students be X yuan, then

64.90<