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The seventh grade mathematics interesting problem volume one
1. Someone wrote a program, starting from 1, alternately doing multiplication or addition (the first time can be addition or multiplication), adding 2 or 3 to the result of the last operation each time; Multiply the result of the last operation by 2 or 3, such as 30, and you can get:1+3 = 4 * 2 = 8+2 =10 * 3 = 30. How can I get: 100 times of 2+97 times of 2-2?

Answer: 1+3=4+2=2 thrice -2=2 thrice +2-2=(2 thrice +2 -2) * 2 =...= =2 100 times +2 97 times -2 97 times =2 65438.

This poem was written by Xu Ziyun, a mathematician in Qing Dynasty. Please figure out how many monks there are in this poem.

The majestic ancient temple is in the clouds. I wonder how many monks there are.

364 bowls, see if they are used up.

Three people eat a bowl and four people eat a bowl of soup.

Excuse me, sir, how many monks are there in the temple?

Answer: three people eat one bowl: then one person eats the third bowl.

Four people eat a bowl of soup: one person eats soup with a quarter bowl.

A total of1/3+1/4 = 7/12 bowls for each person.

Let * * * have x monks, according to the meaning of the question:

7/ 12X=364

Solution, X=624.

3. Two boys each ride a bicycle, starting from two places that are 20 miles apart (1 mile+1.6093 km). At the moment they set off, a fly on the handlebar of one bicycle began to fly straight to another bicycle. As soon as it touched the handlebar of another bicycle, it immediately turned around and flew back. The fly flew back and forth, between the handlebars of two bicycles, until the two bicycles met. If every bicycle runs at a constant speed of 10 miles per hour and flies fly at a constant speed of 15 miles per hour, how many miles will flies fly?

Answer: The speed of each bicycle is 10 miles per hour. After 1 hour, the two will meet at the midpoint of the distance of 2O miles. The speed of a fly is 15 miles per hour, so in 1 hour, it always flies 15 miles.

4. Sunzi Suanjing is one of the top ten famous arithmetical classics in the early Tang Dynasty, and it is an arithmetic textbook. It has three volumes. The first volume describes the system of counting, the rules of multiplication and division, and the middle volume illustrates the method of calculating scores and Kaiping with examples, which are all important materials for understanding the ancient calculation in China. The second book collects some arithmetic problems, and the problem of "chickens and rabbits in the same cage" is one of them. The original question is as follows: let pheasant (chicken) rabbits be locked together, with 35 heads above and 94 feet below. Male rabbit geometry?

A: If X is the pheasant number and Y is the rabbit number, then there is

x+y=b,2x+4y=a

Solution: y = b/2-a,

x=a-(b/2-a)

According to this set of formulas, it is easy to get the answer to the original question: 12 rabbits, 22 pheasants.

Let's try to run a hotel with 80 suites and see how knowledge becomes wealth.

According to the survey, if we set the daily rent as 160 yuan, we can be full; And every time the rent goes up in 20 yuan, three guests will be lost. Daily expenses for services, maintenance, etc. Each occupied room is calculated in 40 yuan.

Question: How can we set the price to be the most profitable?

A: The daily rent is 360 yuan.

Although 200 yuan was higher than the full price, we lost 30 guests, but the remaining 50 guests still brought us 360*50= 18000 yuan. After deducting 40*50=2000 yuan for 50 rooms, the daily net profit is 16000 yuan. When the customer is full, the net profit is only 160*80-40*80=9600 yuan.

6. Mathematician Weiner's age: The cube of my age this year is four digits, and the fourth power of my age is six digits. These two numbers only use all ten digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. How old is Weiner?

Answer: Let Wiener's age be X. First, the cube of age is four digits, which defines a range. The cube of 10 is 1000, the cube of 20 is 8000, and the cube of 2 1 is 926 1, which is a four-digit number; The cube of 22 is10648; So 10 =

7. The natural numbers of 1, 2, 3, 4 1987 ... 1986, 1987 are evenly arranged in a big circle, counting from 1: every 1 cross 2 and. Cross out 5 and 6 every 4, so that two numbers are crossed out every other number, and then circle. Q: How many numbers are left in the end?

Answer: 663

8. Stick a gold paper edge with the same width on the periphery of a landscape painting with a length of 90cm and a width of 40cm to make a wall chart. If the landscape painting area is required to account for 72% of the whole wall chart area, what should be the width of the gold paper edge?

Answer: (90+2X)(40+2X)*72%=90*40.

(90+2X)(40+2X)=3600/0.72

3600+ 180X+80X+4X2=5000

4X2+260X- 1400=0

(4X-20)(X+70)=0

Get 4x-20=0 X+70=0.

4*x=20 X=5

X=-70 does not hold.

So X=5CM.

9. Football made of black and white leather blocks, black leather blocks are regular pentagons and white leather blocks are regular hexagons. If there are 32 black and white leather blocks on a ball, please count the number of black and white leather blocks.

A: Equivalence:

The number of edges used in white skin and black skin = the number of edges used in black skin and white skin.

Set: there is a white skin X.

3x=5(32-x)

The solution is x=20.

10. There are ten identical black socks in Shuang Yi and ten identical white socks in Shuang Yi in the drawer. If you open a drawer in the dark and reach for your socks, how many socks do you have to take out to make sure you get a pair?

Answer: 3

1 1. Xiao Zhao, Xiaoqian, Sun Xiao and Xiao Li discuss which team won the final of the football match. Xiao Zhao said, "Team D will lose, but Team C can win." Penny said, "Team A and Team C are better than Team B, and there will be losses at the same time." Sun Xiao said: "Team A, Team B and Team C can all win." Xiao Li said, "Team A lost, while Team C and Team D obviously won."

They have said which team won. Can you guess which team won the championship?

Answer: Xiao Zhao, Xiao Gan, Sun Xiao and Xiao Li discussed which team won the final of the football match. Xiao Zhao said, "Team D will lose, but Team C can win." Penny said, "Team A, Team C will win and Team B will lose at the same time." Sun Xiao said: "Team A, Team B and Team C can all win." Xiao Li said, "Team A lost, while Team C and Team D obviously won."

Xiao Zhao's words indicate that Team D lost.

Small money means that team B lost.

Sun Xiao's words indicate that Team D lost.

Xiao Li's words showed that Team A lost.

So, Team C won.

12. If three line segments with lengths A, B and C can form a triangle, can the root numbers A, B and C of the line segments form a triangle?

If yes or no, please prove it.

If not, please give an example.

Answer: Yes.

Assuming that A is the smallest and C is the largest, the necessary and sufficient condition for abc to form a triangle is A+B >; c;

At this time, the comparison between √a+√b and √c is actually the comparison between √ a+b+2√ab and C (both sides are squares). a+b is already greater than C, and obviously a triangle can be formed.

13. A farmer met the devil, and the devil said, "I have an idea that will make you rich! As long as you cross the bridge behind me, your money will double. When you come back, your money will double every time you cross the bridge, but you must make sure that you give me a steel plate every time after you double your money. The farmer was overjoyed and immediately crossed the bridge. After crossing the bridge three times, there is only one steel plate left in his pocket, and he will pay the devil nothing left. Please use a single item containing a to indicate the number of steel plates in farmers' initial pockets.

Answer: let the initial amount be X.

2[2(2x-a)-a]-a=0

X=7a/8 to solve this equation.

14. Three students came home from school and saw a yellow car on the road. When they walked on, they heard that the car hit an old man and ran away. But no one wrote down the license plate number of this car. When the police asked the three middle school students, they all said that the license plate number was four digits. One of them remembers that the first two digits of the number are the same, another remembers that the last two digits of the number are the same, and the third remembers that the four digits are exactly the same.

A: Four digits can be expressed as

a× 1000+a× 100+b× 10+b

= a× 1 100+b× 1 1

= 1 1×(a× 100+b)

Because A × 100+B must be divisible by 1 1, A+B = 1 1 is brought into the above formula.

Four digits =1/kloc-0 /× (a×100+(1-a))

= 1 1×(a×99+ 1 1)

= 1 1× 1 1×(9a+ 1)

As long as 9a+ 1 is a complete square number.

Verified by a = 2, 3, 4, 5, 6, 7, 8, 9,

9a+ 1= 19、28、27、46、55、64、73 .

So there is only one solution with a = 7; b=4 .

Therefore, the four digits are 7744 =112× 82 = 88× 88.

15. It is known that 1 plus 3 equals to the second power of 4, 1 plus 3 plus 5 equals to the third power of 9, 1 plus 3 plus 7= 16 equals to the second power of 4, 1 plus 3 plus 5 equals to the second power of 25 equals to 5, and1plus 3 plus 9 equals to the second power of 5. ......

& lt 1 & gt; Imitate the above example and calculate 1 plus 2 plus 3 plus 5 plus 7 plus ... plus 99 equals?

& lt2> According to the above law, please use the natural number n(n is greater than or equal to 1) to express the general law.

Answer:

& lt2>1+3+5+...+n = the square of [(n-1)/2+1].

16. Once, a cat caught 20 mice and arranged them in a row. The cat announced its decision: first, eat the odd-numbered mice, then renumber the remaining teachers by 1, 2, 3, 4 ... and then eat all the odd-numbered mice. Repeat this operation and the last mouse will be released. A clever mouse listened and immediately chose a position. Finally, it was him, and the cat let him go!

Do you know where this clever little mouse is standing?

Answer: Rank 16. 1 remains divisible by 2, 2 (the square of 2) remains divisible by 4, 3 (the third power of 2) remains divisible by 8, and 4 (the fourth power of 2) remains divisible by 16, so only 16 will not be eaten.

17. 1/( 1*2*3)+ 1/(2*3*4)+ 1/(3*4*5)+…+ 1/(98*99* 100)

Answer:1(1* 2 * 3)+1(2 * 3 * 4)+1(3 * 4 * 5)+…+1(98 * 99).

=( 1- 1/2- 1/3)+( 1/2- 1/3- 1/4)+( 1/3- 1/4- 1/5)+...... 1/98- 1/99- 1/ 100

= 1- 1/ 100

=99/ 100

Remarks:1(1* 2 * 3) =1-1/3.

18. Xiao Wei and Xiaoming exchanged activities during the summer vacation. Xiao Wei said, "I attended the summer camp for science and technology and went out for a week. The sum of the dates of these seven days is 84. Do you know what date I left? " Xiao Ming said: "I stayed at my uncle's house for seven days during the holiday, and the date and number of months were also 84." Guess what date I went home? "

Answer: Question 1: Let the departure date be X.

X+X+ 1+X+2+X+3+X+4+X+5+X+6 = 84

X=9

Xiao Wei set out on the 9 th.

Question 2: Because it is a summer vacation activity, it can only be held in July and August.

Set the date back to x.

rank

7+X+X- 1+X-2+X-3+X-4+X-5+X-6 = 84

or

8+X+X- 1+X-2+X-3+X-4+X-5+X-6 = 84

The first formula solves X= 14.

The result of the second formula is not an integer.

So I can only get home in July 14.

19. A school has three classes: A, B and C. Class A has 4 more girls than Class B, and Class B has/kloc-0 more girls than Class C. If the first students of Class A are transferred to Class B, the first students of Class B are transferred to Class C, and the first students of Class C are transferred to Class A at the same time, the number of girls in the three classes is exactly equal. It is known that there are two girls in the first group of Class C. How many girls are there in the first group of Class A and Class B?

Answer: We assume that there are M and N girls in the first group of Class A and Class B respectively. There are x girls in Class B, so there are x+ 1, and there are x+5 girls in Class A, with an average of x+2 (calculated by variation). Class c:-2+n = (x+2)-X.

Class a: +2-m=(x+2)-(x+5) can get m=5 n=4.

20. There is a reservoir, which has a certain water flow per unit time and is also discharging water. According to the current flow rate, the water in the reservoir can be used for 40 days. Due to the recent rainfall in the reservoir area, the amount of water flowing into the reservoir has increased by 20%. If the discharged water volume is also increased by 10%, it can still be used for 40 days. Q: If the water is discharged according to the original discharge, how many days can it be used?

Answer: Let the total water volume of the reservoir be X, and the daily water inflow and water outflow are M and N respectively.

Then x/(n-m) = 40 = x/[n (1+10%)-m (1+20%)] needs x/[n-m( 1+20%)].

You can simplify n=2m x=40m and bring it into the second formula to get x=50 days.

2 1. A hotel first sets the temperature of air conditioners A and B to 1 degree. Results Air conditioner A saves 27 degrees more electricity than air conditioner B every day, and then the equipment of air conditioner B is cleaned, so that the total power saving of air conditioner B is 1.1times that of air conditioner A after the temperature is raised, while the power saving of air conditioner A remains unchanged.

Answer: Assuming that only after the temperature increases by 1 degree, two air conditioners A and B will save X and Y degrees every day.

X-Y=27,

X+ 1. 1Y=405

X=207

Y= 180

A and B air conditioners save electricity by 207 180 degrees each day.

22. The existing barren hills in Hongmian Village are 1 1,000 hectares, and the greening rate is 80%. There is no need to afforest 300 hectares of fertile land. The greening rate of trees on the river slope of X hectares this year is 20%, so the greening rate of all land in Hongmian Village will reach 60%. How many hectares of river slopes are there?

Answer: (x * 20%+1000 * 80%)/(1000+300+x) = 60%.

(0.2*x+800)/( 1300+x)=0.6

0.2*x+800=780+0.6*x

X=50 hectares

23. A piece of paper is 0.06 cm thick, and the distance from the earth to the moon is 3.85 * 10 5 km.

Xiao Ming said, if you cut this paper into two equal parts, pile up the paper cut into two equal parts and cut it into two equal parts. If it is repeated, the height of all the papers will be greater than the distance from the moon to the earth.

Xiao Gang said, I don't believe what Xiao Ming said.

Is Xiao Ming's statement correct Why?

Answer: 40 cuts are higher than 3.85 *10.5 km.

2 40 * 0.06/100000 = 6.597 *105km.

Xiao Ming is right, but the paper must be big enough, otherwise it won't be cut several times.

24. There are 27 pearls, one of which is fake, but its appearance is the same as that of a real pearl, only a little lighter than that of a real pearl. Q: If you weigh it with a balance at least a few times (without weights), you can definitely find out the fake pearls?

Answer: 3 times

For the first time, divide 27 pearls into 3 equal parts, take 2 of them and weigh them at both ends of the balance. If the balance is tilted, consider 9 light pearls; if not, consider 9 unweighted pearls. In the same way, divide nine pearls into three equal parts, weigh two at each end of the balance, and get three "suspicious" pearls again. Take out two scales. If the balance is skewed, the light one is defective ~ otherwise, it is not called defective.

25. Like China, Egypt is also a famous ancient civilization in the world. The ancient Egyptians treated scores differently. Generally, they only use fractions whose numerator is 1, such as1115 for 2/5 and 1/4 for 65438. 1/90。 1/9 1, where whether to add 10, and add a sign to make it sum to-1, if it exists, please write down the number of 10, if not, please explain the reason.

Answer: Solution:

- 1=- 1/5- 1/6- 1/8- 1/9- 1/ 10- 1/ 12- 1/ 15- 1/ 18- 1/20- 1/24

Two solutions:

1- 1/2+ 1/2- 1/3+ 1/3- 1/4+ 1/4- 1/5+ 1/5- 1/6+ 1/6- 1/7+ 1 /7- 1/8+ 1/ 8- 1/9+ 1/9- 1/ 10= 1- 1/ 10

So:

1/2+ 1/6+ 1/ 12+ 1/20+ 1/30+ 1/42+ 1/56+ 1/72+ 1/90+ 1/ 10= 1

Namely:

- 1/2- 1/6- 1/ 12- 1/20- 1/30- 1/42- 1/56- 1/72- 1/90- 1/ 10=- 1

1. Two boys each ride a bicycle, starting from two places 20 miles apart (1 mile +0.6093 km) and riding in a straight line. At the moment they set off, a fly on the handlebar of one bicycle began to fly straight to another bicycle. As soon as it touched the handlebar of another bicycle, it immediately turned around and flew back. The fly flew back and forth, between the handlebars of two bicycles, until the two bicycles met. If every bicycle runs at a constant speed of 10 miles per hour and flies fly at a constant speed of 15 miles per hour, how many miles will flies fly?

answer

The speed of each bicycle is 10 miles per hour, and the two will meet at the midpoint of the distance of 2O miles after 1 hour. The speed of a fly is 15 miles per hour, so in 1 hour, it always flies 15 miles.

Many people try to solve this problem in a complicated way. They calculate the first distance between the handlebars of two bicycles, then return the distance, and so on, and calculate those shorter and shorter distances. But this will involve the so-called infinite series summation, which is very complicated advanced mathematics. It is said that at a cocktail party, someone asked John? John von neumann (1903 ~ 1957) is one of the greatest mathematicians in the 20th century. ) Put forward this question, he thought for a moment, and then gave the correct answer. The questioner seems a little depressed. He explained that most mathematicians always ignore the simple method to solve this problem and adopt the complex method of summation of infinite series.

Von Neumann had a surprised look on his face. "However, I use the method of summation of infinite series," he explained.

2. A fisherman, wearing a big straw hat, sat in a rowboat and fished in the river. The speed of the river is 3 miles per hour, and so is his rowing boat. "I must row a few miles upstream," he said to himself. "The fish here don't want to take the bait!"

Just as he started rowing upstream, a gust of wind blew his straw hat into the water beside the boat. However, our fisherman didn't notice that his straw hat was lost and rowed upstream. He didn't realize this until he rowed the boat five miles away from the straw hat. So he immediately turned around and rowed downstream, and finally caught up with his straw hat drifting in the water.

In calm water, fishermen always row at a speed of 5 miles per hour. When he rowed upstream or downstream, he kept the speed constant. Of course, this is not his speed relative to the river bank. For example, when he paddles upstream at a speed of 5 miles per hour, the river will drag him downstream at a speed of 3 miles per hour, so his speed relative to the river bank is only 2 miles per hour; When he paddles downstream, his paddle speed will interact with the flow rate of the river, making his speed relative to the river bank 8 miles per hour.

If the fisherman lost his straw hat at 2 pm, when did he get it back?

answer

Because the velocity of the river has the same influence on rowing boats and straw hats, we can completely ignore the velocity of the river when solving this interesting problem. Although the river is flowing and the bank remains motionless, we can imagine that the river is completely static and the bank is moving. As far as rowing boats and straw hats are concerned, this assumption is no different from the above situation.

Since the fisherman rowed five miles after leaving the straw hat, he certainly rowed five miles back to the straw hat. Therefore, compared with rivers, he always paddles 10 miles. The fisherman rowed at a speed of 5 miles per hour relative to the river, so he must have rowed 65,438+00 miles in 2 hours. So he found the straw hat that fell into the water at 4 pm.

This situation is similar to the calculation of the speed and distance of objects on the earth's surface. Although the earth rotates in space, this motion has the same effect on all objects on its surface, so most problems about speed and distance can be completely ignored.

3. An airplane flies from city A to city B, and then returns to city A. In the absence of wind, the average ground speed (relative ground speed) of the whole round-trip flight is 100 mph. Suppose there is a persistent strong wind blowing from city A to city B. If the engine speed is exactly the same as usual during the whole round-trip flight, what effect will this wind have on the average ground speed of the round-trip flight?

Mr. White argued, "This wind will not affect the average ground speed at all. In the process of flying from City A to City B, strong winds will accelerate the plane, but in the process of returning, strong winds will slow down the speed of the plane by the same amount. " "That seems reasonable," Mr. Brown agreed, "but if the wind speed is 100 miles per hour. The plane will fly from city A to city B at a speed of 200 miles per hour, but the speed will be zero when it returns! The plane can't fly back at all! " Can you explain this seemingly contradictory phenomenon?

answer

Mr. White said that the wind increases the speed of the plane in one direction by the same amount as it decreases the speed of the plane in the other direction. That's right. But he said that the wind had no effect on the average ground speed of the whole round-trip flight, which was wrong.

Mr. White's mistake is that he didn't consider the time taken by the plane at these two speeds.

It takes much longer to return against the wind than with the wind. In this way, it takes more time to fly when the ground speed is slow, so the average ground speed of round-trip flight is lower than when there is no wind.

The stronger the wind, the more the average ground speed drops. When the wind speed is equal to or exceeds the speed of the plane, the average ground speed of the round-trip flight becomes zero, because the plane cannot fly back.

4. Sunzi Suanjing is one of the top ten famous arithmetical classics in the early Tang Dynasty, and it is an arithmetic textbook. It has three volumes. The first volume describes the system of counting, the rules of multiplication and division, and the middle volume illustrates the method of calculating scores and Kaiping with examples, which are all important materials for understanding the ancient calculation in China. The second book collects some arithmetic problems, and the problem of "chickens and rabbits in the same cage" is one of them. The original question is as follows: let pheasant (chicken) rabbits be locked together, with 35 heads above and 94 feet below.

Male rabbit geometry?

The solution of the original book is; Let the number of heads be a and the number of feet be b, then b/2-a is the number of rabbits and a-(b/2-a) is the number of pheasants. This solution is really great. When solving this problem, the original book probably adopted the method of equation.

Let x be the pheasant number and y be the rabbit number, then there is

x+y=b,2x+4y=a

Get a solution

y=b/2-a,

x=a-(b/2-a)

According to this set of formulas, it is easy to get the answer to the original question: 12 rabbits, 22 pheasants.

Let's try to run a hotel with 80 suites and see how knowledge becomes wealth.

According to the survey, if we set the daily rent as 160 yuan, we can be full; And every time the rent goes up in 20 yuan, three guests will be lost. Daily expenses for services, maintenance, etc. Each occupied room is calculated in 40 yuan.

Question: How can we set the price to be the most profitable?

A: The daily rent is 360 yuan.

Although 200 yuan was higher than the full price, we lost 30 guests, but the remaining 50 guests still brought us 360*50= 18000 yuan. After deducting 40*50=2000 yuan for 50 rooms, the daily net profit is 16000 yuan. When the customer is full, the net profit is only 160*80-40*80=9600 yuan.

Of course, the so-called "learned through investigation" market was actually invented by myself, so I entered the market at my own risk.

6 Mathematician Weiner's age, the whole question is as follows: The cube of my age this year is four digits, and the fourth power of my age is six digits. These two numbers only use all ten digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. How old is Weiner? Answer: this question is difficult at first glance, but it is not. Let Wiener's age be X. First, the cube of age is four digits, which defines a range. The cube of 10 is 1000, the cube of 20 is 8000, and the cube of 2 1 is 926 1, which is a four-digit number; The cube of 22 is10648; So 10 =

Uniformly arranged 1, 2,3,4 1987 natural numbers ... 1986, 1987 form a big circle, counting from 1: every 1 crosses 2 and 3; Cross out 5 and 6 every 4, so that two numbers are crossed out every other number, and then circle. Q: How many numbers are left in the end?

Answer: 663

It is known that 1 plus 3 equals to the power of 2 of 4, 1 plus 3 plus 5 equals to the power of 3 of 9, 1 plus 3 plus 5 plus 7 equals to the power of 4, 1 plus 3 plus 5 plus 7 equals to the power of 5, and so on. ......

& lt 1 & gt; Imitate the above example and calculate 1 plus 2 plus 3 plus 5 plus 7 plus ... plus 99 equals?

& lt2> According to the above law, please use the natural number n(n is greater than or equal to 1) to express the general law.

& lt 1 & gt; 1+3+5+...+99 = 50 squared

& lt2>1+3+5+...+n = the square of [(n-1)/2+1].