1, A, B and C are all reading the same book. There are 100 stories in the book. Everyone starts with a story and then reads it in order. It is known that A has read 75 articles, B has read 60 articles and C has read 52 articles. So how many stories have A, B and C read?

First of all, we can look at two of them, such as A and B. In order to ensure that both of them read as little as possible, we must first try different reading methods, so both of them read at least 75+60- 100=35 stories, and then C read 52 stories. First of all, he should try not to read the same stories as these 35 stories, but to be related, so he should try to read them with A.

2. There are three mountains and five mountains in China, among which the five mountains refer to Mount Taishan in Dongyue, Mount Hengshan in Nanyue, Mount Huashan in Xiyue, Mount Hengshan in Beiyue and Mount Songshan in Zhongyue. A teacher took photos of the five mountains and marked them with numbers. He asked five students to distinguish them. Each student names two, and the students answer as follows: A: 2 is Songshan, 3 is Huashan, and B:.

The teacher found that all five students were only half right, so what should be the correct statement?

Answer:

Assuming that the first half of A is right and the second half is wrong, then 2 is Mount Tai and 3 is not Huashan; Because everyone said half sentence is right and half sentence is wrong, it can be concluded that the first half sentence of E is wrong and the second half sentence is right, that is, 2 is not Huashan and 5 is Taishan. This contradicts A's saying that "2 is Mount Tai", so the assumption is wrong.

So we can know that the first half of what A said is wrong and the second half is right, that is, 3 is Huashan; From what Wu said, the second is not Huashan, and the fifth is Taishan; According to C, 5 is not Mount Tai, 1 is Hengshan; From b, 4 is not Hengshan, 2 is Songshan; According to Ding, 3 is not Songshan Mountain and 4 is Hengshan Mountain, so the correct statement is: 1 is Hengshan Mountain, 2 is Songshan Mountain, 3 is Huashan Mountain, 4 is Hengshan Mountain and 5 is Mount Tai.

3. Prove that++is between and.

Analysis × 10 =

× 1 1= < + +…+ < × 1 1=

4. Six digits are multiples of six. How many such six figures are there?

Solution Because 6 = 2× 3 and 2 and 3 are coprime, this integer can be divisible by both 2 and 3. Inferring from the fact that six digits can be divisible by two, A can take five values: 0, 2, 4, 6 and 8. From the fact that six digits can be divisible by 3, it is inferred that 3+A+B+A+B+A = 3+3A+2B.

Can be divisible by 3, so 2B can be divisible by 3. B can take four values: 0, 3, 6 and 9. Because B can take four values, A can take five values, and the topic does not require A≠B, so there are 5× 4 = 20 qualified six digits * *.

5. Choose four numbers from the five numbers of 0, 2, 3, 6 and 7. How many can four numbers that are divisible by 8 and do not repeat?

16 analysis.

Hints: 6320, 3720, 2360, 2760, 6032, 3072, 2736, 7632,

7320,6720,7360,3760,7032,6072,2376,3672。

6. Once upon a time, there were three monks. One told the truth, one lied, and the other sometimes told the truth and sometimes lied. One day, a wise man met these three monks. He asked the first monk, "Which monk is behind you?" He Shang replied: "It's true." He asked the second monk, "Who are you?" Answer: "Sometimes telling the truth, sometimes telling lies." He asked the third monk, "Which monk is in front of you?" The third dealer replied, "liar." According to their answers, the wise man immediately distinguished which monk they were. Please tell the wise man's answer.

Answer: Suppose the first monk answers the truth, that is, the second monk is a monk who "tells the truth", but the second monk says that he is "sometimes telling the truth and sometimes telling lies", which creates contradictions. So the first monk's answer is not valid, that is, the second monk is not a monk who tells the truth, and of course he will not be a "monk who tells the truth" himself, so the third monk can only be a monk who tells the truth. So the third monk answered the truth, that is, the second monk was "lying", indicating that the first monk sometimes told the truth and sometimes lied.

7. Both sisters are 40 years old this year. When my sister is now the same age as my sister, her age is exactly half of her age. How old is her sister?

The age difference between the two sisters is 3 times and 2 times respectively, that is, the age ratio is 3∶2.

8. On the circular runway, two people run in the same direction in the same place and meet every 16 minutes. If their speed is constant and they run in opposite directions in the same place and meet every 8 minutes, how long will it take for Party A and Party B to complete a lap?

Suppose the distance is 1, the speed difference between Party A and Party B is 0, the sum of Party A and Party B's speeds is 0, the fast speed is 0, and the slow speed is 0. It takes minutes to finish a lap.

9. The speed of a ship in still water is 25km/h, and it takes 6 hours to return to its original place after sailing 2 10/0km along the river?

Water speed: (2 10÷6)-25= 10 (km/h)

Time required to return to the original place: 210 ÷ (25-10) =14 (hours).

10 and 46305 are multiplied by the natural number a, and the product is the square of the integer. Find the smallest a and this integer.

a = 3×5×7 = 105； 46305× 105=22052。

Tip: All prime factors of a complete square number are even powers.

1 1. As shown in the figure, the triangle ABC is divided into two parts: A (shaded part) and B. What is the area of B?

Connect.

∵ ,

∴ ,

Say it again,

∴ ,∴ , .

12. Mom walks from home to work at a speed of 100 meter per minute. After a few minutes, Xiaohua runs from home to catch up with her mother.

As a result, I caught up with my mother far from home. How many meters does Xiaohua run per minute?

My mother left in minutes (meters). In the process of Xiaohua catching up with her mother, her mother left again (meters). The time for her to walk is (minutes), which is the time for Xiaohua to catch up with her mother. I also know that the distance Xiaohua runs is meters, and then according to speed = distance ÷ time, we can find out how many meters Xiaohua runs per minute, that is, Xiaohua's speed: (meters

13, the kindergarten bought a lot of plastic toys for white rabbits, pandas and giraffes, and each child chose two at random, so no matter how they choose, two out of every seven children always choose the same toy. Try to explain the truth.

Choose two toys from three kinds, and the matching methods can only be the following six kinds: (rabbit, rabbit), (rabbit, panda), (rabbit, giraffe), (panda, panda), (panda, giraffe), (giraffe, giraffe). Considering that each collocation method is a drawer and seven children are the objects, according to the principle of 1, at least two objects should be placed in the same drawer, that is to say, at least two people have chosen toys with the same collocation method and the same toy.

14 and 99 are written as 1 ~ 99 respectively. A draws a card from it, B draws another card from it, and so on.

Go down. If the last two numbers are prime numbers, A wins; If the last two numbers are not prime numbers, B wins.

Ask A how to draw cards to win.

A draws 99, and the remaining numbers are combined in pairs (1, 2) (3,4) ... (97,98). No matter what number B draws, A draws another number in the same group. This leaves two numbers in the same group, which are adjacent to each other, and A wins.

15, 100 monk, 140 steamed stuffed bun, 1 big monk, 1 little monk, 1 steamed stuffed bun. Q: How many monks are there?

This topic comes from China's famous ancient topic "The problem of dividing steamed buns among hundreds of monks". If the big monk and the little monk are regarded as chickens and rabbits respectively, and the steamed stuffed bun is regarded as legs, then the problem of chickens and rabbits in the same cage can be solved by hypothetical methods.

Assuming that 100 people are big monks, then * * * needs 300 steamed buns, which is 300- 140 = 160 (one) more than the actual situation. Now, if young monks are replaced by big monks, the total number will remain the same, but the number of steamed buns will be reduced by 3- 1 = 2 (one), because 160 ÷ 2 = 80, so there are 80 young monks and big monks.

100-80 = 20 (person).

Similarly, it can also be assumed that 100 people are young monks, so students might as well try it themselves.

In the following example, we only give a hypothetical method.

16、

Answer: The original formula ()

17, as shown in the figure, the area of the triangle is, up, point up, sum, and intersect with the point. What is the area of the quadrilateral?

A: Connection,

According to the dovetail theorem,,,

Create a copy, then a copy, a copy,

share

Share it.

therefore

18,,, is a prime number less than,, and find these three prime numbers.

Answer: Because the sum of three prime numbers is even, these three prime numbers must be odd and even, where even numbers can only be, and the sum of the other two odd prime numbers is, and because these three numbers are all less than, they can only be sum, so these three prime numbers are,,.

19, 6 people each carry a bucket to the faucet to receive water. It takes 5 minutes, 4 minutes, 3 minutes, 10 minutes, 7 minutes, and 6 minutes for the tap to fill the bucket of 6 people. Only this faucet is available now. How to arrange the order of taking water for these six people to minimize their total waiting time? What is the shortest time?

Answer: When the first person picks up the water, six people, including himself, are waiting, and when the second person picks up the water, five people are waiting; When the sixth person picks up the water, only 1 person is waiting. It can be seen that the more people wait, the shorter the water receiving time should be, so the total waiting time will be the least. Therefore, the water receiving time should be arranged in order from less to more, and the shortest time is (minutes).

20. There is a cuboid container with a length of 30 cm, a width of 20 cm and a height of 10 cm. The water depth inside is 6 cm (the largest surface is the bottom). If the container is tightly covered (watertight) and then erected to the left (the smallest surface is the bottom), what is the water depth inside?

Answer: V=30×20×6=3600 (cubic centimeter) H = 3600 ÷ (20×10) =18 (centimeter).

2 1, four students played table tennis singles. After a few innings, the PE teacher asked them how many innings they had played. Four students answered 1, 2, 3 and 3 games respectively, and the teacher said, "Some of you must remember wrong." Excuse me: How did the teacher know? (Hint: Consider from parity)

The total number of four-person games will increase by two, so the total number of four-person games must be even, but in this dialogue, four students replied that the game of 1, 2, 3, 3 can't be compared with the nine games.

22. Both Party A and Party B go from Party A to Party B at the same time. In the first three hours, Party A repaired the car 1 hour, so Party B was 4 kilometers ahead of Party A ... After another three hours, A was ahead of B 17 kilometers, so find the speed of both.

Answer: After 3 hours, A drove more than B: 4+ 17=2 1 km.

Every hour, a is greater than B: 2 1÷3=7 kilometers.

In the first three hours, if Party A does not repair the car, it can drive 2 1 km more than Party B..

A car repair 1 hour, 4 kilometers behind B.

Explain that A lost 2 1 hour in repairing cars1+4 = 25km.

The speed is 25 kilometers per hour.

The speed of B is 25-7 = 18km per hour.

23. The master and apprentice produced similar parts, and the land started two hours earlier than the master. After two hours of production, the master found that 20 parts were made less than the apprentice. They were born for another 2 hours. On the contrary, the master is more than the apprentice 10. How much does the owner produce per hour?

Answer: In the next two hours, the master was more than the apprentice: 10+20=30.

The master is more productive than the apprentice per hour: 30÷2= 15.

If the instruction starts at the same time, the first four hours,

The master is more productive than the apprentice: 15×4=60.

The master is 2 hours less than the apprentice and produces 20 less than the apprentice.

Explain that the master can produce in 2 hours: 20+60=80.

Main output per hour: 80÷2=40.

Apprentices produce per hour: 40- 15=25.

24. Party A produces 12 pieces per hour, and Party B produces 8 pieces per hour. Once, Party A and Party B produced the same number of parts at the same time. As a result, Party A completed the task five hours ahead of Party B. Q: How many parts did a * * * produce?

A: If A also produces according to the time of B, it can produce more than B:

5× 12=60

Every hour, A produces more than B: 12-8=4.

Production time of B: 60÷4= 15 hours.

The number of Party A and Party B is the same, which is 15×8= 120.

25. Write a number 1993 before 28 to get a multi-digit:19931993 ...1993199328, if this multi-digit can be

(9+3)-( 1-9)=2

8-2=6

6+2n≡0(mod 1 1)

The minimum n is 8, that is, write 8 1993 before 28, which is a number of 4×8+2=34.

26. Saw a cube-shaped wood block with a side length of 1m into three pieces along the horizontal direction, each piece into four pieces with any size, and each piece into five small pieces with any size, and * * * got 60 cuboids with various sizes, as shown in the figure below. What is the sum of the surface areas of these 60 cuboids?

The original cube has six outer surfaces, and the area of each surface is 1× 1 = 1 (square meter). No matter how many pieces are sawed later, the 6 square meters of the outer surface of these six pieces are always counted in the surface area of the subsequent small pieces of wood. Considering each saw, you will get two sides of 1 square meter, 65438+.

Now a * * * saw: 2+3+4 = 9 (knife),

The area given by a * * * is 2× 9 = 18 (square meter).

So the total surface area is 6+(2+3+4) × 2 = 24 (square meters).

In this problem, as long as you understand that each saw will get two surfaces of one square meter, then you can find out how many saws you have and the total surface area.

Write 30 as the sum of several consecutive natural numbers: 30 = 4+5+6+7+8 = 9+10+1.

Then writing 2002 as the sum of several natural numbers can be:

2002=_________________________

Thinking: We know that the summation formula of continuous N natural numbers is as follows:

Suppose the first number is a, then the nth number is a+n- 1, and their sum is (a+a+n- 1)*n/2, that is, (2a+n- 1)n/2.

So 2002 = (2a+n- 1) n/2.

(2a+n- 1)n = 4004 = 2 * 2 * 7 * 1 1 * 13

We find that 2a+n- 1 is even when n is odd; When n is even, 2a+n- 1 is odd. In other words, even a factor of 2 cannot be separated.

(1).n=4, then a=499, namely 2002 = 499+500+50 1+502.

(2).n=4*7=28, then a=58, which means 2002=58+59+60+...+84+85.

(3).n=4* 1 1=44, then a=24, that is, 2002=24+25+26+...+66+67.

(4).n=4* 13=52, then a= 13, that is, 2002 =13+14+15+...+63+64.

(5).n=4*7* 1 1=308, then a=- 147, and it is discarded.

When n takes a larger value, A no longer has a solution.

So there are four solutions to this problem.

28. What are the natural numbers with odd divisors less than 50?

Thinking: Any natural number can be expressed as the product of two natural numbers: n = a× b, where a, b and n are all natural numbers. (the prime number p can be expressed as: P = P× 1)

In other words, the divisors of natural numbers all appear in pairs. If the divisor is odd, there is only one case, that is, a = b, that is, n is a complete square number.

So the solution of this problem is: 1, 4,9,16,25,36,49.

29. There are three kinds of teacups, each of which is 5 yuan, 7 yuan and 9 yuan. Sharla Cheung bought several cups of each of the three kinds, and the quantities were not equal to each other. * * * spent 52 yuan. If the price of each kind of teacup is reduced to 2 yuan, then he only needs to spend 36 yuan. How many cups did he buy for 9 yuan?

Idea: If the price is reduced, 2 yuan will pay 52-36 = 16 yuan less, so a * * * bought 8 cups.

Suppose 9 yuan bought X, 7 yuan bought Y and 5 yuan bought (8-x-y).

Column equation: 9x+7y+5(8-x-y)=52.

The relation is 2x+y=6.

There are two possibilities: x = 1, y = 4；; x=2，y=2

Because the numbers are complementary and equal, 9 yuan has 1, 7 yuan has 4, and 5 yuan has 3.

30. At five o'clock in the World Cup group match in China, fans began to enter the stadium. Before entering the venue, there are already fans waiting in line, assuming that the number of fans arriving every minute after 5 o'clock is fixed. Then open six entrances, and after 40 minutes, there will be no fans waiting in line. If you open four entrances, there will be no fans waiting in line after 80 minutes. How many entrances must be opened at least to make no fans wait after 20 minutes?

Idea: Suppose that there are X people checking in at each port every minute, Y people queuing every minute, and A people are already queuing.

40*6x=40y+a

80*4x=80y+a

Subtract the two expressions to get y = 2x and a = 160x.

20 minutes: 20 * NX = 20y+A, and the substitution result is: 20nx = 40x+ 160x, n = 10.

Open 10 import.

3 1. There are three problems in math classroom exercises. The teacher writes one first and then writes one every five minutes. Regulation: (1) When the teacher writes a new question, if the original question has not been finished, every student must immediately stop and turn to a new one. (2) When a problem is finished, the teacher will turn to the adjacent unfinished problem without writing a new one. How many possibilities are there to complete these three questions in different order?

List five situations.

When Wang Ming returned to his home 800 meters away from home, his sister and a puppy ran towards him. Wang Ming walks 40 meters per minute, his sister runs 50 meters per minute, and his dog runs 160 meters per minute. After meeting Wang Ming, the puppy has been shuttling back and forth between Wang Ming and his sister at the same speed. How many meters did the puppy run when Wang Ming and his sister were 80 meters away?

Thinking: When the distance is 80 meters, a * * * left: (800-80) ÷ (40+50) = 8 minutes.

The puppy ran away: 8×160 =1280m.

33. A truck from A to B will arrive six hours late if it travels 60 kilometers per hour, and three hours early if it travels 80 kilometers per hour. How many kilometers is it between A and B?

Suppose it takes t hours to be on time, then

60*(t+6)=80*(t-3)

60*t+360=80*t-240

20t=600

t=30

Then the distance between Party A and Party B is 60 * (30+6) = 60 * 36 = 2 160km.

34. According to 10 balls with the same appearance, only one ball is defective. Please weigh it three times with the balance to find out the defective products.

Solution: Divide 10 balls into three, three and 1 4 groups, and express the four groups of balls and their weights as A, B, C and D respectively. Weigh Group A and Group B on two plates of the balance, and then

(1) If A=B, both A and B are genuine, and then called B and C. If B=C, it is obvious that the ball in D is defective; If B > C, the defective product is in c, and the defective product is lighter than the genuine product. Then take out two balls in C and weigh them, and you can draw a conclusion. If b < c, we can also draw a conclusion by imitating the situation of b > C.

(2) if A > B, then both c and d are credible. If b and c are called again, there can be no B=C or B < C (B > C). Why? If B=C, the defective product is in A, and the defective product is heavier than the genuine product. Then take out two balls in A and weigh them, and you can draw a conclusion. If b < c, you can also draw a conclusion before imitation.

(3) if a < b, similar to the case of a > b, we can draw a conclusion through analysis.

35. Figure (1) and Figure (2) are two large rectangles with the same shape and size. As shown in Figure (3), four small rectangles are placed in each large rectangle, and the diagonal area is empty. It is known that the length of a large rectangle is 6 cm wider than its width. Q: Figure (1) and Figure (2). How much bigger?

Analysis: The circumference of diagonal area in figure (1) is exactly equal to the circumference of large rectangle, and the circumference of diagonal area in figure (2) is obviously smaller than that of large rectangle. The difference between the two is 2? AB .

Seen from the vertical direction of figure (2), AB = A-CD The length of the rectangle in figure (2) is A+2B and the width is 2B+CD, so (A+2B)-(2B+CD) = A-CD = 6 (cm). Therefore, the circumference of the diagonal area in the figure is (1).

36. Find the area of trapezoidal ABCD in the figure, where BC=56 cm. (Unit: cm)

Answer: According to the trapezoid area formula, there are: S ladder = 1/2×(AB+CD)×BC, and since triangles ABC and CDE are isosceles right triangles, AB=BE, CD=CE, that is, S ladder = 1/2× (AB+CD )× BC =

37. There is a number: 1 1 1. . . . . . 1()222。 . . . . . 2. () is preceded by 100 1s, and () is followed by 100 2s, which can be divisible by 13. What's the number in ()?

1

38. There are some red and white balls. If you take out 1 red ball and 1 white ball at a time, when the red ball is finished, there are still 50 white balls left. If you take/kloc-0 red balls and 3 white balls at a time, and there are 50 red balls left when the white balls are taken away, how many red balls and white balls are there?

(3× 50+50) ÷ (3-1) =100-red

100+50= 150_ white

100+ 150=250

39, calculation:

Primitive formula

.

40, calculation:

Primitive formula

.

4 1. In the multiplication formula on the left, I, Xue, Shu and Le each represent four different numbers. If "music" stands for "9", then "I" stands for _ _, "number" stands for _ _, and "learning" stands for _ _.

Solution: "Le" stands for 9, and it can be deduced that "Xue" stands for 1 and "number" stands for 6; The product is ten digits, and the first two digits are all 6, so it can be inferred that "I" stands for 8.

Note: This question is a change in the form of a question written by Mr. Tan on May 25th 1992. To infer what numbers "music", "learning" and "number" represent respectively, we can get the result immediately by using the knowledge of "square mantissa property of natural numbers" and carry. It will be a bit difficult to push "I" a few more times.

Need to use the valuation method:

Because 800002 < 6616161< 900002.

So 8≤ I ≤9 Obviously, "I" can only be 8.

42. On a long wire, the yellow beetle climbs from the right end to the left end at a speed of centimeters per minute, while the red beetle and the blue beetle climb from the left end to the right end at a speed of centimeters per minute. When is the red beetle right between the blue beetle and the yellow beetle?

At 8: 30, the yellow beetle was at the left end1200-15 *10 =1050 (cm).

Suppose in another t minutes, the red beetle is between the blue beetle and the yellow beetle. At this time, the distance between the red beetle and the blue beetle is (13-1) t cm, and the distance between the red beetle and the yellow beetle is [1050-(13+15) t] cm, and the equation can be obtained: (. So 35 minutes from 8:30, that is, 9:05, the red beetle is right between the blue beetle and the yellow beetle.

43. A list of numbers, what is the sum of all the scores of these 239 numbers that are not integers?

Analysis: It will be difficult to find non-integers directly and then add them up. You can think of it another way, first add them all up and then subtract the whole number!

Is an integer, the molecule must be a multiple of 12, and in 1~239, the multiple of 12 is 12, 24, 36, 48...228.

So, the sum of all the scores is

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