2.11112222 pieces are arranged in a long square. The number of pieces in each row is more than that in each vertical column 1. How many pieces are there in each row of this long square?
Party A and Party B walk along the railway at the same speed. It took 8 seconds for a train to pass by Party A, and 7 seconds to pass by Party B after leaving Party A for 5 minutes. How many minutes has it taken since Party B met the train?
Xiaoming solved a problem between 7 and 8 o'clock. At first, the minute hand and the hour hand were just in a straight line. When the problem is solved, both hands coincide. When did Xiao Ming start to solve the problem? How long did it take Xiao Ming to solve this problem?
5. Two oil drums, A and B, each contain oil15kg. The salesman sold 65,438+04 kilograms. Later, the salesman poured part of the oil of the one with more oil left into bucket B, which doubled the oil in bucket B; Then pour a part from barrel B to barrel A, so that the oil in barrel A doubles. At this time, the oil in barrel A is exactly three times that in barrel B. Q: How many kilograms of oil did the salesman sell from each of the two barrels?
6. For a project, it takes 12 hours for Party A to do it alone, and 18 hours for Party B to do it alone. If Party A does 65,438+0 hours first, then Party B takes over 65,438+0 hours, and then Party A takes over 65,438+0 hours.
7, school cleaning, assign a number of people to clean the glass, two of them each wipe 4 pieces, and the rest each wipe 5 pieces, leaving 12 pieces; If everyone wipes 6 pieces, it's just finished. How many people clean the glass and how many pieces of glass are there?
8. The ball fell into a cylindrical barrel filled with water. The diameter of the ball is 12 cm, and the diameter of the barrel bottom is 60 cm. Two thirds of the ball is immersed in water (below). After the ball fell into the water, how many centimeters did the water level in the bucket rise?
9. Randomly place 10 points in a regular triangle with a side length of1m.. Prove that the distance between at least two points is not greater than 1/3m.
10. The number of people who won the first prize in a math contest was originally 10, and the number of people who won the second prize was 20. Now the last four people in the first prize are transferred to the second prize, so that the average score of students who won the second prize will increase by 1 point, and the score of the first prize will increase by 3 points. The average score of the first prize is more and less than that of the second prize?
Problem solving process:
1. Add three numbers after 865 to form a six-digit number, which can be divisible by 3, 4 and 5 respectively, and the number should be as small as possible. (divisible by number)
Analysis: suppose that the six digits after the number is added up are. Because this six-digit number can be divisible by 3, 4 and 5 respectively, the following three conditions should be met:
First, the sum of numbers (8+6+5+A+B+C) is a multiple of 3.
Second, the last two digits are multiples of 4.
Third, the last number c is 0 or 5.
Solution: Let the required six digits be. According to the meaning of the question:, and c can only take 0 or 5
The single digit of a number divisible by 4 cannot be 5.
∴c can only get 0. So b can only take one of 0, 2, 4, 6 and 8.
∵, and (8+6+5) divided by 3 is 1,
∴ A+B divided by 3 equals 2.
In order to satisfy the meaning of "the smaller the value, the better", take a=0 and b=2.
The required six digits are 865020.
2.11112222 pieces are arranged in a long square. The number of pieces in each row is more than that in each vertical column 1. How many pieces are there in each row of this long square?
analyse
The number of chess pieces in each horizontal row is more than that in each vertical row 1.
The number of rows and columns should be two adjacent natural numbers.
Solution:11112222 = 3333× 3334.
The answer is 3334.
Party A and Party B walk along the railway at the same speed. It took 8 seconds for a train to pass by Party A, and 7 seconds to pass by Party B after leaving Party A for 5 minutes. How many minutes has it taken since Party B met the train? (Travel problems)
Analysis requires that when A and B meet in a few minutes, they must find out the relationship between their distance and their speed, which is related to the movement of the train. The distance between a and b can only be found by the movement of the train. The running time of the train is known, so it is necessary to find out its speed, at least the proportional relationship between it and the speeds of A and B. Because this problem is difficult, it is explained in detail as follows:
① Find the relationship between the train speed v and the speed v of Party A and Party B, and let the train length be L, then:
(i) It takes 8 seconds for the train to pass through A, and this process is a catching-up problem: therefore, L = (V train -V people) × 8; ( 1)
(2) It takes 7 seconds for the train to pass through B, and this process is a meeting problem: therefore, l=(V train +V people) ×7. (2)
From (1) and (2): 8(V car -V people) = 7 (V car +V people),
Therefore, V car = 15V people.
(2) The distance between locomotive encounter A and locomotive encounter B is:
(8+5×60)×(V car +V people) =308× 16V people =4928V people.
③ Find out the distance between Party A and Party B when the locomotive meets Party B. ..
It takes (8+5×60) seconds for the locomotive to meet with B. Therefore, when the locomotive meets with B, the distance between A and B is: 4928V people -2 (8+5× 60) V people =43 12V people.
(4) How many minutes will A and B meet?
Xiaoming solved a problem between 7 and 8 o'clock. At first, the minute hand and the hour hand were just in a straight line. When the problem is solved, both hands coincide. When did Xiao Ming start to solve the problem? How long did it take Xiao Ming to solve this problem? (Clock face strokes)
To analyze how long it took Xiao Ming to solve this problem, we must first find out when Xiao Ming began to solve the problem and when he finished it.
(1) When Xiao Ming started to solve the problem:
Because when Xiao Ming started to solve the problem, the minute hand and the hour hand were just on a straight line, that is, the included angle between the minute hand and the hour hand was 180. At this time, the minute hand is 60×( 180÷360)=30 squares behind the hour hand, and at 7 o'clock, the minute hand is 5× 7 = 35 squares behind the hour hand.
(2) At the end of Xiao Ming's problem solving:
Because at the end of Xiao Ming's problem solving, the two needles just coincide, so we should divide our hands from 7 o'clock to this moment.
In this way, we can calculate the time it takes Xiao Ming to solve the problem.
Solution: Find the moment when Xiaoming started to solve the problem;
Let Xiao Ming come back after solving the problem;
5. Two oil drums, A and B, each contain oil15kg. The salesman sold 65,438+04 kilograms. Later, the salesman poured part of the oil of the one with more oil left into bucket B, which doubled the oil in bucket B; Then pour a part from barrel B to barrel A, so that the oil in barrel A doubles. At this time, the oil in barrel A is exactly three times that in barrel B. Q: How many kilograms of oil did the salesman sell from each of the two barrels?
The key to solve the problem is to find out how many kilograms of oil are in drums A and B. It is known that "drums A and B each contain 15 kilograms of oil, and the salesman sold 14 kilograms". We can find the remaining oil of two oil drums A and B * *14 = 65438.
After finding out the last kilograms of oil in the two oil drums A and B, we can find out the kilograms of oil in the two oil drums A and B before the oil is poured into the two oil drums B through backward calculation and drawing, so as to find out how many kilograms are sold from the two oil drums.
Solution: ① How many kilograms are left in two barrels of oil?
15× 2-14 =16 (kg)
② How many kilograms of oil is left in barrel B? 16 ÷ (3+ 1) = 4 (kg)
③ How many kilograms is left in a barrel of oil? 4× 3 = 12 (kg)
Draw a picture with reverse deduction as follows:
How many kilograms is a barrel of oil sold? 15- 1 1 = 4 (kg)
⑤ How many kilograms of oil does barrel B sell? 15-5 = 10 (kg)
Answer: Barrel A sells 4 kg of oil and barrel B sells 10 kg of oil.
6. For a project, it takes 12 hours for Party A to do it alone, and 18 hours for Party B to do it alone. If Party A does 1 hour first, then Party B takes over 1 hour, and then Party A takes over 1 hour, (engineering problem)
How many hours does it take to analyze the requirements? It is conceivable to redistribute these hours: A do 1 hour, B do 1 hour, which is equivalent to 1 hour of cooperation, that is, every 2 hours is equivalent to 1 hour of cooperation. In this way, first roughly calculate how many times this two hours has been carried out, and the remaining problems will be solved.
7, school cleaning, assign a number of people to clean the glass, two of them each wipe 4 pieces, and the rest each wipe 5 pieces, leaving 12 pieces; If everyone wipes 6 pieces, it's just finished. How many people clean the glass and how many pieces of glass are there? (profit and loss problem)
Solution: If two people each wipe 4 pieces, the rest wipe 5 pieces, and the rest 12 pieces. It can be seen that if everyone rubs 5 pieces, the remaining 12-(5-4)×2 = 10 pieces, and everyone rubs 6 pieces, just right. It can be seen that each person can wipe one more piece of the remaining 10 piece.
A: There are 10 people cleaning the glass, and there are 60 pieces of glass.
8. The ball fell into a cylindrical barrel filled with water. The diameter of the ball is 12 cm, and the diameter of the barrel bottom is 60 cm. Two thirds of the ball is immersed in water (below). After the ball fell into the water, how many centimeters did the water level in the bucket rise?
Solution: the volume of the ball:
9. Randomly place 10 points in a regular triangle with a side length of1m.. Proof: at least 2 points.
Divide each side of a regular triangle into three equal parts, connect the points as shown in the figure, and divide the regular triangle into three parts.
At least two points fall in the same small triangle (or side). In the same small regular triangle.
10. The number of people who won the first prize in a math contest was originally 10, and the number of people who won the second prize was 20. Now the last four people in the first prize are transferred to the second prize, so that the average score of students who won the second prize will increase by 1 point, and the score of the first prize will increase by 3 points. The average score of the first prize is more and less than that of the second prize?
Solution: according to the meaning of the problem
Average score of the top six = average score of the top ten +3.
This shows that when calculating the average score of the top ten people, the top six people * * * add 3×6= 18 (points) to make up for the scores of the last four people, so the average score of the last four people is less than that of the top ten people.
18÷4=4.5 (point).
When the last four people were adjusted to the second prize, the second prize * * * was 20+4=24 (people), with an increase per capita.