Fifth grade Mathematical Olympiad (in this tutorial / is ÷ * is ×)
Inclusion and exclusion
1. There are 40 students in a class, 15 of them 18 people participated in the math group, 18 people participated in the aircraft model group, and 10 people participated in both groups. So how many people don’t participate in either group?
Explanation: The two groups *** have (15+18)-10=23 (people),
The number of people who did not participate is 40-23=17 (people)
p>
Answer: 17 people did not participate in either group.
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2. 45 students in a class took the final exam. After the results were announced, 10 students got full marks in mathematics, and 3 students got full marks in both mathematics and Chinese. , 29 people did not get full marks in these two subjects. So how many people have perfect scores in Chinese?
Answer: 45-29-13=9 (persons)
Answer: There are 9 people who got full marks in Chinese.
3. 50 students stood in a line facing the teacher. The teacher first asked everyone to count 1, 2, 3,..., 49, 50 from left to right; then asked the students who reported the number to be a multiple of 4 to turn back, and then asked the students to report the number to be a multiple of 6. Turn backwards. Question: How many students are there currently facing the teacher?
Solution: There are 12 50/4 quotients for multiples of 4, 8 50/6 quotients for multiples of 6, and 4 50/12 quotients for multiples of both 4 and 6.
The number of people who turned backward in multiples of 4 = 12, and the number of people who turned backward in multiples of 6 = 8 people, of which 4 people turned backward and 4 people turned back.
Number of students facing teachers = 50-12 = 38 (people)
Answer: There are still 38 students facing teachers.
4. At the entertainment party, 100 students drew lottery tickets with labels ranging from 1 to 100. The rules for awarding prizes based on lottery ticket tag numbers are as follows: (1) If the tag number is a multiple of 2, 2 pencils will be awarded; (2) If the tag number is a multiple of 3, 3 pencils will be awarded; (3) If the tag number is a multiple of 2, Prizes can be claimed repeatedly in multiples of 3; (4) All other tag numbers will be awarded 1 pencil. So how many pencils are there as prizes prepared by the fair for this event?
Solution: There are 50 100/2 quotients for multiples of 2, 33 100/3 quotients for multiples of 3, and 16 100/6 quotients for multiples of 2 and 3 people.
The *** preparation for receiving 2 branches (50-16)*2=68, the *** preparation for receiving 3 branches (33-16)*3=51, the *** preparation for repeated collection 16*(2+3)=80, prepare the rest 100-(533-16)*1=33
***Need 68+51+833=232 (supports)
*** p>
Answer: There are 232 prize pencils prepared by the entertainment club for this event.
5. There is a 180 cm long rope. Make a mark every 3 cm from one end and every 4 cm. Then cut the marked place. Ask how many pieces the rope was cut into?
Explanation: The mark of 3 cm: 180/3=60, and in the end, without marking, 60-1=59 pieces
The mark of 4 cm: 180/4=45, 45 -1=44, repeated marks: 180/12=15, 15-1=14, so there are actually 59+44-14=89 marks in the middle of the rope.
After cutting 89 times, it becomes 89+1=90 segments
Answer: The rope was cut into 90 segments.
6. There were many paintings on display at the Donghe Primary School Art Exhibition, 16 of which were not from sixth grade, and 15 were not from fifth grade. Now we know that there are 25 paintings by *** in the fifth and sixth grades. How many paintings by *** in other grades?
Explanation: There are 16 in Grades 1, 2, 3, 4, and 5, there are 15 in Grades 1, 2, 3, 4, and 6, and there are 25 in Grades 5 and 6.
So the total number is (16+15+25)/2=28 (frames), and the grades 1, 2, 3, and 4 have 28-25=3 (frames)
p>Answer: There are 3 paintings from other grades.
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7. There are a number of cards. Each card has a number written on it. It is a multiple of 3 or a multiple of 4. Among them, the card marked with a multiple of 3 There are 2/3, cards marked with multiples of 4 account for 3/4, and there are 15 cards marked with multiples of 12. So, how many cards are there in a day?
Answer: The multiples of 12 are 2/3+3/4-1=5/12, 15/(5/12)=36 (pieces)
Answer: These cards There are 36 cards in a ***.
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8. Among the natural numbers from 1 to 1000, they are neither divisible by 5 nor divisible by 7 How many are there?
Solution: There are 200 1000/5 quotients for multiples of 5, 142 1000/7 quotients for multiples of 7, and 28 1000/35 quotients for both 5 and 7.
There are 20142-28=314 multiples of 5 and 7.
1000-314=686
Answer: There are 686 numbers that are neither divisible by 5 nor divisible by 7.
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9. Students in Class 3 of fifth grade participate in extracurricular interest groups, and each student participates in at least one activity. Among them, 25 people participated in the nature interest group, 35 people participated in the art interest group, 27 people participated in the Chinese interest group, 12 people participated in the Chinese and art interest groups, and 8 people participated in the nature and art interest groups. Naturally, 9 people also participated in Chinese interest groups, and 4 people participated in all three subject interest groups: Chinese, art, and science. Find the number of students in this class.
Solution: 25+35+27-(8+12+9)+4=62 (persons)
Answer: The number of students in this class is 62.
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10. As shown in Figure 8-1, it is known that the areas of the three circles A, B, and C are all 30. A and B, B and C, The areas of the overlapping parts of A and C are 6, 8, and 5 respectively, and the total area covered by the three circles is 73. Find the area of ??the shaded part.
Solution: The area of ??the overlapping parts of A, B, and C = 73+ (6+8+5)-3*30=2
The area of ??the shaded part = 73-(6 +8+5)+2*2=58
Answer: The area of ??the shaded part is 58.
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-- Author: abc
-- Release time: 2004-12-12 15:45:02
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11. There are 46 students in a fourth grade class participating in 3 extracurricular activities. Among them, 24 people participated in the math group and 20 people participated in the Chinese group. The number of people participating in the art group was 3.5 times the number of people who participated in both the math group and the art group, and 7 times the number of people who participated in all three activities. The number of people who also participate in the Chinese group is twice the number of people who participate in all three groups. There are 10 people who participate in both the math group and the Chinese group. Find the number of people participating in the literary and art group.
Explanation: Suppose the number of people participating in the literary group is X, 24+2X-(X/305+2/7*X+10)+X/7=46, the solution is /p>
Answer: The number of people participating in the literary and art group is 21.
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-- Author: abc
-- Release time: 2004-12-12 15:45:43
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12. There are 100 books in the library, and those who borrow books need to sign on the books. It is known that among the 100 books, there are 33, 44 and 55 books signed by A, B and C respectively. Among them, 29 books are signed by both A and B, and 25 books are signed by both A and C. There are 36 books signed by B and C. Ask how many books in this batch of books are at least not borrowed by anyone from A, B, or C?
Explanation: The number of books that three people have read together is: A + B + C - (A, B + A, C + B, C) + A, B, C = 33 + 44 + 55 - (29 +25+36) + A, B, and C = 42 + A, B, and C. When A, B, and C are the largest, the three of them have read the most books, because A and C have only read 25 books together, which is more than A, B, and B and C* **have read very few of them, so A, B, and C have both read at most 25 books together.
The three of them have read at most 42+25=67 (books), and they have read at least 100-67=33 (books).
Answer : There are at least 33 books in this batch that have not been borrowed by any of A, B, and C.
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-- Author: abc
-- Release time: 2004-12-12 15:46:53
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13. As shown in Figure 8-2, five equally long line segments form a five-pointed star. If exactly 1994 points on each line segment are dyed red, then what is the minimum number of red points on this five-pointed star?
Solution: There are 5*1994=9970 red dots on the right side of the five lines. If a red dot is placed on all intersections, the red dots will be the least. These five lines have 10 intersections, so There are at least 9970-10=9960 red dots
Answer: There are at least 9960 red dots on this five-pointed star.
Related pictures for this topic are as follows:
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-- Author: abc
-- Release time: 2004-12-12 15:47:12
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14. A, B, and C water 100 pots of flowers at the same time. It is known that A has watered 78 pots, B has watered 68 pots, and C has watered 58 pots. How many pots of flowers have been watered by all three people?
Explanation: A and B must have 78+68-100=46 pots that have been watered together, and C has 100-58=42 that have not been watered, so all three people have watered at least 46 -42=4 (pots)
Answer: There are at least 4 pots of flowers that have been watered by 3 people.
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-- Author: abc
-- Release time: 2004-12-12 15:52:54
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15. A, B, and C are all reading the same story book. There are 100 stories in the book. Everyone starts with a certain story and reads forward in order. It is known that A has read 75 stories, B has read 60 stories, and C has read 52 stories. So what is the minimum number of stories that A, B, and C have read together?
Explanation: B and C*** have read at least 652-100=12 stories together. A must read these 12 stories no matter where he starts.
Answer: A, B, and C*** have read at least 12 stories together.
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-- Author: abc
-- Release time: 2004-12-12 15:53:43
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15. A, B, and C are all reading the same story book. There are 100 stories in the book. Everyone starts with a certain story and reads from there in order. It is known that A has read 75 stories, B has read 60 stories, and C has read 52 stories. So what is the minimum number of stories that A, B, and C have read together?
Explanation: B and C*** have read at least 652-100=12 stories together. A must read these 12 stories no matter where he starts.
Answer: A, B, and C*** have read at least 12 stories together.
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-- Author: cxcbz
-- Release time: 2004-12-13 21:53:23
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The following is a quote from abc’s speech at 2004-12-12 15:42:17:
8. Among the natural numbers from 1 to 1000, they cannot be divided by 5. , how many numbers are there that are not divisible by 7?
Solution: There are 200 1000/5 quotients for multiples of 5, 142 1000/7 quotients for multiples of 7, and 28 1000/35 quotients for both 5 and 7. There are 20142-28=314 multiples of 5 and 7.
1000-314=686
Answer: There are 686 numbers that are neither divisible by 5 nor divisible by 7.
The division in the question should be integer division.
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-- Author: cxcbz
-- Release time: 2004 -12-13 21:56:00
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The following is a quote from abc’s speech at 2004-12-12 15:45:02:
11. There are 46 students in a fourth grade class participating in 3 extracurricular activities. Among them, 24 people participated in the math group and 20 people participated in the Chinese group. The number of people participating in the art group was 3.5 times the number of people who participated in both the math group and the art group, and 7 times the number of people who participated in all three activities. The number of people who also participate in the Chinese group is twice the number of people who participate in all three groups. There are 10 people who participate in both the math group and the Chinese group. Find the number of people participating in the literary and art group.
Explanation: Suppose the number of people participating in the literary group is X, 24+2X-(X/305+2/7*X+10)+X/7=46, the solution is /p>
Answer: The number of people participating in the literary and art group is 21.
1. In the third class of fourth grade, 19 people subscribed to "Youth Digest", 24 people subscribed to "Learn and Play", and 13 people subscribed to both.
How many people subscribe to "Youth Digest" or "Learn and Play"?
2. There are 58 people learning piano in the kindergarten, 43 people learning painting, and 37 people learning both piano and painting. How many of them only learn piano and only learn painting
< p>People?3. Among the natural numbers from 1 to 100:
(1) How many numbers are there that are multiples of 2 and 3?
(2) How many numbers are multiples of 2 or 3?
(3) How many numbers are there that are multiples of 2 but not multiples of 3?
4. The statistics of the mid-term exam results of a certain class of mathematics and English are as follows: 12 people scored 100 points in English, 10 people scored 100 points in mathematics, two subjects
There were 3 people who got 100 points in both subjects, and 26 people who didn't get 100 points in both subjects. How many students are there in this class?
5. There are 50 people in the class, 32 people can ride bicycles, 21 people can roller skate, and 8 people can do both. How many people can’t do both?
6. There are 42 students in a class, 30 students participate in the sports team, and 25 students participate in the art team, and each student participates in at least one team.
How many people are participating in both teams in this class?
Answers to the test questions
1. There are 19 people in the third class of fourth grade who subscribe to "Youth Digest", 24 people who subscribe to "Learn and Play", and 13 people who subscribe to both. people. How many people subscribe to "Youth Digest"
or "Learn and Play"?
19 + 24—13 = 30 (people)
Answer: There are 30 people who subscribe to "Youth Digest" or "Learn and Play".
2. There are 58 people learning piano in the kindergarten, 43 people learning painting, and 37 people learning both piano and painting. How many of them only learn piano and only learn painting
< p>People?Number of people who only learn piano: 58—37 = 21 (people)
Number of people who only learn painting: 43—37 = 6 (people)
3. Among the natural numbers from 1 to 100:
(1) How many numbers are there that are multiples of 2 and multiples of 3?
If it is a multiple of 3 and a multiple of 2, it must be a multiple of 6
100÷6 = 16...4
Therefore, it is a multiple of 2 There are 16 multiples of 3
(2) How many numbers are multiples of 2 or 3?
100÷2 = 50, 100÷3 = 33...1
50 + 33—16 = 67 (pieces)
Therefore, it is 2 There are 67 numbers that are multiples or multiples of 3.
(3) How many numbers are there that are multiples of 2 but not multiples of 3?
50—16 = 34 (numbers)
Answer: There are 34 numbers that are multiples of 2 but not multiples of 3.
4. The statistics of the mid-term exam results of a certain class of mathematics and English are as follows: 12 people scored 100 points in English, 10 people scored 100 points in mathematics, two subjects
There were 3 people who got 100 points in both subjects, and 26 people who didn't get 100 points in both subjects. How many students are there in this class?
12 + 10—3 + 26 = 45 (people)
Answer: There are 45 students in this class.
5. There are 50 people in the class, 32 people can ride bicycles, 21 people can roller skate, and 8 people can do both. How many people can’t do both?
50—(30 + 21—8) = 7 (people)
Answer: There are 7 people who can’t do both.
6. There are 42 students in a class, 30 students participate in the sports team, and 25 students participate in the art team, and each student participates in at least one team.
How many people are participating in both teams in this class?
30 + 25—42 = 13 (people)
Answer: There are 13 people participating in both teams in this class.
A certain class of students took the entrance exam, and the number of students who got full marks was as follows: 20 people in mathematics, 20 people in Chinese, 20 people in English, 8 people got full marks in mathematics and English, and 7 people got full marks in mathematics and Chinese. Among people, there are 9 people who got perfect marks in Chinese and English, and 3 people who didn’t get perfect marks in three subjects. How many people can this class have at most? What is the minimum number of people?
The analysis and solution are shown in Figure 6. Students with perfect scores in mathematics, Chinese, and English are included in this class. Suppose there are y people in this class, represented by a rectangle. A, B, and C represent mathematics, People who get full marks in Chinese and English are known to have A∩C=8, A∩B=7, B∩C=9. A∩B∩C=X.
From the inclusion-exclusion principle, we have
Y=A+B+c-A∩B-A∩C-B∩C+A∩B∩C+3
That is, y=2220-7-8-9+x+3=39+x.
Next we examine how to find the maximum and minimum values ??of y.
It can be seen from y=39+x that when x takes the maximum value, y also takes the maximum value; when x takes the minimum value, y also takes the minimum value. The number of people who got full marks, so the number of them must not exceed the number of people who got full marks in two subjects, that is, x ≤ 7, x ≤ 8 and x ≤ 9, from which we get x ≤ 7. On the other hand, the students who got full marks in mathematics are Maybe no one got full marks in Chinese, which means there are no students who got full marks in all three subjects, so x≥0, so 0≤x≤7.
When x takes the maximum value 7, y has the maximum value 39+7=46. When x takes the minimum value 0, y has the minimum value 39+0=39.
Answer: This class has a maximum of 46 people and a minimum of 39 people.