Classic Mathematical Olympiad questions and answers for primary school studentsThe following are some classic Olympiad math questions and answers for primary school students:
** 1. Profits and losses **
1. [Question: How many children are there in total? Each child has 10 watermelons, 9 less, 8 more than 7.]
- ** Answer **: According to the formula,"First add more or less, then divide by the distribution to find the difference." If there are 9 fewer and 7 more, the first sum is [9 + 7=16]; if each person is allocated 10 and each person is allocated 8, the difference is [10 - 8 = 2]. Finally, divide by the number of people.
** 2. Train itinerary **
1. [Title: There are two trains, one is 102 meters long and travels 20 meters per second, and the other is 120 meters long and travels 17 meters per second.] The two trains were traveling in the same direction. How many seconds would it take for the first train to catch up with the second train and for the two trains to leave?
- ** Answer **: From catching up to leaving, the distance of catching up is the sum of the two train leaders, which is <102 + 120 = 222> m. The speed difference is <20 - 17 = 3> m/s. Time = distance difference/speed difference, so <222> div3 = 74> s.
2. [Title: A person's walking speed is 2 meters per second.] A train came from behind and overtook him by 10 seconds. It was known that the train was 90 meters long. Ask for the speed of the train.
- [Answer: The distance that a train can travel is 90 meters, the time is 10 seconds, and the speed of a person is 2 meters per second.] Assuming that the train speed is <v> m/s, according to distance = speed x time,<(v - 2)<times10 = 90>, the solution is <v = 11> m/s.
3. [Title: There are two trains moving in the same direction at the same time. After 12 seconds, the fast train overtakes the slow train.] The express train traveled 18 meters per second, while the slow train traveled 10 meters per second. If the two trains were traveling in the same direction, the express train would overtake the slow train in 9 seconds. Find the length of the two trains.
- ** Answer **: When moving head to head, the distance that the express train passes over the slow train is the length of the express train, which is [(18 - 10)<times12 = 96] m; when the rear of the train is moving head to head, the distance that the express train passes over the slow train is the length of the slow train, which is [(18 - 10)<times9 = 72] m.
4. [Title: A train needs 40 seconds to cross a 440-meter bridge, and 30 seconds to cross a 310-meter tunnel at the same speed.] What was the speed and length of this train?
- ** Answer **: Let the train's length be x meters and its speed be v meters per second. According to the relationship between the distance and time through the bridge and tunnel, the equation system can be obtained.(\begin{cases} 40v = 440 + x30v = 310 + x\end{cases}}), subtract the second equation from the first equation to get\(10v = 130'), the solution is [v = 13') m/s. Substitute [v = 13')] into [30v = 310 + x], the solution is [390 = 310+x].
5. ** Title **: In order to measure the speed and length of the train, Xiaoying used a watch to record the time it took for the train to pass in front of her was 15 seconds, and Xiaomin used another watch to record the time it took for the train to pass the first telephone pole from the front to the second telephone pole at the back was 20 seconds. It was known that the distance between two utility poles was 100 meters. Find the length and speed of the train.
- ** Answer **: Let the train's length be x meters and its speed be v meters per second. The train passes in front of us, the distance it travels is the length of the train (x = 15v\); The distance from the front of the car to the back of the car is the length of the car plus the distance between the poles.(x + 100 = 20v_), substitute_(x = 15v_) into_(x + 100 = 20v_) to get_(15v +100 = 20v_). The solution is_(v = 20_) m/s, the length of the car is_(x = 15, time20 = 300_) m, and the speed is_(20,time3.6 = 72_) km/h.
6. [Title: A train needs 40 seconds to cross a 530-meter bridge, and 30 seconds to cross a 380-meter cave at the same speed.] How many meters is the speed and length of this train?
- ** Answer **: Let the train's length be x meters and its speed be v meters per second. We can get the equation set of\(\begin{cases} 40v = 530 + x'30v = 380 + x 'end{cases}). Subtracting the first equation from the second equation gives\(10v = 150), which is solved as\(v = 15) m/s. Substituting\(v = 15) into\(30v = 380 + x) gives\(450 = 380 + x), which is solved as\(x = 70) m.
7. [Title: The two of them walked along the trail beside the railway line, starting from two places and traveling at the same speed.] A train came, and it took 10 seconds for the entire train to pass by A. Three minutes later, B encountered a train. The entire train passed by B in just nine seconds. How long after the train left B did the two meet?
- ** Answer **: Let the speed of the train be <v> 1>, the speed of the person be <v> 2>, and the length of the train be <L>. According to the relationship between the train and A, L=(v1v2)<times10>, and the relationship between the train and B, L=(v1v2)<times9>, we can get <(v1v2)<times10>=(v1v2)<times9>, which is simplified to <(v1v2 = 19v2>). 3 minutes after the train leaves A, it meets B. The distance between A and B and the distance between A and B is (180v1 - 180v2). The time after the train leaves B is set to be <t>, then <(v2 + v2)t= 180v1 - 180v2>. Substituting <v1 = 19v2> into <2v2t = 180Times19v2 - 180v2>, the solution is <t = 1710> seconds = 28.5 minutes.
8. [Question: Two trains, one 120 meters long and traveling at 20 meters per second, the other 160 meters long and traveling at 15 meters per second. The two trains are moving in opposite directions. How many seconds does it take for the two trains to meet and leave?]
- ** Answer **: The two trains are moving in opposite directions. The distance from the front of the train to the back of the train is the sum of the two train leaders, which is the sum of the speed of the train. The speed is the sum of the speed of the train. The time is the sum of the speed of the train, so the speed of the train is the sum of the speed of the train.
9. [Title: A person's walking speed is 2 meters per second.] A train came from behind and passed him in ten seconds. Given that the train is 90 meters long, find the speed of the train.
- ** Answer **: The distance the train can travel over a person is 90 meters, the time is 10 seconds, and the speed of a person is 2 meters per second. Assuming that the train speed is <v> m/s, according to distance = speed x time,<(v - 2)<times10 = 90>, the solution is <v = 11> m/s.
10. [Title: A and B are walking along the railway at the same speed. A train takes 8 seconds to pass by A and meets B 5 minutes later. It only takes 7 seconds to pass by B. How many minutes will it take for B and B to meet each other after B meets the train?]
- ** Answer **: Let the speed of the train be <v> 1>, the speed of the person be <v> 2>, and the length of the train be <L>. According to the relationship between the train and A, L=(v1v2)<times8>, and the relationship between the train and B, L=(v1v2)<times7>, we can get <8(v1v2)=7(v1v2)>, which is simplified to <v1v2>. The train will meet B 5 minutes after leaving A. The distance between A and B and the distance between A and B in these 5 minutes is (300v1 - 300v2), from the time B meets the train, the time when A and B meet is set to be (t), then ((v2.1 + v2.1)t = 300v1 - 300v2.1). Substituting <<v1>= 15v2> into <<2v2.1>= 300.1> time15v2 - 300v2>, the solution is <<t = 2100>> seconds = 35 minutes.
** 3. Average problem **
1. [Question: Cai Chen's average score in the final exam was 89 in politics, Chinese, mathematics, English, and biology.] The average score for politics and mathematics was 91.5. The average score for Chinese and English was 84. The average score for politics and English was 86 points, and English was 10 points higher than Chinese. He asked Cai Chen what his marks for each subject should be for this exam?
- ** Answer **: If the Chinese test score is x, then the English test score is x + 10. According to the average score of Chinese and English, the equation could be written as [(x + x + 10)<div2 = 84], and the solution was [x = 79]. The English score was [89]. The average score for politics and English is 86 points, so the political score is 86/89 = 83/. The average score for politics and mathematics is 91.5, so the math score is 91.5'time2 - 83 = 100 '. The biology score was [89 'times5-(79 + 89+83 + 100)=94].
2. [Title: A and B cotton fields, with an average yield of 185 catties of seed cotton per mu] The A cotton field had 5 mu, with an average yield of 203 catties of seed cotton per mu. The B cotton field had an average yield of 170 catties of seed cotton per mu. How many mu was the B cotton field?
- ** Answer **: Suppose B has a cotton field of x mu. According to the total yield equal-countable equation, the solution is x = 6 mu.
3. [Question: Given that the sum of eight consecutive odd numbers is 144, find these eight consecutive odd numbers.]
- ** Answer **: If the first odd number is <x>, then the eight consecutive odd numbers are <x,x + 2,x+4,x + 6,x+8,x + 10,x+12,x + 14>. According to the sum of the 144 equations, x+(x + 2)+(x+4)+(x + 6)+(x+8)+(x + 10)+(x+12)+(x + 14)=144, the solution was x = 7. The eight consecutive odd numbers were 7, 9, 11, 13, 15, 17, 19, and 21.
4. ** Question **: A sugar is 8.8 yuan per kilogram, B sugar is 7.2 yuan per kilogram. How much B sugar should be mixed with 5 kilograms of A sugar to make the price of each kilogram of sugar 8.2 yuan?
- ** Answer **: Assuming that you need [B] sugar in a kilogram. According to the constant countable equation of the total price before and after mixing, the solution is [8.8'times5 +7.2x=(5 + x)'times8.2].
5. [Question: Buy five sheep from the canteen. Take out two sheep each time and weigh them. There are ten different weights (kg):47, 50, 51, 52, 53, 54, 55, 57, 58, 59.] How many kilograms did each of the five sheep weigh?
- ** Answer **: Let the weight of the five sheep from light to heavy be a, b, c, d, e. Each sheep was called together with the other four sheep once, so each sheep was called four times. Then, 4(a + b + c + d+e)=(47 + 50+51+52+53+54+55+57+58+59)=536, so 4 (a + b + c + d+e = 134). The lightest combination is a + b = 47, and the heaviest combination is d + e = 59, so c = 134- 47 + 59 =28. Because a + c = 50, then a = 50 - 28 = 22, b = 47 - 22 = 25, c+e = 58, then e = 58 - 28 = 30, d = 59 - 30 = 29.
** 4. Arithmetic Sequence Problem **
1. [Question: Below is a string of numbers arranged according to a pattern: 2, 5, 8, 11, 14…] What was the 1995th item?
- ** Answer **: This is an arithmetic sequence. The first term is <a1 = 2>, and the tolerance is <d = 3>. According to the general term formula of arithmetic sequence,<a_n= a1 +(n - 1)d>, the 1995th term is <a_{1995}=2+(1995 - 1)><times3 = 2+ 1994><times3 = 5984>.
2. [Question: Among the natural numbers starting from 1, what is the 100th number that cannot be completely divided by 3?]
- ** Answer **: 1, 2, 3, 4, 5, 6, 7,.... Starting from 1, form a group of three numbers. The first two numbers in each group cannot be divided by 3. If there are two numbers in a group, there will be a group of 100 numbers. Each group has three numbers, and there will be a total of 50. Then the 100th number that cannot be divided by 3 is 150 - 1 = 149.
** 5. The problem of cows eating grass **
1. [Title: A patch of green grass that grows at a constant rate every day.] This patch of grass could feed twenty-seven cows for six days or twenty-three cows for nine days. How many days could this patch of grass feed twenty-one cows?
- ** Answer **: Suppose a cow eats one unit of grass a day. First, find the amount of grass that grows every day. Twenty-seven cows ate for six days, eating a total of 27,000 units of grass; Twenty-three cows ate for nine days, eating a total of 20,000 units of grass. (207- 162 = 45 units. This is the amount of grass that grows in a day, so the grass grows in a day. The original amount of grass is in units of [162 - 6'times15 = 72]. Assuming that 21 cows eat for [x] days, the equation [21x = 72+15x] can be solved for [x = 12] days.
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