• A runner ran 250 meters in 36 seconds. Find the average speed of the runner over the course. Give your answer in kilometers per hour and explain the algorithm for solving the problem. 13
• The plot has the shape of a rectangle with sides of 30 meters and 20 meters. The owner fenced off a square aviary with a side of 12 meters on the site. Find the area of ​​the rest of the plot. Give your answer in square meters and write an algorithm for solving the problem. 15
• The angle at the vertex opposite the base of an isosceles triangle is 30°. The side of the triangle is 11. Find the area of ​​this triangle. Write down the solution to the problem. 11
• In a cylindrical vessel, the liquid level reaches 48 cm. At what height will the liquid level be if it is poured into a second cylindrical vessel, the diameter of which is 2 times greater than the diameter of the first? Explain the solution to the problem. 20
• City N has 150,000 inhabitants. Among them, 15% are children and teenagers. Among adults, 45% do not work (pensioners, students, housewives, etc.). How many adult residents work? Describe the solution to the problem. 21
• Notepad in the store costs 22 rubles. How many rubles will the buyer pay for 70 notebooks if, when buying more than 50 notebooks, the store makes a 5% discount on the cost of the entire purchase? Write a solution to the problem. 20
• A meter of rope in the store costs 19 rubles. How many rubles will the buyer pay for 60 meters of rope if, when buying more than 50 meters of rope, the store makes a 5% discount on the cost of the entire purchase? Write an algorithm for solving the problem. 22

Solving Olympiad problems in elementary school

Caterpillar movement.

It is impossible to ignore an interesting old problem:
On Sunday at 6:00 am, the caterpillar decided to climb to the top of a 12-foot tree. During the day, she managed to rise 4 feet, and at night, in her sleep, she slid 3 feet. When will the caterpillar reach the top?
Let's find out how many feet the caterpillar manages to climb in a day.
4 - 3 = 1 (ft).
The answer is asked that the caterpillar will rise 12 feet in 12 days. But this answer is incorrect, since the last crawl of the caterpillar should not be taken into account.
12 - 4 = 8 (ft).
8 days have passed. The caterpillar rose 8 feet. On the ninth day it will rise 12 feet and by 6 pm on Monday it will reach the top.
Answer: next Monday in a week by 6 pm it will reach the top.
It is important for the students to understand that when the caterpillar reaches the top, at that point the counting of time stops. She has reached her goal and it doesn't matter if she goes down or not.
For the first task, it is better to choose the option where the height of the column is small, and with the help of the picture you can trace the entire path of the caterpillar.
A snail climbs up a pole 10 meters high. During the day it rises by 5 m, and at night it drops by 4 m. In how many days will the snail reach the top of the pillar?

The drawing shows that it will take 6 days before the snail reaches the top of the tree. It is also necessary to write down the arithmetic method of the solution:
1. 5 - 4 \u003d 1 (m) - a snail rises in a day.
2. 10 - 5 = 5 (m) - you need to pass the snail without the last rise.
3. 5: 1 \u003d 5 (days) - the caterpillar will need to go 5 m.
4. 5 + 1 \u003d 6 (days) - the caterpillar needs to climb to the top of the tree, because on the last sixth day the caterpillar will immediately rise 5 m and reach the top.
In the literature, I met several problems that can be considered variants of this problem.
1. A snail crawls along a pillar 20 m high. Every day it rises 2 m. And every night it descends 1 m. In how many days will it reach the top?
2. The height of the pillar is 10 m. An ant rises 4 m up on it during the day, and falls 2 m down during the night. How many days will it take for the ant to crawl to the top of the pillar?
3. A snail crawls along a vertical pillar 6 m high. During the day it rises 4 m, during the night it falls 3 m. How many days will it take for it to reach the top?
4. A snail climbs up a pole 100 meters high. During the day she rises 5 m along the pillar, during the night she descends 4 m. How many days will it take for her to climb to the top of the pillar?
5. Every day a snail crawls 7 meters up the wall and descends 4 meters down at night. On what day will it, starting from the ground, reach the roof of a house whose height is 19 m?
6. A worm crawls along a linden trunk. At night it rises 4m up, and during the day it falls 2m down. On the eighth night the worm reached the top of the tree. How tall is the lime tree?
7. At 6 o'clock in the morning on Monday, the caterpillar began to crawl up a tree 12 m high. During the day (until 18 o'clock) it rose 4 m, and during the night it descended 3 m. When will it reach the top?
8. Petya, taking a step in a second, goes as follows: 2 steps forward, step back. How many seconds does it take him to walk 20 steps?
9. A caterpillar crawls along the trunk of an apple tree. In the first hour it rose by 10 cm, in the second it dropped by 4 cm, in the third it rose again, etc. How many cm will the caterpillar rise in 11 hours?
10. Dwarf Putalka goes to the cage with the tiger. Every time he takes 2 steps forward, the tiger growls and the dwarf takes a step back. How long will it take him to reach the cage if there are 5 steps to it, and Putalka takes one step in 1 second?
11. At 6 o'clock on Sunday, the caterpillar began to crawl up the tree. During the day, that is, until 6 p.m., she crawled to a height of 5 m, and during the night she descended to 2 meters. On what day and hour will she be at a height of 9 meters?
12. Vitya watches a spider, which climbs on a cobweb to the top of a tree 12 meters high. Moreover, it rises like this: it rises 5 meters in a day, and drops 4 meters in a dream at night. How many days will a spider rise to the top?
13. A snail moves along a vertical column 6 m high. During the day she climbs 4 m, at night she slides down 3 m in her sleep. How many days will she need to get to the top?

In the USE of the basic level there is a task for ingenuity under No. 20. Most of these problems are fairly easy to solve. Let's distribute the tasks presented in the open USE bank by type and give them a conditional name:

Consider the first four types.

Type 1.

The grasshopper jumps along the coordinate line in any direction by a single segment per jump. The grasshopper starts jumping from the origin. How many different points on the coordinate line are there that the grasshopper can reach after making exactly 11 jumps?

Decision . Note that the grasshopper eventually can only appear at points with odd coordinates,as the number of jumps he makes is odd.

The maximum grasshopper can be at the pointswhose modulus does not exceed eleven. Thus, the grasshopper can end up at points: -11, -9, -7, -5, -3, -1, 1, 3, 5, 7, 9 and 11;total 12 points.

Answer: 12

Tasks for independent solution.

• The hare jumps along the coordinate line in any direction for a unit segment per jump. How many different points on the coordinate line are there that the hare can reach after making exactly 6 jumps, starting from the origin?

• The sparrow jumps along a straight line in any direction. The length of the jump is equal to one segment. How many points are there that a sparrow can reach after making 5 jumps?

• The grasshopper jumps along the coordinate line in any direction for a unit segment per jump. How many different points on the coordinate line are there that the grasshopper can reach after making exactly 12 jumps, starting from the origin?

Type 2.

Task 1.A snail crawls up a tree 4 m in a day, and slides 3 m in a night. The height of a tree is 10 m. In how many days will a snail crawl to the top of a tree for the first time?

Decision . During the day, the snail crawls up to 4 meters, and during the night it slides down 3 meters. In total, she will crawl a meter in a day. In six days it will rise to a height of six meters. And the next day, she will already be at the top of the tree.

Answer: 7

Task 2. An oil company is drilling a well for oil production, which, according to geological exploration, lies at a depth of 3 km. During the working day, drillers go 300 meters deep, but during the night the well “silts up” again, that is, it is filled with soil by 30 meters. How many working days will oil workers drill a well to the depth of oil?

Decision . During the day, the well increases by 300 - 30 = 270 m. By the beginning of the eleventh working day, the oilmen will drill 2700 meters. During the eleventh working day, the oil workers will drill another 300 meters, that is, they will reach a depth of 3 km.

Answer: 11

Task 3. As a result of the flood, the pit was filled with water up to a level of 2 meters. The construction pump continuously pumps out water, lowering its level by 20 cm per hour. Groundwater, on the contrary, raises the water level in the pit by 5 cm per hour. How many hours of pump operation will the water level in the pit drop to 80 cm?

Decision . In an hour, the water level in the pit decreases by 20 - 5 \u003d 15 cm. It is necessary to pump out 2 100 - 80 \u003d 120 cm of water. Consequently, the water level in the pit will drop to 80 cm in 120: 15 = 8 hours.

Answer: 8

Task 4. A full bucket of water with a volume of 8 liters is poured into a tank with a volume of 38 liters every hour, starting at 12 o'clock. But there is a small gap in the bottom of the tank, and 3 liters flow out of it in an hour. At what point in time (in hours) will the tank be completely filled.

Decision . By the end of each hour, the volume of water in the tank increases by 8 − 3 = 5 liters. After 6 hours, that is, at 18 hours, there will be 30 liters of water in the tank. At 6 pm, 8 liters of water will be added to the tank and the volume of water in the tank will become 38 liters.

Answer: 18

Decide for yourself.

• A snail crawls 4 m up a tree in a day, and slides 1 m in a night. The height of a tree is 13 m. How many days does it take for a snail to crawl to the top of a tree for the first time?

• A snail crawls 4 m up a tree in a day, and slides 2 m in a night. The height of a tree is 26 m. How many days does it take for a snail to crawl to the top of a tree for the first time?

• A snail crawls 3 m up a tree in a day, and slides 2 m in a night. The height of a tree is 28 m. How many days will it take for a snail to crawl to the top of a tree for the first time?

Type 3.

Task 1. Sasha invited Petya to visit, saying that he lives in the seventh entrance in apartment No. 462, but he forgot to say the floor. Approaching the house, Petya discovered that the house had seven floors. What floor does Sasha live on? (On all floors, the number of apartments is the same, the numbers of apartments in the building start from one.)

Decision . Since there are at least 462 apartments in the first 7 entrances, there are at least 462 in each entrance: 7 = 66 apartments. Therefore, on each of the 7 floors in the entrance there are at least 9 apartments.

Let there be 9 apartments on each landing. Then there are only 9 · 7 · 7 = 441 apartments in the first seven entrances, and apartment 462 will be in the eighth entrance, which contradicts the condition.

Let there be 10 apartments on each site. Then in the first seven entrances 10 · 7 · 7 = 490 apartments, and in the first six - 420. Consequently, apartment 462 is in the seventh entrance. She is the 42nd in a row, since there are 10 apartments on the floor, she is located on the fifth floor.

If there were 11 apartments on each site, then there would be 11 · 7 · 6 = 462 apartments in the first six entrances, that is, 462 apartments in the sixth entrance, which contradicts the condition.

So Sasha lives on the fifth floor.

Answer: 5

Task 2. All entrances of the house have the same number of floors, and each floor has the same number of apartments. At the same time, the number of floors in the house is greater than the number of apartments per floor, the number of apartments per floor is greater than the number of entrances, and the number of entrances is more than one. How many floors are there in a building if there are 110 apartments in total?

Decision. The number of apartments, floors and entrances can only be an integer.

Note that the number 110 is divisible by 2, 5 and 11. Therefore, the house should have 2 entrances, 5 apartments and 11 floors.

Answer: 11

Decide for yourself.

• Sasha invited Petya to visit, saying that he lives in the eighth entrance in apartment No. 468, but he forgot to say the floor. Approaching the house, Petya discovered that the house had 12 floors. What floor does Sasha live on? (On all floors, the number of apartments is the same, the numbers of apartments in the building start from one.)

• Sasha invited Petya to visit, saying that he lives in the twelfth entrance in apartment No. 465, but he forgot to say the floor. Approaching the house, Petya discovered that the house had five floors. What floor does Sasha live on? (On all floors, the number of apartments is the same, the numbers of apartments in the building start from one.)

• Katya and her friend Lena went to visit Sveta, knowing that she lives in the 364th apartment in the 6th entrance. Approaching the house, they found that the house had 16 floors. What floor does Sveta live on? (On all floors the number of apartments is the same, apartment numbers start from one).

• Igor decided to do his homework in mathematics with Kolya and went to his house, knowing that he lives next to the house, in the fifth entrance and in apartment 206. Approaching the house, Igor discovered that it had nine floors. What floor does Kolya live on? (On all floors the number of apartments is the same, the numbers of apartments in the building start from one).

• All entrances of the house have the same number of floors, and each floor has the same number of apartments. At the same time, the number of floors in the house is greater than the number of apartments per floor, the number of apartments per floor is greater than the number of entrances, and the number of entrances is more than one. How many floors are there in a building if there are 170 apartments in total?

Type 4.

In the exchange office, you can perform one of two operations:

• for 2 gold coins get 3 silver and one copper;

• for 5 silver coins, get 3 gold and one copper.

Nicholas had only silver coins. After several visits to the exchange office, he had fewer silver coins, no gold coins, but 50 copper coins appeared. By how much did Nicholas's number of silver coins decrease?

Decision . Let Nikolai first perform x operations of the second type, and then y operations of the first type. Since after several operations there were no gold coins left, andthe number of copper coins increased by 50, we compose and solve the system of equations:

Then there were 3y -5x = 90 - 100 = -10 silver coins, that is, 10 less.

Answer: 10

Decide for yourself.

• for 3 gold coins get 4 silver and one copper;for 6 silver coins get 4 gold and one copper.Nicholas had only silver coins. After visiting the exchange office, he had fewer silver coins, no gold coins, but 35 copper coins appeared. By how much did Nicholas's number of silver coins decrease?

• In the exchange office, you can perform one of two operations:behind2 goldeget coins3 silvereand one copper;behind5 get silver coins3

A snail crawls up a tree `4` m in a day, and slides down `2` m in a night. The height of a tree is `14` m. How many days will it take for a snail to crawl to the top of the tree for the first time? Source: USE 2017. Mathematics. A basic level of. 30 training options for examination papers. Ed. Yashchenko I.V. / M .: 2017. - 160 p. ( option number 9)

Decision:

If you calculate how many meters the snail moves in exactly one day and divide the height of the tree by this number, the answer will be wrong. Because the snail could get to the top of the tree during the day, and then crawl down during the night. In addition, if you solve the problem about a snail and a tree in this way, it turns out that at some point the snail crawls higher than the top of the tree is. Therefore, this approach cannot be used. We will solve the problem gradually.

First day the snail crawled up to `4` meters. This height is less than the height of the tree, it turns out that the snail did not reach the given height on the first day. During the night, she went down to `2` meters, which means that she rose in a day to a height of `4−2=2` meters.

On the second day the snail crawled to a height: `2+4=6` meters and descended at night to `2` meters: `6-2=4` meters.

For the third day:
rose to a height of `4+4=8` meters;
descended to a height of `8-2=6` meters.

For the fourth day:
rose to a height of `6+4=10` meters;
descended to a height of `10-2=8` meters.

For the fifth day:
rose to a height of `8+4=12` meters;
descended to a height of `12-2=10` meters.

For the sixth day:
rose to a height of `10+4=14` meters.

Thus, for the first time the snail will crawl to a height of `14` meters on the sixth day.

Task 20 Basic level of the exam

1) A snail crawls 4 m up a tree in a day, and slides 1 m in a night. The height of a tree is 13 m. In how many days will a snail crawl to the top of a tree for the first time?(4-1 \u003d 3, the morning of the 4th day will be at a height of 9m, and 4m will crawl in a day. Answer: 4 )

2) A snail crawls 4 m up a tree in a day, and slides 3 m in a night. The height of a tree is 10 m. In how many days will a snail crawl to the top of a tree for the first time?Answer: 7

3) A snail climbs 3 m up a tree in a day, and descends 2 m in a night. The height of a tree is 10 m. How many days will it take for a snail to climb to the top of a tree?Answer: 8

4) On the stick are marked transverse lines of red, yellow and green. If you cut a stick along the red lines, you get 15 pieces, if along the yellow lines - 5 pieces, and if along the green lines - 7 pieces. How many pieces do you get if you cut a stick along the lines of all three colors? ( If you cut a stick along red lines, you get 15 pieces, therefore, lines - 14. If you saw a stick along yellow lines - 5 pieces, therefore, lines - 4. If you saw it along green lines - 7 pieces, lines - 6. Total lines: 14 + 4 + 6 = 24 lines. Answer: 25 )

5) On the stick are marked transverse lines of red, yellow and green. If you saw the stick along the red lines, you get 5 pieces, if along the yellow lines - 7 pieces, and if along the green lines - 11 pieces. How many pieces will you get if you cut a stick along the lines of all three colors?Answer : 21

6) Transverse lines of red, yellow and green are marked on the stick. If you cut a stick along the red lines, you get 10 pieces, if along the yellow lines - 8 pieces, if along the green lines - 8 pieces. How many pieces will you get if you cut a stick along the lines of all three colors?Answer : 24

7) In the exchange office, you can perform one of two operations:

for 2 gold coins get 3 silver and one copper;

for 5 silver coins, get 3 gold and one copper.

Nicholas had only silver coins. After several visits to the exchange office, he had fewer silver coins, no gold coins, but 50 copper coins appeared. By how much did Nicholas's number of silver coins decrease? Answer: 10

8) At the exchange office, you can perform one of two operations:

· for 2 gold coins get 3 silver and one copper;

· For 5 silver coins, get 3 gold and one copper.

Nicholas had only silver coins. After several visits to the exchange office, he had fewer silver coins, no gold coins, but 100 copper coins appeared. By how much did Nicholas's number of silver coins decrease? ? Answer: 20

9) In the exchange office, you can perform one of two operations:

2) for 6 silver coins, get 4 gold and one copper.

Nikola had only silver coins. After visiting the exchange office, he had fewer silver coins, no gold coins, but 35 copper coins appeared. By how much did Nikola's number of silver coins decrease?Answer: 10

10) In the exchange office, you can perform one of two operations:

1) for 3 gold coins get 4 silver and one copper;

2) for 7 silver coins, get 4 gold and one copper.

Nikola had only silver coins. After visiting the exchange office, he had fewer silver coins, no gold coins, but 42 copper coins appeared. By how much did Nikola's number of silver coins decrease?Answer: 30

11) In the exchange office, you can perform one of two operations:

1) for 4 gold coins get 5 silver and one copper;

2) for 8 silver coins, get 5 gold and one copper.

Nicholas had only silver coins. After several visits to the exchange office, he had fewer silver coins, no gold coins, but 45 copper coins appeared. By how much did Nicholas's number of silver coins decrease?Answer: 35

12) There are 50 mushrooms in the basket: mushrooms and milk mushrooms. It is known that among any 28 mushrooms there is at least one camelina, and among any 24 mushrooms at least one mushroom. How many mushrooms are in the basket?( According to the condition of the problem: (50-28)+1=23 - must be redheads. ( 50-24)+1=27 - must be gruzdey. Answer: mushrooms in the basket 27 .)

13) There are 40 mushrooms in the basket: mushrooms and milk mushrooms. It is known that among any 17 mushrooms there is at least one camelina, and among any 25 mushrooms at least one mushroom. How many mushrooms are in the basket? (According to the condition of the problem: (40-17)+1=24 - must be redheads. ( 40-25)+1=16 24 .)

14) the basket contains 30 mushrooms: mushrooms and milk mushrooms. It is known that among any 12 mushrooms there is at least one camelina, and among any 20 mushrooms at least one mushroom. How many mushrooms are in the basket?(According to the condition of the problem: (30-12)+1=19 - must be redheads. ( 30-20)+1=11 - must be gruzdey. Answer: saffron milk caps in a basket 19 .)

15) There are 45 mushrooms in the basket: mushrooms and milk mushrooms. It is known that among any 23 mushrooms there is at least one camelina, and among any 24 mushrooms at least one mushroom. How many mushrooms are in the basket?( According to the condition of the problem: (45-23)+1=23 - must be redheads. ( 45-24)+1=22 - must be gruzdey. Answer: saffron milk caps in a basket 23 .)

16) There are 25 mushrooms in the basket: mushrooms and milk mushrooms. It is known that among any 11 mushrooms there is at least one camelina, and among any 16 mushrooms at least one mushroom. How many mushrooms are in the basket? (Since among any 11 mushrooms at least one is a mushroom, then there are no more than 10 mushrooms. Since among any 16 mushrooms at least one is a mushroom, then there are no more than 15 mushrooms. And since there are 25 mushrooms in the basket, there are exactly 10 mushrooms, and Ryzhikov exactly Answer:15.

17) The owner agreed with the workers that they would dig a well for him on the following terms: for the first meter he would pay them 4,200 rubles, and for each next meter - 1,300 rubles more than for the previous one. How much money will the owner have to pay the workers if they dig a well 11 meters deep?(Answer: 117700)

18) The owner agreed with the workers that they would dig a well for him on the following conditions: for the first meter he would pay them 3,700 rubles, and for each next meter - 1,700 rubles more than for the previous one. How much money will the owner have to pay the workers if they dig a well 8 meters deep? (77200 )

19) The owner agreed with the workers that they are digging a well on the following terms: for the first meter he will pay them 3,500 rubles, and for each next meter - 1,600 rubles more than for the previous one. How much money will the owner have to pay the workers if they dig a well 9 meters deep? (89100 )

20) The owner agreed with the workers that they would dig a well for him on the following conditions: for the first meter he would pay them 3,900 rubles, and for each next meter he would pay 1,200 rubles more than for the previous one. How many rubles will the owner have to pay to the workers if they dig a well 6 meters deep?(41400)

21) The trainer advised Andrey to spend 15 minutes on the treadmill on the first day of classes, and on each next lesson to increase the time spent on the treadmill by 7 minutes. How many sessions will Andrey spend on the treadmill for a total of 2 hours and 25 minutes if he follows the advice of the trainer? (5 )

22) The coach advised Andrey to spend 22 minutes on the treadmill on the first day of training, and on each next session to increase the time spent on the treadmill by 4 minutes until it reaches 60 minutes, and then continue to train for 60 minutes every day. In how many sessions, starting from the first one, Andrey will spend 4 hours and 48 minutes on the treadmill? (8 )

23) There are 24 seats in the first row of the cinema hall, and in each next row there are 2 more than in the previous one. How many seats are in the eighth row? (38 )

24) The doctor prescribed the patient to take the medicine according to the following scheme: on the first day he should take 3 drops, and on each next day - 3 drops more than on the previous one. Having taken 30 drops, he drinks 30 drops of the medicine for another 3 days, and then reduces the intake by 3 drops daily. How many vials of medicine should a patient buy for the entire course of treatment if each contains 20 ml of medicine (which is 250 drops)?(2) the sum of an arithmetic progression with the first term equal to 3, the difference equal to 3 and the last term equal to 30.; 165 + 90 + 135 = 390 drops; 3+ 3( n -1)=30; n =10 and 27- 3( n -1)=3; n =9

25) The doctor prescribed the patient to take the medicine according to the following scheme: on the first day he should take 20 drops, and on each next day - 3 drops more than on the previous one. After 15 days of taking the patient takes a break of 3 days and continues to take the medicine according to the reverse scheme: on the 19th day he takes the same number of drops as on the 15th day, and then reduces the dose by 3 drops daily until the dosage becomes less than 3 drops per day. How many vials of medicine should a patient buy for the entire course of treatment if each contains 200 drops? (7 ) drinks 615 + 615 + 55 = 1285; 1285: 200 = 6.4

26) In a household appliance store, sales of refrigerators are seasonal. In January, 10 refrigerators were sold, and in the next three months, 10 refrigerators were sold. Since May, sales have increased by 15 units compared to the previous month. Since September, sales began to decrease by 15 refrigerators every month compared to the previous month. How many refrigerators did the store sell in a year?(360) (5*10+2*25+2*40+2*55+70=360

27) On the surface of the globe, 12 parallels and 22 meridians were drawn with a felt-tip pen. Into how many parts did the drawn lines divide the surface of the globe?

( 13 22= 286)

28) On the surface of the globe, 17 parallels and 24 meridians were drawn with a felt-tip pen. Into how many parts did the drawn lines divide the surface of the globe?A meridian is an arc of a circle connecting the North and South Poles. A parallel is a circle lying in a plane parallel to the plane of the equator.( 18 24 = 432)

29) What is the smallest number of consecutive numbers you need to take so that their product is divisible by 7?(2) If the condition of the problem sounded like this: “What is the smallest number of consecutive numbers you need to take so that their product guaranteed divisible by 7? Then it would be necessary to take seven consecutive numbers.

30) What is the smallest number of consecutive numbers you need to take so that their product is divisible by 9?(2)

31) The product of ten consecutive numbers is divided by 7. What can be the remainder?(0) Among 10 consecutive numbers, one of them will necessarily be divisible by 7, so the product of these numbers is a multiple of seven. Therefore, the remainder when divided by 7 is zero.

32) The grasshopper jumps along the coordinate line in any direction for a unit segment per jump. How many different points on the coordinate line are there that the grasshopper can reach after making exactly 6 jumps, starting from the origin? (the grasshopper can end up at points: -6, -4, -2, 0, 2, 4 and 6; only 7 points.)

33) The grasshopper jumps along the coordinate line in any direction for a unit segment per jump. How many different points on the coordinate line are there that the grasshopper can reach after making exactly 12 jumps, starting from the origin? (the grasshopper can end up at points: -12, -10, -8, -6, -4, -2, 0, 2, 4, 6, 8, 10 and 12; total 13 points.)

34) The grasshopper jumps along the coordinate line in any direction for a unit segment per jump. How many different points on the coordinate line are there that the grasshopper can reach after making exactly 11 jumps, starting from the origin?(may appear at points: -11, -9, -7, -5, -3, -1, 1, 3, 5, 7, 9 and 11; 12 points in total.)

35) The grasshopper jumps along the coordinate line in any direction for a unit segment per jump. How many different points on the coordinate line are there that the grasshopper can reach after making exactly 8 jumps, starting from the origin?

Note that the grasshopper can only end up at points with even coordinates, since the number of jumps it makes is even. The maximum grasshopper can be at points, the module of which does not exceed eight. Thus, the grasshopper can end up at the points: -8, -6, -2 ; −4, 0.2 , 4, 6, 8 total 9 points.