Task 20 Basic level of the Unified State Exam

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

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

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

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

5) The stick is marked with transverse lines of red, yellow and green. If you cut a stick along the red lines, you will 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) The stick is marked with transverse lines of red, yellow and green. If you cut a stick along the red lines, you will get 10 pieces, if along the yellow lines - 8 pieces, if along the green - 8 pieces. How many pieces will you get if you cut a stick along the lines of all three colors?Answer : 24

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

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

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

Nicholas only had silver coins. After several visits to the exchange office, his silver coins became smaller, no gold coins appeared, 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 you get 3 silver and one copper;

· for 5 silver coins you get 3 gold and one copper.

Nicholas only had silver coins. After several visits to the exchange office, his silver coins became smaller, no gold coins appeared, but 100 copper coins appeared. How much did Nicholas's number of silver coins decrease? ? Answer: 20

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

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

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

10) At 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 you get 4 gold and one copper.

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

11) At 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 you get 5 gold and one copper.

Nicholas only had silver coins. After several visits to the exchange office, his silver coins became smaller, no gold coins appeared, 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: saffron milk caps and milk mushrooms. It is known that among any 28 mushrooms there is at least one saffron milk cap, and among any 24 mushrooms there is at least one milk mushroom. How many milk mushrooms are there in the basket?( According to the problem conditions: (50-28)+1=23 - there must be saffron milk caps. ( 50-24)+1=27 - there must be milk mushrooms. Answer: milk mushrooms in a basket 27 .)

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

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

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

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

17) The owner agreed with the workers that they would dig him a well under the following conditions: for the first meter he would pay them 4,200 rubles, and for each subsequent 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 him a well under the following conditions: for the first meter he would pay them 3,700 rubles, and for each subsequent 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 would dig a well under the following conditions: for the first meter he would pay them 3,500 rubles, and for each subsequent 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 him a well under the following conditions: for the first meter he would pay them 3,900 rubles, and for each subsequent meter he would pay 1,200 rubles more than for the previous one. How many rubles will the owner have to pay 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 at each subsequent lesson to increase the time spent on the treadmill by 7 minutes. In how many sessions will Andrey spend a total of 2 hours and 25 minutes on the treadmill if he follows the trainer’s advice? (5 )

22) The trainer advised Andrey to spend 22 minutes on the treadmill on the first day of classes, and at each subsequent lesson 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, will Andrey spend a total of 4 hours and 48 minutes on the treadmill? (8 )

23) There are 24 seats in the first row of the cinema, and each next row has 2 more seats than 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 regimen: on the first day he should take 3 drops, and on each subsequent day - 3 drops more than on the previous day. 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 bottles of medicine should a patient buy for the entire course of treatment, if each bottle contains 20 ml of medicine (which is 250 drops)?(2) amount 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 regimen: on the first day he should take 20 drops, and on each subsequent day - 3 drops more than the previous one. After 15 days of use, 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 daily reduces the dose by 3 drops until the dosage becomes less than 3 drops per day. How many bottles of medicine should a patient buy for the entire course of treatment, if each bottle contains 200 drops? (7 ) will drink 615 + 615 + 55 = 1285 ;1285: 200 = 6.4

26) In a household appliances store, the volume of sales of refrigerators is 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 volume began to decrease by 15 refrigerators each month relative 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 are drawn with a felt-tip pen. How many parts did the drawn lines divide the surface of the globe into?

( 13 22= 286)

28) On the surface of the globe, 17 parallels and 24 meridians were drawn with a felt-tip pen. How many parts did the drawn lines divide the surface of the globe into?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 equator.( 18 24 = 432)

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

30)What is the smallest number of consecutive numbers that must be taken so that their product is divisible by 9?(2)

31) The product of ten consecutive numbers is divided by 7. What can the remainder be equal to?(0) Among 10 consecutive numbers, one of them will definitely 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) A grasshopper jumps along a coordinate line in any direction for a unit segment per jump. How many different points are there on the coordinate line at which the grasshopper can end up after making exactly 6 jumps, starting from the origin? (the grasshopper may end up at points: −6, −4, −2, 0, 2, 4 and 6; only 7 points.)

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

34) A grasshopper jumps along a coordinate line in any direction for a unit segment per jump. How many different points are there on the coordinate line at which the grasshopper can end up 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 are there on the coordinate line at which the grasshopper can end up 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 whose modulus does not exceed eight. Thus, the grasshopper may end up at points: −8, −6, -2 ; −4, 0.2, 4, 6, 8 for a total of 9 points.

• A runner ran 250 m in 36 seconds. Find the average speed of the runner over the distance. 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 enclosure with a side of 12 meters on the property. Find the area of ​​the remaining part 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 lateral 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 the diameter of the first? Explain the solution to the problem. 20
• City N has 150,000 inhabitants. Among them, 15% are children and adolescents. Among adults, 45% do not work (pensioners, students, housewives, etc.). How many adult residents work? Describe the solution to the problem. 21
• A notepad in the store costs 22 rubles. How many rubles will a buyer pay for 70 notebooks if, when purchasing more than 50 notebooks, the store gives a 5% discount on the cost of the entire purchase? Write the solution to the problem. 20
• A meter of rope in a store costs 19 rubles. How many rubles will the buyer pay for 60 meters of rope, if when purchasing more than 50 meters of rope the store gives a 5% discount on the cost of the entire purchase? Write an algorithm for solving the problem. 22

In the basic level Unified State Examination there is a task for ingenuity under number 20. Most of these problems are solved quite simply. Let's distribute the tasks presented in the open Unified State Exam bank by type and give them a conventional name:

Let's look at the first four types.

Type 1.

The grasshopper jumps along a coordinate line in any direction a unit segment in one jump. The grasshopper begins to jump from the origin. How many different points are there on the coordinate line at which the grasshopper can end up after making exactly 11 jumps?

Solution . Note that the grasshopper in the end can only end up at points with odd coordinates,because the number of jumps he makes is odd.

The maximum grasshopper can be at pointsthe module of which 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;only 12 points.

Answer: 12

Problems for independent solution.

• The hare jumps along the coordinate line in any direction for a unit segment per jump. How many different points are there on the coordinate line at which the hare can end up 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 a unit segment. How many points are there at which a sparrow can end up after making 5 jumps?

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

Type 2.

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

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

Answer: 7

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

Solution . During the day, the well increases by 300 − 30 = 270 m. By the beginning of the eleventh working day, oil workers will drill 2,700 meters. During the eleventh working day, 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 to a level of 2 meters. The construction pump continuously pumps out water, lowering its level by 20 cm per hour. Subsoil water, on the contrary, increases the water level in the pit by 5 cm per hour. How many hours of pump operation will it take for the water level in the pit to drop to 80 cm?

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

Answer: 8

Problem 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 from 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?

Solution . At 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 o’clock, there will be 30 liters of water in the tank. At 6 p.m., 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 up a tree 4 m in a day, and slides 1 m up a tree during the night. The height of the tree is 13 m. How many days will it take for the snail to crawl to the top of the tree for the first time?

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

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

Type 3.

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

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

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

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

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

So Sasha lives on the fifth floor.

Answer: 5

Problem 2. All entrances of the house have the same number of floors, and each floor has the same number of apartments. In this case, the number of floors in the house is greater than the number of apartments on the floor, the number of apartments on the floor is greater than the number of entrances, and the number of entrances is more than one. How many floors are in the building if there are 110 apartments in total?

Solution. 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 lived in the eighth entrance in apartment No. 468, but forgot to say the floor. Approaching the house, Petya discovered that the house was 12 stories high. What floor does Sasha live on? (On all floors the number of apartments is the same; apartment numbers in the building begin with one.)

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

• Katya and her friend Lena went to visit Sveta, knowing that she lived in apartment 364 in the 6th entrance. When they approached the house, they discovered that the house was 16 stories high. What floor does Sveta live on? (On all floors the number of apartments is the same, apartment numbers begin with one).

• Igor decided to do 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 was nine stories high. What floor does Kolya live on? (The number of apartments on all floors is the same; apartment numbers 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. In this case, the number of floors in the house is greater than the number of apartments on the floor, the number of apartments on the floor is greater than the number of entrances, and the number of entrances is more than one. How many floors are in the building if there are 170 apartments in total?

Type 4.

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

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

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

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

Solution . 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 has increased by 50, let’s create and solve a 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 you get 4 silver and one copper;for 6 silver coins you get 4 gold and one copper.Nicholas only had silver coins. After visiting the exchange office, his silver coins became smaller, no gold coins appeared, but 35 copper coins appeared. By how much did Nicholas's number of silver coins decrease?

• At the exchange office you can perform one of two operations:for2 goldseget coins3 silvereand one copper;for5 get silver coins3

During the day, a snail crawls up a tree `4` m, and during the night it slides down `2` m. The height of the tree is `14` m. How many days will it take for the snail to crawl to the top of the tree for the first time? Source: Unified State Exam 2017. Mathematics. Basic level. 30 training options exam papers. Ed. Yashchenko I.V. / M.: 2017. - 160 pp. ( option No. 9)

Solution:

If you calculate how many meters a snail moves in exactly one day and divide the height of the tree by this number, the answer will be incorrect. Because the snail could reach 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. Therefore, this approach cannot be used. We will solve the problem gradually.

On the first day The snail crawled `4` meters. This height is less than the height of the tree, so it turns out that the snail did not reach the given height on the first day. During the night it descended by `2` meters, which means that it rose during the day to a height of `4−2=2` meters.

On the second day the snail crawled to a height of: `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.

On 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.

Solving Olympiad problems in elementary school

Caterpillar movement.

We cannot ignore an interesting ancient problem:
On Sunday at 6 a.m., the caterpillar decided to climb to the top of a 12-foot-tall 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 a caterpillar can climb in a day.
4 – 3 = 1 (ft).
The answer is that the caterpillar will rise 12 feet in 12 days. But this answer is incorrect, because it is not necessary to take into account the last crawl of the caterpillar.
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 that students understand that when the caterpillar reaches the top, at that point the counting of time stops. She has reached her goal and it no longer matters whether she goes down or not.
For the first task, it is better to choose an option where the height of the pillar is small, and with the help of a drawing you can trace the entire path of the caterpillar.
A snail climbs up a 10-meter-high pole. During the day it rises by 5 m, and at night it drops by 4 m. How many days will it take the snail to reach the top of the column?

The picture 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 for solving:
1. 5 – 4= 1(m) – the snail rises in a day.
2. 10 – 5 = 5 (m) – the snail needs to pass without the last lift.
3. 5: 1 = 5 (days) – the caterpillar will need to travel 5 m.
4. 5 + 1 =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 came across several problems that can be considered variants of this problem.
1. A snail crawls along a pole 20 m high. Every day it rises 2 m. And every night it falls 1 m. In how many days will it reach the top?
2. The height of the pillar is 10 m. An ant climbs it 4 m up during the day, and falls 2 m down during the night. How many days will it take the ant to crawl to the top of the pillar?
3. A snail is crawling along a vertical pole 6 m high. During the day it rises 4 m, during the night it falls 3 m. How many days will it take her to reach the top?
4. A snail is climbing up a pole 100 m high. During the day she climbs the pillar 5 m, during the night she falls 4 m. How many days will it take her to climb to the top of the pillar?
5. Every day a snail crawls 7 m up the wall and at night goes down 4 m. 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 the trunk of a linden tree. At night it rises 4m up, and during the day it drops 2m down. On the eighth night the worm reached the top of the tree. How tall is the linden 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 climbed 4 m, and during the night it descended 3 m. When will it reach the top?
8. Petya, taking a step per second, walks as follows: 2 steps forward, one step back. How many seconds will it take him to walk 20 steps?
9. A caterpillar crawls along the trunk of an apple tree. In the first hour it rose 10 cm, in the second it dropped 4 cm, in the third it rose again, etc. How many cm will the caterpillar rise in 11 hours?
10. The gnome Confusion 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 Confused 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., it crawled to a height of 5 m, and during the night it descended to 2 meters. On what day and time will it be at a height of 9 meters?
12. Vitya watches a spider that rises on a cobweb to the top of a tree 12 m high. Moreover, it rises like this: during the day it rises 5 meters, and at night in a dream it drops 4 m. How many days will it take the spider to rise to the top?
13. A snail is moving along a vertical column 6 m high. During the day she rises 4 m, at night in her sleep she slides 3 m. How many days will it take her to get to the top?