Collection for preparing for the exam ( a basic level of)

Job prototype #20

1. 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?

2. On the stick are marked transverse lines of red, yellow and Green colour. 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?

3. There are 40 mushrooms in the basket: mushrooms and milk mushrooms. It is known that among any 17 mushrooms there is at least one mushroom, and among any 25 mushrooms - at least one mushroom. How many mushrooms are in the basket?

4. 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?

5. 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?

6. 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?

7. 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?

8. There are 30 mushrooms in the basket: 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?

9.

1) for 2 gold coins get 3 silver and one copper;

2) 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?

10. In a home 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?

11. 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?

12. The list of tasks of the quiz consisted of 25 questions. For each correct answer, the student received 7 points, for an incorrect answer, 10 points were deducted from him, and if there was no answer, they were given 0 points. How many correct answers were given by the student who scored 42 points, if it is known that he was wrong at least once?

13. 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?

14. 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 100 copper coins appeared. By how much did Nicholas's number of silver coins decrease?

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?

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

17. 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?

18. 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?

19. Sasha invited Petya to visit, saying that he lives in the tenth entrance in apartment No. 333, but he forgot to say the floor. Approaching the house, Petya discovered that the house had nine 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.)

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

1) for 5 gold coins get 6 silver and one copper;

2) for 8 silver coins, get 6 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 55 copper coins appeared. By how much did Nicholas's number of silver coins decrease?

21. The trainer advised Andrey to spend 22 minutes on the treadmill on the first day of training, and on each subsequent session, 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?

22. Every second a bacterium divides into two new bacteria. It is known that the entire volume of one glass of bacteria is filled in 1 hour. In how many seconds will the glass be half filled with bacteria?

23. The restaurant menu has 6 types of salads, 3 types of first courses, 5 types of second courses and 4 types of dessert. How many salad, first, second and dessert lunch options can diners at this restaurant choose?

24. 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?

25. In how many ways can two identical red dice, three identical green dice and one blue dice be lined up?

26. The product of ten consecutive numbers is divided by 7. What can be the remainder?

27. 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?

28. The list of tasks of the quiz consisted of 33 questions. For each correct answer, the student received 7 points, for an incorrect answer, 11 points were deducted from him, and if there was no answer, they were given 0 points. How many correct answers were given by the student who scored 84 points, if it is known that he was wrong at least once?

29. On the surface of the globe, 13 parallels and 25 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.

30. There are four gas stations on the ring road: A, B, C and D. The distance between A and B is 35 km, between A and C is 20 km, between C and D is 20 km, between D and A is 30 km (all distances measured along the ring road in the shortest direction). Find the distance between B and C. Give your answer in kilometers.

31. 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 numbering of apartments in the building starts from one.)

32. There are 30 mushrooms in the basket: 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?

33. The owner agreed with the workers that they were digging a well on the following terms: for the first meter he would 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?

34. Sasha invited Petya to visit, saying that he lives in the tenth entrance in apartment No. 333, but he forgot to say the floor. Approaching the house, Petya discovered that the house had nine floors. What floor does Sasha live on? (The number of apartments on each floor is the same, the numbers of apartments in the building start from one.)

35. 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)?

36. The rectangle is divided into four smaller rectangles by two straight cuts. The perimeters of three of them, starting from the top left and proceeding clockwise, are 24, 28 and 16. Find the perimeter of the fourth rectangle.

37. There are four gas stations on the ring road: A, B, C and D. The distance between A and B is 50 km, between A and C is 30 km, between C and D is 25 km, between D and A is 45 km (all distances measured along the ring road along the shortest arc).

Find the distance (in kilometers) between B and C.

38. 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?

39. A group of tourists overcame a mountain pass. They covered the first kilometer of the ascent in 50 minutes, and each next kilometer passed 15 minutes longer than the previous one. The last kilometer before the summit was completed in 95 minutes. After a ten-minute rest at the top, the tourists began their descent, which was more gentle. The first kilometer after the summit was covered in an hour, and each next one is 10 minutes faster than the previous one. How many hours did the group spend on the entire route if the last kilometer of the descent was covered in 10 minutes.

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

For 3 gold coins, get 4 silver and one copper;

For 7 silver coins, get 4 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 42 copper coins appeared. By how much did Nicholas's number of silver coins decrease?

41. 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 will you get if you cut a stick along the lines of all three colors?

42. 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?

43. 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?

44. 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.

45. 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?

46. What is the smallest number of consecutive numbers that must be taken so that their product is divisible by 7?

47. 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?

48. 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?

49. On the globe, 17 parallels (including the equator) and 24 meridians were drawn with a felt-tip pen. Into how many parts do the lines drawn divide the surface of the globe?

50. 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?

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.

Answers to the prototype task number 20

4. Answer: 117700

14. Answer: 77200

19. Answer: 3599

29. Answer: 89100

Yakovleva Natalya Sergeevna
Job title: mathematic teacher
Educational institution: MKOU "Buninskaya secondary school"
Locality: Bunino village, Solntsevsky district, Kursk region
Material name: article
Subject:"Methods for solving tasks No. 20 USE in mathematics basic level"
Publication date: 05.03.2018
Chapter: complete education

Single State exam is on this moment the only

form of final certification of graduates high school. And receiving

certificate of secondary education is not possible without successful passing the exam By

mathematics. Mathematics is not only important subject, But

and quite complex. Mathematical skills are far

not all children, but from successful delivery exam depends on their future fate.

Graduate teachers ask the question again and again: “How can I help

student in preparation for the exam and successfully pass it? In order to

the graduate received a certificate enough to pass the basic level of mathematics. A

the success of the exam is directly related to how the teacher speaks

methodology for solving various problems. I bring to your attention examples

solving task No. 20 mathematics basic level FIPI 2018 under

edited by M.V. Yashchenko.

1 .On the tape on opposite sides of the middle, two stripes are marked: blue and

red. If the tape is cut along the red strip, then one part will be 5 cm

longer than the other. If the tape is cut along the blue strip, then one part will be on

15 cm longer than the other. Find the distance between red and blue

stripes.

Solution:

Let a cm be the distance from the left end of the ribbon to the blue stripe, in cm

distance from the right end of the tape to the red stripe, cm distance

between the stripes. It is known that if the tape is cut along the red strip, then

one part is 5 cm longer than the other, that is, a + c - b \u003d 5. If cut by

blue stripe, then one part will be 15 cm longer than the other, which means that in + s -

a=15. We add two equalities term by term: a + c-b + c + c-a \u003d 20, 2c \u003d 20, c \u003d 10.

2 . Arithmetic mean 6 different natural numbers equals 8. On

how much should the largest of these numbers be increased so that the average

arithmetic has increased by 1.

Solution: Since the arithmetic mean of 6 natural numbers is 8,

so the sum of these numbers is 8*6=48. Arithmetic mean of numbers

increased by 1 and became equal to 9, and the number of numbers did not change, which means that

the sum of the numbers becomes equal to 9*6=54. To find how much one has increased

from the numbers, you need to find the difference 54-48=6.

3. The cells of the 6x5 table are painted in black and white. Pairs of neighbors

26 cells of different colors, pairs of neighboring black cells 6. How many pairs

neighboring white cells.

Solution:

In each horizontal line, 5 pairs of neighboring cells are formed, which means that

there will be 5*5=25 pairs of neighboring cells horizontally. Vertically

4 pairs of neighboring cells are formed, that is, a total of pairs of neighboring cells along

vertical will be 4*6=24. In total, 24+25=49 pairs of neighboring cells are formed. From

there are 26 pairs of different colors, 6 pairs of black, therefore there will be 49 pairs of white

26-6 = 17 par.

Answer: 17.

4. On the counter of the flower shop are three vases of roses: white, blue and

red. There are 15 roses to the left of the red vase and 12 to the right of the blue vase.

roses. There are 22 roses in total in vases. How many roses are in the white vase?

Solution: Let x roses be in a white vase, y roses be in a blue vase, z roses be in

red. According to the condition of the problem, there are 22 roses in vases, that is, x + y + z = 22. It is known

that to the left of the red vase, that is, there are 15 roses in the blue and white ones, which means that x + y \u003d 15. A

to the right of the blue vase, that is, there are 12 roses in the white and red vase, so x + z = 12.

Got:

Let's add the 2nd and 3rd equalities term by term: x+y+x+ z=27 or 22+x=27, x=5.

5 .Masha and the Bear ate 160 cookies and a jar of jam, starting and finishing

simultaneously. At first Masha ate jam, and Bear biscuits, but in some

moment they changed. The bear eats both 3 times faster than Masha.

How many cookies did the Bear eat if they ate the jam equally.

Solution: Since Masha and the Bear started eating cookies and jam

at the same time and finished at the same time, and they ate one product, and then

another, and according to the condition of the problem, the Bear eats both 3 times faster than

Masha means the Bear devoured food 9 times faster than Masha. Then let x

Masha ate the cookies, and the Bear ate 9 cookies. It is known that they ate everything

160 cookies. We get: x + 9x \u003d 160, 10x \u003d 160, x \u003d 16, which means that the bear ate

16*9=144 cookies.

6. Several consecutive pages fell out of the book. Last number

pages before the dropped sheets 352. Number of the first page after

of the dropped sheets is written with the same numbers, but in a different order.

How many sheets fell out?

Solution: Let x sheets fall out, then the number of dropped pages is 2x, then

is an even number. The number of the first dropped page is 353. The difference between

the number of the first dropped page and the first page after the dropped

must be an even number, which means that the number after the dropped sheets will be

523. Then the number of dropped sheets will be equal to (523-353):2=85.

7. About natural numbers A, B, C it is known that each of them is greater than 5, but

less than 9. Think of a natural number, then multiply by A, add B and

subtracted C. We got 164. What number was conceived?

Solution: Let x be a natural number, then Ax+B-C=164, Ax=

164 - (B-C), since the numbers A, B, C more 5 but less than 9, then -2≤B-C≤2,

so Ax = 166; 165; 164;163;162. Of the numbers 6,7,8, only 6 is

Average general education

UMK line G. K. Muravina. Algebra and beginnings mathematical analysis(10-11) (deep)

Line UMK Merzlyak. Algebra and the Beginnings of Analysis (10-11) (U)

Mathematics

Preparation for the exam in mathematics ( profile level): tasks, solutions and explanations

We analyze tasks and solve examples with the teacher

Examination paper profile level lasts 3 hours 55 minutes (235 minutes).

Minimum Threshold- 27 points.

The examination paper consists of two parts, which differ in content, complexity and number of tasks.

The defining feature of each part of the work is the form of tasks:

• part 1 contains 8 tasks (tasks 1-8) with a short answer in the form of an integer or a final decimal fraction;
• part 2 contains 4 tasks (tasks 9-12) with a short answer in the form of an integer or a final decimal fraction and 7 tasks (tasks 13-19) with a detailed answer (full record of the decision with the rationale for the actions performed).

Panova Svetlana Anatolievna, mathematic teacher the highest category schools, 20 years of work experience:

"In order to receive school certificate, the graduate must pass two mandatory exams in USE form, one of which is mathematics. In accordance with the Development Concept mathematics education V Russian Federation The USE in mathematics is divided into two levels: basic and specialized. Today we will consider options for the profile level.

Task number 1- checks with USE participants the ability to apply the skills acquired in the course of 5-9 classes in elementary mathematics in practical activities. The participant must have computer skills, be able to work with rational numbers, be able to round decimals be able to convert one unit of measurement to another.

Example 1 An expense meter was installed in the apartment where Petr lives cold water(counter). On the first of May, the meter showed an consumption of 172 cubic meters. m of water, and on the first of June - 177 cubic meters. m. What amount should Peter pay for cold water for May, if the price of 1 cu. m of cold water is 34 rubles 17 kopecks? Give your answer in rubles.

Solution:

1) Find the amount of water spent per month:

177 - 172 = 5 (cu m)

2) Find how much money will be paid for the spent water:

34.17 5 = 170.85 (rub)

Answer: 170,85.

Task number 2- is one of the simplest tasks of the exam. The majority of graduates successfully cope with it, which indicates the possession of the definition of the concept of function. Task type No. 2 according to the requirements codifier is a task for using acquired knowledge and skills in practical activities and Everyday life. Task No. 2 consists of describing, using functions, various real relationships between quantities and interpreting their graphs. Task number 2 tests the ability to extract information presented in tables, diagrams, graphs. Graduates need to be able to determine the value of a function by the value of the argument with various ways of specifying the function and describe the behavior and properties of the function according to its graph. It is also necessary to be able to find the largest or smallest value from the function graph and build graphs of the studied functions. The mistakes made are of a random nature in reading the conditions of the problem, reading the diagram.

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Example 2 The figure shows the change in the exchange value of one share of a mining company in the first half of April 2017. On April 7, the businessman purchased 1,000 shares of this company. On April 10, he sold three-quarters of the purchased shares, and on April 13 he sold all the remaining ones. How much did the businessman lose as a result of these operations?

Solution:

2) 1000 3/4 = 750 (shares) - make up 3/4 of all purchased shares.

6) 247500 + 77500 = 325000 (rubles) - the businessman received after the sale of 1000 shares.

7) 340,000 - 325,000 = 15,000 (rubles) - the businessman lost as a result of all operations.

Answer: 15000.

Task number 3- is a task of the basic level of the first part, checks the ability to perform actions with geometric shapes on the content of the course "Planimetry". Task 3 tests the ability to calculate the area of ​​a figure on checkered paper, the ability to calculate degree measures of angles, calculate perimeters, etc.

Example 3 Find the area of ​​a rectangle drawn on checkered paper with a cell size of 1 cm by 1 cm (see figure). Give your answer in square centimeters.

Solution: To calculate the area of ​​this figure, you can use the Peak formula:

To calculate the area of ​​this rectangle, we use the Peak formula:

S= B +	G
	2

where V = 10, G = 6, therefore

S = 18 +	6
	2

Answer: 20.

See also: Unified State Examination in Physics: solving vibration problems

Task number 4- the task of the course "Probability Theory and Statistics". The ability to calculate the probability of an event in the simplest situation is tested.

Example 4 There are 5 red and 1 blue dots on the circle. Determine which polygons are larger: those with all red vertices, or those with one of the blue vertices. In your answer, indicate how many more of one than the other.

Solution: 1) We use the formula for the number of combinations from n elements by k:

all of whose vertices are red.

3) One pentagon with all red vertices.

4) 10 + 5 + 1 = 16 polygons with all red vertices.

whose vertices are red or with one blue vertex.

whose vertices are red or with one blue vertex.

8) One hexagon whose vertices are red with one blue vertex.

9) 20 + 15 + 6 + 1 = 42 polygons that have all red vertices or one blue vertex.

10) 42 - 16 = 26 polygons that use the blue dot.

11) 26 - 16 = 10 polygons - how many polygons, in which one of the vertices is a blue dot, are more than polygons, in which all vertices are only red.

Answer: 10.

Task number 5- the basic level of the first part tests the ability to solve the simplest equations (irrational, exponential, trigonometric, logarithmic).

Example 5 Solve Equation 2 3 + x= 0.4 5 3 + x .

Solution. Divide both sides of this equation by 5 3 + X≠ 0, we get

2 3 + x	= 0.4 or		2		3 + X	=	2	,
5 3 + X			5				5

whence it follows that 3 + x = 1, x = –2.

Answer: –2.

Task number 6 in planimetry for finding geometric quantities (lengths, angles, areas), modeling real situations in the language of geometry. The study of the constructed models using geometric concepts and theorems. The source of difficulties is, as a rule, ignorance or incorrect application of the necessary theorems of planimetry.

Area of ​​a triangle ABC equals 129. DE- median line parallel to side AB. Find the area of ​​the trapezoid ABED.

Solution. Triangle CDE similar to a triangle CAB at two corners, since the corner at the vertex C general, angle CDE equal to the angle CAB as the corresponding angles at DE || AB secant AC. Because DE is the midline of the triangle by condition, then by property middle line | DE = (1/2)AB. So the similarity coefficient is 0.5. The areas of similar figures are related as the square of the similarity coefficient, so

Hence, S ABED = S Δ ABC – S Δ CDE = 129 – 32,25 = 96,75.

Task number 7- checks the application of the derivative to the study of the function. For successful implementation, a meaningful, non-formal possession of the concept of a derivative is necessary.

Example 7 To the graph of the function y = f(x) at the point with the abscissa x 0 a tangent is drawn, which is perpendicular to the straight line passing through the points (4; 3) and (3; -1) of this graph. Find f′( x 0).

Solution. 1) Let's use the equation of a straight line passing through two given points and find the equation of a straight line passing through points (4; 3) and (3; -1).

(y – y 1)(x 2 – x 1) = (x – x 1)(y 2 – y 1)

(y – 3)(3 – 4) = (x – 4)(–1 – 3)

(y – 3)(–1) = (x – 4)(–4)

–y + 3 = –4x+ 16| · (-1)

y – 3 = 4x – 16

y = 4x– 13, where k 1 = 4.

2) Find the slope of the tangent k 2 which is perpendicular to the line y = 4x– 13, where k 1 = 4, according to the formula:

3) The slope of the tangent is the derivative of the function at the point of contact. Means, f′( x 0) = k 2 = –0,25.

Answer: –0,25.

Task number 8- checks the knowledge of elementary stereometry among the participants of the exam, the ability to apply formulas for finding surface areas and volumes of figures, dihedral angles, compare the volumes of similar figures, be able to perform actions with geometric shapes, coordinates and vectors, etc.

The volume of a cube circumscribed around a sphere is 216. Find the radius of the sphere.

Solution. 1) V cube = a 3 (where A is the length of the edge of the cube), so

A 3 = 216

A = 3 √216

2) Since the sphere is inscribed in a cube, it means that the length of the diameter of the sphere is equal to the length of the edge of the cube, therefore d = a, d = 6, d = 2R, R = 6: 2 = 3.

Task number 9- requires the graduate to transform and simplify algebraic expressions. Task number 9 advanced level Difficulty with short answers. Tasks from the section "Calculations and transformations" in the USE are divided into several types:

transformations of numerical rational expressions;

transformations of algebraic expressions and fractions;

transformations of numerical/letter irrational expressions;

actions with degrees;

transformation of logarithmic expressions;

1. conversion of numeric/letter trigonometric expressions.

Example 9 Calculate tgα if it is known that cos2α = 0.6 and

3π	< α < π.
4

Solution. 1) Let's use the double argument formula: cos2α = 2 cos 2 α - 1 and find

tan 2 α =	1	– 1 =	1	– 1 =	10	– 1 =	5	– 1 = 1	1	– 1 =	1	= 0,25.
	cos 2 α		0,8		8		4		4		4

Hence, tan 2 α = ± 0.5.

3) By condition

3π	< α < π,
4

hence α is the angle of the second quarter and tgα< 0, поэтому tgα = –0,5.

Answer: –0,5.

#ADVERTISING_INSERT# Task number 10- checks the ability of students to use the acquired early knowledge and skills in practical activities and everyday life. We can say that these are problems in physics, and not in mathematics, but all the necessary formulas and quantities are given in the condition. The problems are reduced to solving a linear or quadratic equation, either linear or square inequality. Therefore, it is necessary to be able to solve such equations and inequalities, and determine the answer. The answer must be in the form of a whole number or a final decimal fraction.

Two bodies of mass m= 2 kg each, moving at the same speed v= 10 m/s at an angle of 2α to each other. The energy (in joules) released during their absolutely inelastic collision is determined by the expression Q = mv 2 sin 2 α. At what smallest angle 2α (in degrees) must the bodies move so that at least 50 joules are released as a result of the collision?
Solution. To solve the problem, we need to solve the inequality Q ≥ 50, on the interval 2α ∈ (0°; 180°).

mv 2 sin 2 α ≥ 50

2 10 2 sin 2 α ≥ 50

200 sin2α ≥ 50

Since α ∈ (0°; 90°), we will only solve

We represent the solution of the inequality graphically:

Since by assumption α ∈ (0°; 90°), it means that 30° ≤ α< 90°. Получили, что наименьший угол α равен 30°, тогда наименьший угол 2α = 60°.

Task number 11- is typical, but it turns out to be difficult for students. The main source of difficulties is the construction of a mathematical model (drawing up an equation). Task number 11 tests the ability to solve word problems.

Example 11. During spring break, 11-grader Vasya had to solve 560 training problems to prepare for the exam. On March 18, on the last day of school, Vasya solved 5 problems. Then every day he solved the same number of problems more than the previous day. Determine how many problems Vasya solved on April 2 on the last day of vacation.

Solution: Denote a 1 = 5 - the number of tasks that Vasya solved on March 18, d– daily number of tasks solved by Vasya, n= 16 - the number of days from March 18 to April 2 inclusive, S 16 = 560 - the total number of tasks, a 16 - the number of tasks that Vasya solved on April 2. Knowing that every day Vasya solved the same number of tasks more than the previous day, then you can use the formulas for finding the sum arithmetic progression:

560 = (5 + a 16) 8,

5 + a 16 = 560: 8,

5 + a 16 = 70,

a 16 = 70 – 5

a 16 = 65.

Answer: 65.

Task number 12- check students' ability to perform actions with functions, be able to apply the derivative to the study of the function.

Find the maximum point of a function y= 10ln( x + 9) – 10x + 1.

Solution: 1) Find the domain of the function: x + 9 > 0, x> –9, that is, x ∈ (–9; ∞).

2) Find the derivative of the function:

4) The found point belongs to the interval (–9; ∞). We define the signs of the derivative of the function and depict the behavior of the function in the figure:

The desired maximum point x = –8.

Download for free the work program in mathematics to the line of UMK G.K. Muravina, K.S. Muravina, O.V. Muravina 10-11 Download free algebra manuals

Task number 13- an increased level of complexity with a detailed answer, which tests the ability to solve equations, the most successfully solved among tasks with a detailed answer of an increased level of complexity.

a) Solve the equation 2log 3 2 (2cos x) – 5log 3 (2cos x) + 2 = 0

b) Find all the roots of this equation that belong to the segment.

Solution: a) Let log 3 (2cos x) = t, then 2 t 2 – 5t + 2 = 0,

	log3(2cos x) =	2	⇔		2cos x = 9	⇔		cos x =	4,5	⇔ because |cos x| ≤ 1,
	log3(2cos x) =	1			2cos x = √3			cos x =	√3
		2							2

then cos x =	√3
	2

	x =	π	+ 2π k
		6
	x = –	π	+ 2π k, k ∈ Z
		6

b) Find the roots lying on the segment .

It can be seen from the figure that the given segment has roots

11π	And	13π	.
6		6

Answer: A)	π	+ 2π k; –	π	+ 2π k, k ∈ Z; b)	11π	;	13π	.
	6		6		6		6

Task number 14- advanced level refers to the tasks of the second part with a detailed answer. The task tests the ability to perform actions with geometric shapes. The task contains two items. In the first paragraph, the task must be proved, and in the second paragraph, it must be calculated.

The circumference diameter of the base of the cylinder is 20, the generatrix of the cylinder is 28. The plane intersects its bases along chords of length 12 and 16. The distance between the chords is 2√197.

a) Prove that the centers of the bases of the cylinder lie on the same side of this plane.

b) Find the angle between this plane and the plane of the base of the cylinder.

Solution: a) A chord of length 12 is at a distance = 8 from the center of the base circle, and a chord of length 16, similarly, is at a distance of 6. Therefore, the distance between their projections on a plane parallel to the bases of the cylinders is either 8 + 6 = 14, or 8 − 6 = 2.

Then the distance between chords is either

= = √980 = = 2√245

= = √788 = = 2√197.

According to the condition, the second case was realized, in which the projections of the chords lie on one side of the axis of the cylinder. This means that the axis does not intersect this plane within the cylinder, that is, the bases lie on one side of it. What needed to be proven.

b) Let's denote the centers of the bases as O 1 and O 2. Let us draw from the center of the base with a chord of length 12 the perpendicular bisector to this chord (it has a length of 8, as already noted) and from the center of the other base to another chord. They lie in the same plane β perpendicular to these chords. Let's call the midpoint of the smaller chord B, greater than A, and the projection of A onto the second base H (H ∈ β). Then AB,AH ∈ β and, therefore, AB,AH are perpendicular to the chord, that is, the line of intersection of the base with the given plane.

So the required angle is

∠ABH = arctan	AH	= arctg	28	= arctg14.
	BH		8 – 6

Task number 15- an increased level of complexity with a detailed answer, checks the ability to solve inequalities, the most successfully solved among tasks with a detailed answer of an increased level of complexity.

Example 15 Solve the inequality | x 2 – 3x| log 2 ( x + 1) ≤ 3x – x 2 .

Solution: The domain of definition of this inequality is the interval (–1; +∞). Consider three cases separately:

1) Let x 2 – 3x= 0, i.e. X= 0 or X= 3. In this case, this inequality becomes true, therefore, these values ​​are included in the solution.

2) Let now x 2 – 3x> 0, i.e. x∈ (–1; 0) ∪ (3; +∞). In this case, this inequality can be rewritten in the form ( x 2 – 3x) log 2 ( x + 1) ≤ 3x – x 2 and divide by a positive expression x 2 – 3x. We get log 2 ( x + 1) ≤ –1, x + 1 ≤ 2 –1 , x≤ 0.5 -1 or x≤ -0.5. Taking into account the domain of definition, we have x ∈ (–1; –0,5].

3) Finally, consider x 2 – 3x < 0, при этом x∈ (0; 3). In this case, the original inequality will be rewritten in the form (3 x – x 2) log 2 ( x + 1) ≤ 3x – x 2. After dividing by a positive expression 3 x – x 2 , we get log 2 ( x + 1) ≤ 1, x + 1 ≤ 2, x≤ 1. Taking into account the area, we have x ∈ (0; 1].

Combining the obtained solutions, we obtain x ∈ (–1; –0.5] ∪ ∪ {3}.

Answer: (–1; –0.5] ∪ ∪ {3}.

Task number 16- advanced level refers to the tasks of the second part with a detailed answer. The task tests the ability to perform actions with geometric shapes, coordinates and vectors. The task contains two items. In the first paragraph, the task must be proved, and in the second paragraph, it must be calculated.

In an isosceles triangle ABC with an angle of 120° at the vertex A, a bisector BD is drawn. Rectangle DEFH is inscribed in triangle ABC so that side FH lies on segment BC and vertex E lies on segment AB. a) Prove that FH = 2DH. b) Find the area of ​​the rectangle DEFH if AB = 4.

Solution: A)

1) ΔBEF - rectangular, EF⊥BC, ∠B = (180° - 120°) : 2 = 30°, then EF = BE due to the property of the leg opposite the angle of 30°.

2) Let EF = DH = x, then BE = 2 x, BF = x√3 by the Pythagorean theorem.

3) Since ΔABC is isosceles, then ∠B = ∠C = 30˚.

BD is the bisector of ∠B, so ∠ABD = ∠DBC = 15˚.

4) Consider ΔDBH - rectangular, because DH⊥BC.

2x	=	4 – 2x
2x(√3 + 1)		4

1	=	2 – x
√3 + 1		2

√3 – 1 = 2 – x

x = 3 – √3

EF = 3 - √3

2) S DEFH = ED EF = (3 - √3 ) 2(3 - √3 )

S DEFH = 24 - 12√3.

Answer: 24 – 12√3.

Task number 17- a task with a detailed answer, this task tests the application of knowledge and skills in practical activities and everyday life, the ability to build and explore mathematical models. This task is a text task with economic content.

Example 17. The deposit in the amount of 20 million rubles is planned to be opened for four years. At the end of each year, the bank increases the deposit by 10% compared to its size at the beginning of the year. In addition, at the beginning of the third and fourth years, the depositor annually replenishes the deposit by X million rubles, where X - whole number. Find highest value X, at which the bank will add less than 17 million rubles to the deposit in four years.

Solution: At the end of the first year, the contribution will be 20 + 20 · 0.1 = 22 million rubles, and at the end of the second - 22 + 22 · 0.1 = 24.2 million rubles. At the beginning of the third year, the contribution (in million rubles) will be (24.2 + X), and at the end - (24.2 + X) + (24,2 + X) 0.1 = (26.62 + 1.1 X). At the beginning of the fourth year, the contribution will be (26.62 + 2.1 X), and at the end - (26.62 + 2.1 X) + (26,62 + 2,1X) 0.1 = (29.282 + 2.31 X). By condition, you need to find the largest integer x for which the inequality

(29,282 + 2,31x) – 20 – 2x < 17

29,282 + 2,31x – 20 – 2x < 17

0,31x < 17 + 20 – 29,282

0,31x < 7,718

x <	7718
	310

x <	3859
	155

x < 24	139
	155

The largest integer solution to this inequality is the number 24.

Answer: 24.

Task number 18- a task of an increased level of complexity with a detailed answer. This task is intended for competitive selection to universities with increased requirements for the mathematical preparation of applicants. Exercise high level complexity is not a task for applying one solution method, but for a combination various methods. For the successful completion of task 18, in addition to solid mathematical knowledge, a high level of mathematical culture is also required.

At what a system of inequalities

	x 2 + y 2 ≤ 2ay – a 2 + 1
	y + a ≤ |x| – a

has exactly two solutions?

Solution: This system can be rewritten as

	x 2 + (y– a) 2 ≤ 1
	y ≤ |x| – a

If we draw on the plane the set of solutions to the first inequality, we get the interior of a circle (with a boundary) of radius 1 centered at the point (0, A). The set of solutions of the second inequality is the part of the plane that lies under the graph of the function y = | x| – a, and the latter is the graph of the function
y = | x| , shifted down by A. The solution of this system is the intersection of the solution sets of each of the inequalities.

Therefore, two solutions this system will have only in the case shown in Fig. 1.

The points of contact between the circle and the lines will be the two solutions of the system. Each of the straight lines is inclined to the axes at an angle of 45°. So the triangle PQR- rectangular isosceles. Dot Q has coordinates (0, A), and the point R– coordinates (0, – A). In addition, cuts PR And PQ are equal to the circle radius equal to 1. Hence,

QR= 2a = √2, a =	√2	.
	2

Answer: a =	√2	.
	2

Task number 19- a task of an increased level of complexity with a detailed answer. This task is intended for competitive selection to universities with increased requirements for the mathematical preparation of applicants. A task of a high level of complexity is not a task for applying one solution method, but for a combination of different methods. To successfully complete task 19, you must be able to search for a solution by choosing different approaches from among the known, modifying the studied methods.

Let sn sum P members of an arithmetic progression ( a p). It is known that S n + 1 = 2n 2 – 21n – 23.

a) Give the formula P th member of this progression.

b) Find the smallest modulo sum S n.

c) Find the smallest P, at which S n will be the square of an integer.

Solution: a) Obviously, a n = S n – S n- 1 . Using this formula, we get:

S n = S (n – 1) + 1 = 2(n – 1) 2 – 21(n – 1) – 23 = 2n 2 – 25n,

S n – 1 = S (n – 2) + 1 = 2(n – 1) 2 – 21(n – 2) – 23 = 2n 2 – 25n+ 27

Means, a n = 2n 2 – 25n – (2n 2 – 29n + 27) = 4n – 27.

B) because S n = 2n 2 – 25n, then consider the function S(x) = | 2x 2 – 25x|. Her graph can be seen in the figure.

It is obvious that the smallest value is reached at the integer points located closest to the zeros of the function. Obviously these are points. X= 1, X= 12 and X= 13. Since, S(1) = |S 1 | = |2 – 25| = 23, S(12) = |S 12 | = |2 144 – 25 12| = 12, S(13) = |S 13 | = |2 169 – 25 13| = 13, then the smallest value is 12.

c) It follows from the previous paragraph that sn positive since n= 13. Since S n = 2n 2 – 25n = n(2n– 25), then the obvious case when this expression is a perfect square is realized when n = 2n- 25, that is, with P= 25.

It remains to check the values ​​​​from 13 to 25:

S 13 = 13 1, S 14 = 14 3, S 15 = 15 5, S 16 = 16 7, S 17 = 17 9, S 18 = 18 11, S 19 = 19 13 S 20 = 20 13, S 21 = 21 17, S 22 = 22 19, S 23 = 23 21, S 24 = 24 23.

It turns out that for smaller values P full square is not achieved.

Answer: A) a n = 4n- 27; b) 12; c) 25.

________________

*Since May 2017, the DROFA-VENTANA joint publishing group has been part of the Russian Textbook Corporation. The corporation also included the Astrel publishing house and the LECTA digital educational platform. Alexander Brychkin, a graduate of the Financial Academy under the Government of the Russian Federation, candidate of economic sciences, head of innovative projects of the DROFA publishing house in the field of digital education (electronic forms of textbooks, Russian Electronic School, LECTA digital educational platform) has been appointed General Director. Prior to joining the DROFA publishing house, he held the position of Vice President for Strategic Development and Investments of the EKSMO-AST publishing holding. Today, the Russian Textbook Publishing Corporation has the largest portfolio of textbooks included in the Federal List - 485 titles (approximately 40%, excluding textbooks for correctional schools). The corporation's publishing houses own the most popular Russian schools sets of textbooks on physics, drawing, biology, chemistry, technology, geography, astronomy - areas of knowledge that are needed to develop the country's production potential. The corporation's portfolio includes textbooks and study guides For elementary school awarded the Presidential Prize in Education. These are textbooks and manuals on subject areas that are necessary for the development of the scientific, technical and industrial potential of Russia.

Mysikova Julia

The unified state exam in mathematics at the basic level consists of 20 tasks. Task 20 tests the skills of solving logical problems. The student should be able to apply his knowledge to solve problems in practice, including arithmetic and geometric progression. In this work, we analyze in detail how to solve task 20 of the USE in mathematics at a basic level, as well as examples and methods of solutions based on detailed tasks.

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Tasks for ingenuity of the Unified State Examination in mathematics of a basic level. Tasks No. 20 Mysikova Yulia Alexandrovna, student 11 "A" socio-economic class Municipal educational institution"Average comprehensive school No. 45"

Snail on a tree Solution. A snail crawls up a tree by 3 m per day, and descends by 2 m during the night. In total, it moves 3 - 2 = 1 meter per day. In 7 days it will rise by 7 meters. On the eighth day, she will crawl up another 3 meters and for the first time will be at a height of 7 + 3 = 10 (m), i.e. at the top of the tree. Answer: 8 A snail crawls up a tree by 3 m in a day, and descends by 2 m in a night. The height of a tree is 10 m. How many days will it take for a snail to crawl from the base to the top of the tree?

Gas Stations Solution. Let's draw a circle and arrange the points (gas stations) so that the distances correspond to the condition. Note that all distances between points A, C and D are known. AC=20, AD=30, CD=20. Mark point A. From point A clockwise mark point C, remember that AC=20. Now we will mark the point D, which lies at a distance of 30 from A, this distance cannot be plotted clockwise from A, since then the distance between C and D will be 10, and by condition CD = 2 0. So from A to D you need to move counterclockwise, mark point D. Since CD=20, the length of the entire circle is 20+30+20=70. Since AB=35, then point B is diametrically opposite to point A. The distance from C to B will be 35-20=15. Answer: 15. There are four gas stations on the ring road: A, B, C and D. The distance between A and B is 35 km, between A and C - 20 km, between C and D - 20 km, between D and A - 30 km (all distances are measured along the ring road in the shortest direction). Find the distance between B and C. Give your answer in kilometers.

In the cinema hall Solution. 1 way. We just count how many seats are in the rows up to the eighth: 1 - 24 2 - 26 3 - 28 4 - 30 5 - 32 6 - 34 7 - 36 8 - 38. Answer: 38. There are 24 seats in the first row of the cinema hall, and in each next row on 2 more than the previous one. How many seats are in the eighth row? 2 way. We notice that the number of places in the rows is an arithmetic progression with the first term in 24 and the difference is equal to 2. According to the formula of the nth term of the progression, we find the eighth term a 8 = 24 + (8 - 1) * 2 = 38. Answer: 38.

Mushrooms in a basket Solution. From the condition that among any 27 mushrooms there is at least one mushroom, it follows that the number of mushrooms is not more than 26. From the second condition, that among any 25 mushrooms there is at least one mushroom, it follows that the number of mushrooms is not more than 24. Since there are 50 mushrooms in total, then there are 24 mushrooms, and 26 milk mushrooms. Answer: 24. There are 50 mushrooms in the basket: mushrooms and milk mushrooms. It is known that among any 27 mushrooms there is at least one camelina, and among any 25 mushrooms at least one mushroom. How many mushrooms are in the basket?

Cubes in a row Solution. If we number all the cubes with numbers from one to six (not taking into account that there are cubes of different colors), then we get the total number of permutations of the cubes: Р(6)=6*5*4*3*2*1=720 Now remember that there are 2 red cubes and rearranging them (P(2)=2*1=2) will not give a new way, so the resulting product must be reduced by 2 times. Similarly, we recall that we have 3 green cubes, so we will have to reduce the resulting product by another 6 times (P (3) \u003d 3 * 2 * 1 \u003d 6) So, we get the total number of ways to arrange the cubes 60. Answer: 60 In how many ways can two identical red cubes, three identical green cubes and one blue cube be placed in a row?

On the treadmill The trainer advised Andrey to spend 15 minutes on the treadmill on the first day of training, and on each next session 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? Solution. 1 way. We note that we need to find the sum of an arithmetic progression with the first term 15 and the difference equal to 7. According to the formula for the sum of the n first terms of the progression S n = (2a 1 + (n-1) d) * n / 2 we have 145 = (2 * 15 + (n–1)*7)*n/2, 290=(30+(n–1)*7)*n, 290=(30+7n–7)*n, 290=(23+7n)*n , 290=23n+7n 2 , 7n 2 +23n-290=0, n=5 . Answer: 5. 2 way. More labor intensive. 1-15-15 2-22-37 3-29-66 4-36-102 5-43-145. Answer: 5.

Changing coins Task 20. 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? Solution. Let Nikolai first perform x operations of the second type, and then y operations of the first type. Then we have: Then there were 3y silver coins -5x = 90 - 100 = -10 i.e. 10 less. Answer: 10

The owner agreed the decision. It is clear from the condition that the sequence of prices for each excavated meter is an arithmetic progression with the first member a 1 = 3700 and the difference d=1700. The sum of the first n members of an arithmetic progression is calculated by the formula S n = 0.5 (2a 1 + (n - 1) d) n. Substituting the original data, we get: S 10 \u003d 0.5 (2 * 3700 + (8 - 1) * 1700) * 8 \u003d 77200. Thus, the owner will have to pay the workers 77,200 rubles. Answer: 77200. The owner agreed with the workers that they would dig a well for him on the following terms: he would pay them 3,700 rubles for the first meter, and 1,700 rubles more for each next meter than for the previous one. How much money will the owner have to pay the workers if they dig a well 8 meters deep?

Water in the pit 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? Solution. As a result of the operation of the pump and flooding with soil water, the water level in the pit decreases by 20-5 = 15 centimeters per hour. It takes 120:15=8 hours to lower the level by 200-80=120 centimeters. Answer: 8.

A tank with a slot In a tank with a volume of 38 liters every hour, starting at 12 o'clock, a full bucket of water with a volume of 8 liters is poured. 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. 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 7 pm, 8 liters of water will be added to the tank and the volume of water in the tank will become 38 liters. Answer: 19.

Well 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? Solution. Given the siltation of the well, 300-30=270 meters pass during the day. This means that 2700 meters will be covered in 10 full days and another 300 meters will be covered in the 11th working day. Answer: 11.

Globe 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? Solution. One parallel divides the surface of the globe into 2 parts. Two to three parts. Three into four parts, etc. 17 parallels break the surface into 18 parts. Let's draw one meridian, and get one whole (not cut) surface. Let's draw the second meridian and we already have two parts, the third meridian will break the surface into three parts, etc. 24 meridians have divided our surface into 24 parts. We get 18*24=432. All lines will divide the surface of the globe into 432 parts. Answer: 432.

Grasshopper jumps 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? Solution: With a little thought, we can see that the grasshopper can only end up at points with even coordinates, since the number of jumps it makes is even. For example, if he makes five jumps in one direction, then in reverse side he will make three jumps and end up at points 2 or −2. The maximum grasshopper can be at points, the module of which does not exceed eight. Thus, the grasshopper can end up at points: -8, -6, -4, -2, 0, 2, 4, 6 and 8; only 9 points. Answer: 9 .

New Bacteria Every second a bacterium divides into two new bacteria. It is known that the entire volume of one glass of bacteria is filled in 1 hour. How many seconds does it take for bacteria to fill half the glass? Solution. Recall that 1 hour = 3600 seconds. Every second there are twice as many bacteria. This means that it takes only 1 second to get a full glass of bacteria from a half glass of bacteria. Therefore, the glass was half filled in 3600-1=3599 seconds. Answer: 3599.

Dividing the numbers The product of ten consecutive numbers is divided by 7. What can be the remainder? Solution. The task is simple, since among ten consecutive natural numbers at least one is divisible by 7. This means that the entire product will be divisible by 7 without a remainder. That is, the remainder is 0. Answer: 0.

Where does Petya live? Task 1. The house Petya lives in has one entrance. There are six apartments on each floor. Petya lives in apartment number 50. What floor does Petya live on? Solution: We divide 50 by 6, we get the quotient 8 and 2 in the remainder. This means that Petya lives on the 9th floor. Answer: 9. Task 2. All entrances of the house have the same number of floors, and all floors have 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 455 apartments in total? Solution: The solution to this problem follows from the decomposition of the number 455 into prime factors. 455 = 13*7*5. So the house has 13 floors, 7 apartments on each floor in the entrance, 5 entrances. Answer: 13.

Task 3. Sasha invited Petya to visit, saying that he lives in the eighth entrance in apartment No. 468, but forgot to say the floor. Approaching the house, Petya discovered that the house had twelve floors. What floor does Sasha live on? (On all floors, the number of apartments is the same, the numbers of apartments in the house start from one.) Solution: Petya can calculate that in a twelve-story building in the first seven entrances there are 12 * 7 = 84 landings. Further, sorting through the possible number of apartments on one site, you can see that there are less than six of them, since 84 * 6 \u003d 504. This is more than 468. This means that there are 5 apartments on each of the sites, then in the first seven entrances 84 * 5 \u003d 420 apartments . 468 - 420 = 48, that is, Sasha lives in apartment 48 in the 8th entrance (if the numbering was from one in each entrance). 48:5 = 9 and 3 remainder. So Sasha's apartment is on the 10th floor. Answer: 10.

Restaurant menu The restaurant menu has 6 types of salads, 3 types of first courses, 5 types of second courses and 4 types of dessert. How many salad, first, second and dessert lunch options can diners at this restaurant choose? Solution. If we number each salad, first, second, dessert, then: with 1 salad, 1 first, 1 second, one of 4 desserts can be served. 4 options. With the second second, there are also 4 options, etc. In total we get 6*3*5*4=360. Answer: 360.

Masha and the Bear The bear ate his half of the jar of jam 3 times faster than Masha, which means he still has 3 times more time left to eat cookies. Because The bear eats cookies 3 times faster than Masha and he still has 3 times more time left (he ate his half jar of jam 3 times faster), then he eats 3⋅3=9 times more cookies than Masha (9 cookies are eaten by the Bear, while Masha only 1 cookie). It turns out that in a ratio of 9:1, Bear and Masha eat cookies. In total, 10 shares are obtained, which means that 1 share is equal to 160:10 \u003d 16. As a result, the Bear ate 16⋅9=144 cookies. Answer: 144 Masha and the Bear ate 160 cookies and a jar of jam, starting and finishing at the same time. At first, Masha ate jam, and the Bear ate cookies, but at some point they changed. The bear eats both three times faster than Masha. How many cookies did the Bear eat if they ate the same amount of jam?

Sticks and lines The stick has red, yellow and green transverse lines. 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 will you get if you cut a stick along the lines of all three colors? Solution. 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, therefore, there will be 25 pieces. Answer: 25

The doctor prescribed 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)? Solution In the first phase of the drops, the number of drops taken per day is an increasing arithmetic progression with the first term equal to 3, the difference equal to 3 and the last term equal to 30. Therefore: Then 3 + 3(n -1)=30; 3+3n-3=30; 3n=30; n =10 , i.e. 10 days have passed according to the scheme of increasing up to 30 drops. We know the formula for the sum of arith. progressions: Calculate S10:

For the next 3 days - 30 drops each: 30 3 = 90 (drops) At the last stage of admission: I.e. 30 -3(n-1)=0; 30 -3n+3=0; -3n=-33; n=11 i.e. 11 days the medication intake decreased. Let's find the arithmetic sum. progressions 4) So, 165 + 90 + 165 = 420 drops in total 5) Then 420: 250 = 42/25 = 1 (17/25) bubbles Answer: you need to buy 2 bubbles

Household Appliance Store 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? Solution. Let's sequentially calculate how many refrigerators were sold for each month and sum up the results: 10 4+(10+15)+(25+15)+(40+15)+(55+15)+(70-15)+ (55- 15)+(40-15)+ (25-15)== 40+25+40+55+70+55+40+25+10=120+110+130=360 Answer: 360.

Boxes Boxes of two types, having the same width and height, are stacked in a warehouse in one row 43 m long, putting them to each other in width. Boxes of one type have a length of 2m, and the other 5m. Which smallest number boxes will be required to fill the entire row without creating empty spaces? Solution you need to find the smallest number of boxes, then => you need to take the largest number big boxes. So 5 7 = 35; 43 - 35 = 8 and 8:2=4; 4+7=11 So there are 11 boxes in total. Answer: 11.

Table The table has three columns and several rows. Each cell of the table was placed with a natural number so that the sum of all the numbers in the first column is 119, in the second - 125, in the third - 133, and the sum of the numbers in each row is greater than 15, but less than 18. How many rows are in the column? Solution. The total sum in all columns = 119 + 125 + 133 = 377 The numbers 18 and 15 are not included in the limit, which means: 1) if the sum in the row = 17, then the number of rows is 377: 17= =22.2 2) if the sum in the line = 16, then the number of lines is 377: 16 = = 23.5 So the number of lines = 23 (because it should be between 22.2 and 23.5) Answer: 23

Quiz and tasks The list of tasks of the quiz consisted of 36 questions. For each correct answer, the student received 5 points, for an incorrect answer, 11 points were deducted from him, and if there was no answer, they were given 0 points. How many correct answers were given by the student who scored 75 points, if it is known that he was wrong at least once? Solution. Method 1: Let X be the number of correct answers y be the number of wrong answers. Then we compose the equation 5x -11y \u003d 75, where 0

A group of tourists A group of tourists overcame a mountain pass. They covered the first kilometer of the ascent in 50 minutes, and each next kilometer passed 15 minutes longer than the previous one. The last kilometer before the summit was completed in 95 minutes. After a ten-minute rest at the top, the tourists began their descent, which was more gentle. The first kilometer after the summit was covered in an hour, and each next one is 10 minutes faster than the previous one. How many hours did the group spend on the entire route if the last kilometer of the descent was completed in 10 minutes? Solution. The group spent 290 minutes climbing the mountain, 10 minutes resting, and 210 minutes descending the mountain. In total, tourists spent 510 minutes on the entire route. Let's translate 510 minutes into hours and get that in 8.5 hours the tourists covered the entire route. Answer: 8.5

Thank you for your attention!

Problem #5922.

The owner agreed with the workers that they were digging a well on the following terms: for the first meter he would 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?

Since the payment for each next meter differs from the payment for the previous one by the same number, we have before us.

In this progression - the payment for the first meter, - the difference in payment for each subsequent meter, - the number of working days.

The sum of the members of an arithmetic progression is found by the formula:

Substitute the data of the problem in this formula.

Answer: 89100.

Problem #5943.

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 100 copper coins appeared. By how much did Nicholas's number of silver coins decrease??

Problem #5960.

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 5 jumps, starting from the origin?

If the grasshopper makes five jumps in one direction (right or left), then it will end up at points with coordinates 5 or -5:

Note that the grasshopper can jump both to the right and to the left. If he makes 1 jump to the right and 4 jumps to the left (for a total of 5 jumps), he will end up at the point with coordinate -3. Similarly, if the grasshopper makes 1 jump to the left and 4 jumps to the right (for a total of 5 jumps), then it will end up at the point with coordinate 3:

If the grasshopper makes 2 jumps to the right and 3 jumps to the left (for a total of 5 jumps), it will end up at the point with coordinate -1. Similarly, if the grasshopper makes 2 jumps to the left and 3 jumps to the right (for a total of 5 jumps), then it will end up at the point with coordinate 1:

Note that if the total number of jumps is odd, then the grasshopper will not return to the origin, that is, it can only hit points with odd coordinates:

There are only 6 of these points.

If the number of jumps were even, then the grasshopper could return to the origin and all points on the coordinate line that it could hit would have even coordinates.

Answer: 6

Problem #5990

A snail climbs up a tree 2 m in a day, and slides down 1 m in a night. The height of the tree is 9 m. How many days will it take for the snail to crawl to the top of the tree?

Note that in this problem one should distinguish between the concept of "day" and the concept of "day".

The question asks exactly how much days the snail will crawl to the top of the tree.

In one day the snail climbs 2 m, and in one day the snail rises to 1 m (it rises by 2 m during the day, and then descends by 1 m during the night).

For 7 days the snail rises to 7 meters. That is, on the morning of the 8th day, she will have to crawl to the top of 2 m. And on the eighth day she will overcome this distance.

Answer: 8 days.

Task number 6010.

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 105 apartments in total?

To find the number of apartments in a house, you need to multiply the number of apartments per floor ( ) by the number of floors ( ) and multiply by the number of entrances ( ).

That is, we need to find ( ) based on the following conditions:

(1)

The last inequality reflects the condition "the number of floors in the building 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."

That is, ( ) is the most more number.

Let's factorize 105 into prime factors:

Taking into account condition (1), .

Answer: 7.

Problem #6036.

There are 30 mushrooms in the basket: 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?

Because among any 12 mushrooms there is at least one camelina(or more) the number of mushrooms must be less than or equal to .

It follows that the number of saffron milk caps is greater than or equal to .

Because among any 20 mushrooms at least one mushroom(or more), the number of saffron milk caps must be less than or equal to

Then we got that, on the one hand, the number of mushrooms is greater than or equal to 19 , and on the other hand, less than or equal to 19 .

Therefore, the number of mushrooms equals 19.

Answer: 19.

Problem number 6047.

Sasha invited Petya to visit, saying that he lives in the seventh entrance in apartment No. 333, but he forgot to say the floor. Approaching the house, Petya discovered that the house had nine floors. What floor does Sasha live on? (The number of apartments on each floor is the same, the numbers of apartments in the building start from one.)

Let on each floor of apartments.

Then the number of apartments in the first six entrances is

Find the maximum natural value that satisfies the inequality ( - the number of the last apartment in the sixth entrance, and it is less than 333.)

From here

The number of the last apartment in the sixth entrance -

The seventh entrance starts from the 325th apartment.

Therefore, apartment 333 is on the second floor.

Answer: 2

Problem number 6060.

On the surface of the globe, 17 parallels and 24 meridians were drawn with a felt-tip pen. Into how many parts do the lines drawn divide the surface of the globe? Meridian is a circular arc that connects the North and south pole. Parallel is a circle lying in a plane parallel to the plane of the equator..

Imagine a watermelon that we cut into pieces.

Having made two cuts from the top point to the bottom (drawing two meridians), we will cut the watermelon into two slices. Therefore, after making 24 cuts (24 meridians), we will cut the watermelon into 24 slices.

Now we will cut each slice.

If we make 1 transverse cut (parallel), then we will cut one slice into 2 parts.

If we make 2 transverse cuts (parallels), then we will cut one slice into 3 parts.

So, having made 17 cuts, we will cut one slice into 18 parts.

So, we cut 24 slices into 18 pieces, and got a piece.

Therefore, 17 parallels and 24 meridians divide the surface of the globe into 432 parts.

Answer: 432.

Problem #6069

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?

If you make 1 cut, you get 2 pieces.

If you make 2 cuts, you get 3 pieces.

In the general case: if you make cuts, you get a piece.

Back: to get pieces, you need to make a cut.

Find the total number of lines along which the stick was cut.

If you cut the stick along the red lines, you get 5 pieces - therefore, there were 4 red lines;

if on yellow - 7 pieces - therefore, there were 6 yellow lines;

and if on green - 11 pieces - therefore, there were 10 green lines.

Hence the total number of lines is . If you cut the stick along all the lines, you get 21 pieces.

Answer: 21.

Problem #9626.

There are four gas stations on the ring road: A, B, B, and D. The distance between A and B is 50 km, between A and C is 40 km, between C and D is 25 km, between D and A is 35 km (all distances are measured along the ring road in the shortest direction). Find the distance between B and C.

Let's see how gas stations can be located. Let's try to arrange them like this:

With such an arrangement, the distance between G and A cannot be equal to 35 km.

Let's try this:

With this arrangement, the distance between A and B cannot be 40 km.

Consider this option:

This option satisfies the condition of the problem.

Answer: 10.

Problem #10041.

The list of tasks of the quiz consisted of 25 questions. For each correct answer, the student received 7 points, for an incorrect answer, 9 points were deducted from him, and if there was no answer, they were given 0 points. How many correct answers were given by the student who scored 56 points, if it is known that he was wrong at least once?

Let the student give correct answers and incorrect ones ( ). Since there may have been more questions that he answered, we get the inequality:

In addition, according to the condition

Since a correct answer adds 7 points, and an incorrect answer subtracts 9, and the student ends up with 56 points, we get the equation:

This equation must be solved in integers.

Since 9 is not divisible by 7, it must be divisible by 7.

Let , then .

In this case, all conditions are met.

Problem #10056.

The rectangle is divided into four small rectangles by two straight cuts. The areas of three of them, starting from the top left and going clockwise, are 15, 18, 24. Find the area of ​​the fourth rectangle.

The area of ​​a rectangle is equal to the product of its sides.

The yellow and blue rectangles have a common side, so the ratio of the areas of these rectangles is equal to the ratio of the lengths of the other sides (not equal to each other).

The white and green rectangles also have a common side, so the ratio of their areas is equal to the ratio of other sides (not equal to each other), that is, the same ratio:

By the property of proportion, we get

From here.

Problem #10071.

The rectangle is divided into four small rectangles by two straight cuts. The perimeters of three of them, starting from the top left and going clockwise, are 17, 12, 13. Find the perimeter of the fourth rectangle.

The perimeter of a rectangle is equal to the sum of the lengths of all its sides.

Let us designate the sides of the rectangles as shown in the figure and express the perimeters of the rectangles in terms of the indicated variables. We get:

Now we need to find what the value of the expression is.

Subtract the second equation from the third equation and add the third. We get:

Simplify the right and left sides, we get:

So, .

Answer: 18.

Problem #10086.

The table has three columns and several rows. Each cell of the table was placed with a natural number so that the sum of all the numbers in the first column is 72, in the second - 81, in the third - 91, and the sum of the numbers in each row is greater than 13, but less than 16. How many rows are there in the table?

Let's find the sum of all the numbers in the table: .

Let the number of rows in the table be .

According to the condition of the problem, the sum of numbers in each line more than 13 but less than 16.

Since the sum of the numbers is a natural number, only two natural numbers satisfy this double inequality: 14 and 15.

If we assume that the sum of the numbers in each row is 14, then the sum of all the numbers in the table is , and this sum satisfies the inequality .

If we assume that the sum of the numbers in each row is 15, then the sum of all the numbers in the table is , and this number satisfies the inequality .

So, a natural number must satisfy the system of inequalities:

The only natural that satisfies this system is

Answer: 17.

It is known about the natural numbers A, B and C that each of them is greater than 4 but less than 8. They guessed a natural number, then multiplied it by A, then added it to the resulting product B and subtracted C. It turned out 165. What number was guessed?

Integers A, B and C can be equal to the numbers 5, 6 or 7.

Let the unknown natural number be .

We get: ;

Let's consider various options.

Let A=5. Then B=6 and C=7, or B=7 and C=6, or B=7 and C=7, or B=6 and C=6.

Let's check: ; (1)

165 is divisible by 5.

The difference between the numbers B and C is either equal to or equal to 0 if these numbers are equal. If the difference is , then equality (1) is impossible. Therefore, the difference is 0 and

Let A=6. Then B=5 and C=7, or B=7 and C=5, or B=7 and C=7, or B=5 and C=5.

Let's check: ; (2)

The difference between the numbers B and C is either equal to or equal to 0 if these numbers are equal. If the difference is equal to or 0, then equality (2) is impossible, since it is an even number, and the sum (165 + even number) cannot be an even number.

Let A=7. Then B=5 and C=6, or B=6 and C=5, or B=6 and C=6, or B=5 and C=5.

Let's check: ; (3)

The difference between the numbers B and C is either equal to or equal to 0 if these numbers are equal. The number 165, when divided by 7, gives a remainder of 4. Therefore, it is also not divisible by 7, and equality (3) is impossible.

Answer: 33

Several consecutive pages fell out of the book. The number of the last page before the dropped sheets is 352, the number of the first page after the dropped sheets is written in the same numbers, but in a different order. How many sheets fell out?

Obviously, the number of the first page after the dropped sheets is greater than 352, so it can be either 532 or 523.

Each dropped sheet contains 2 pages. Therefore, an even number of pages fell out. 352 is an even number. If we add an even number to an even number, we get an even number. Therefore, the number of the last dropped page is an even number, and the number of the first page after the dropped sheets must be odd, that is, 523. Therefore, the number of the last dropped page is 522. Then it fell sheets.

Answer: 85

Masha and the Bear ate 160 cookies and a jar of jam, starting and finishing at the same time. At first, Masha ate jam, and the Bear ate cookies, but at some point they changed. The bear eats both three times faster than Masha. How many cookies did the Bear eat if they ate the same amount of jam?

If Masha and the Bear ate jam equally, and the bear ate three times as much jam per unit time, then he ate jam three times less than Masha. In other words, Masha ate jam three times longer than the Bear. But while Masha was eating jam, the bear was eating cookies. Therefore, the bear ate cookies three times longer than Masha. But the Bear, moreover, ate three times more cookies per unit time than Masha, therefore, in the end, he ate 9 times more cookies than Masha.

Now it's easy to write an equation. Let Masha eat the cookies, then the Bear ate the cookies. Together they ate cookies. we get the equation:

Answer: 144

On the counter of the flower shop there are 3 vases with roses: orange, white and blue. To the left of the orange vase are 15 roses, to the right of the blue vase are 12 roses. There are 22 roses in total in vases. how many roses are in the orange vase?

Since 15+12=27, and 27>22, therefore, the number of flowers in one vase was counted twice. And it's a white vase, because it's supposed to be the vase that's to the right of the blue one and to the left of the orange one. So the vases are in this order:

From here we get the system:

Subtracting the first equation from the third equation, we get O = 7.

Answer: 7

Ten poles are interconnected by wires so that exactly 8 wires extend from each pole. how many wires are strung between these ten pillars?

Solution

Let's simulate the situation. Suppose we have two poles, and they are interconnected by wires so that exactly 1 wire leaves each pole. Then it turns out that 2 wires depart from the poles. But we have this situation:

That is, despite the fact that 2 wires depart from the poles, only one wire is stretched between the poles. This means that the number of extended wires is two times less than the number of outgoing ones.

We get: - the number of outgoing wires.

Number of wires stretched.

Answer: 40

Of the ten countries, seven have signed a friendship treaty with exactly three other countries, and each of the remaining three with exactly seven. How many contracts were signed in total?

This task is similar to the previous one: two countries sign one general agreement. Each contract has two signatures. That is, the number of signed agreements is half as much as the number of signatures.

Find the number of signatures:

Find the number of signed contracts:

Answer: 21

Three rays emanating from the same point divide the plane into three different angles, measured in integer degrees. The largest angle is 3 times the smallest. How many values ​​can the average angle take?

Let the smallest angle be , then largest angle is equal to . Since the sum of all angles is , the mean angle is .

The average angle must be greater than the smallest and less than the largest angle.

We get a system of inequalities:

Therefore, it takes values ​​in the range from 52 to 71 degrees, that is, all possible values.

Answer: 20

Misha, Kolya and Lesha are playing table tennis: the player who loses the game gives way to the player who did not participate in it. As a result, it turned out that Misha played 12 games, and Kolya - 25. How many games did Lesha play?

Solution

It should be explained how the tournament is organized: the tournament consists of a fixed number of games; the player who lost in this game gives way to a player who did not participate in this game. Following the results of the next game, the player who did not take part in it takes the place of the loser. Therefore, each player takes part in at least one of two consecutive games.

Let's find how many games there were.

Since Kolya played 25 games, therefore, at least 25 games were played in the tournament.

Misha played 12 games. Since he definitely took part in every second game, therefore, no more than games were played. That is, the tournament consisted of 25 games.

If Misha played 12 games, then Lesha played the remaining 13.

Answer: 13

At the end of the quarter, Petya wrote down in a row all his marks for one of the subjects, there were 5 of them, and put multiplication signs between some of them. The product of the resulting numbers turned out to be 3495 . What mark does Petya get in a quarter in this subject, if the teacher puts only marks 2, 3, 4 or 5 and the final mark in the quarter is the arithmetic average of all current marks, rounded according to the rounding rules? (For example, 3.2 rounds up to 3; 4.5 rounds up to 5; 2.8 rounds up to 3)

Let's decompose 3495 into prime factors. The last digit of the number is 5, so the number is divisible by 5; The sum of the digits is divisible by 3, so the number is divisible by 3.

Got that

Therefore, Petya's estimates are 3, 5, 2, 3, 3. Let's find the arithmetic mean:

Answer: 3

The arithmetic mean of 6 different natural numbers is equal to 8. By how much should the largest of these numbers be increased so that their arithmetic mean becomes 1 more?

The arithmetic mean is equal to the sum of all numbers divided by their number. Let the sum of all numbers be . By the condition of the problem , therefore .

The arithmetic mean has increased by 1, that is, it has become equal to 9. If one of the numbers has been increased by , then the sum has increased by and has become equal to .

The number of numbers has not changed and is equal to 6.

We get the equality: