Solve 20 basic level quests. Preparation for the exam in mathematics (profile level): tasks, solutions and explanations

Consider such a task plan. We have the following conditions:

Total:N

Of A pieces at least 1 of another type, and of B pieces at least 1 of the first type

Then: (A-1) is the minimum quantity of the first type, and (B-1) is the second.

Then we check: (A-1) + (B-1) \u003dN.

EXAMPLE

DECISION

So: we have 35 fish in total (perches and roach)

Consider the conditions: among any 21 fish there is at least one roach, which means there is at least 1 roach in this condition, therefore (21-1) \u003d 20 is the minimum of perches. Among any 16 fish there is at least one perch, arguing similarly, (16-1) \u003d 15 is the minimum of roach. Now we check: 20 + 15 \u003d 35, that is, we got the total number of fish, which means 20 perches and 15 roach.

ANSWER: 15 roach

Quiz and the number of correct answers

The list of tasks for the quiz consisted of A questions. For each correct answer, the student received a points, for the wrong answer he was cheatedb points, and in the absence of an answer they gave 0 points. How many correct answers did the student who typedN points, if it is known that at least once he was wrong?

We know how many points he earned, we know the cost of a correct and incorrect answer. Based on the fact that at least one incorrect answer was given, then the number of points for correct answers must exceed the number of penalty points forN points. Let x correct answers and wrong ones were given, then:

a*x= N+ b* y

x \u003d (N+ b* y)/a

from this equality it is clear that the number in parentheses must be a multiple of a. With this in mind, we can estimate y (it is also an integer). It should be borne in mind that the number of correct and incorrect answers should not exceed the total number of questions.

EXAMPLE

DECISION:

we introduce the notation (for convenience) x - correct, y - incorrect, then

5 * x \u003d 75 + 11 * y

X \u003d (75 + 11 * y) / 5

Since 75 is divisible by five, then 11 * y must also be divisible by five. Therefore, y can take on values \u200b\u200bthat are multiples of five (5, 10, 15, etc.). take the first value y \u003d 5 then x \u003d (75 + 11 * 5) / 5 \u003d 26 total questions 26 + 5 \u003d 31

Y \u003d 10 x \u003d (75 + 11 * 10) \u003d 37 total answers 37 + 10 \u003d 47 (more than questions) does not fit.

It means that there were 26 correct and 5 incorrect answers in total.

ANSWER: 26 correct answers

Which floor?

Sasha invited Petya to visit, saying that he lived in the entrance of apartment no.N, and forgot to say the floor. Approaching the house, Petya discovered that the housey-storey. What floor does Sasha live on? (On all floors, the number of apartments is the same; apartment numbers in the building start with one.)

DECISION

According to the condition of the problem, we know the apartment number, entrance and the number of floors in the house. Based on these data, an estimate can be made of the number of apartments per floor. Let x be the number of apartments on the floor, then the following condition must be met:

A * y * x must be greater than or equalN

From this inequality we estimate x

First, we take the minimum integer value of x, let it be equal to c, and check: (a-1) * y * c is lessN, and a * y * c is greater than or equal toN.

Having chosen the value x we \u200b\u200bneed, we can easily calculate the floor (at): at \u003d (N-( a-1)* c)/ c, and b is an integer and getting a fractional value, we take the nearest integer (upward)

EXAMPLE

DECISION

Let's estimate the number of apartments on the floor: 7 * 7 * x is greater than or equal to 462, hence x is greater than or equal to 462 / (7 * 7) \u003d 9.42 means the minimum x \u003d 10. We do a check: 6 * 7 * 10 \u003d 420 and 7 * 7 * 10 \u003d 490, as a result, we got that the apartment by number falls into this range. Now we find the floor: (462-6 * 7 * 10) / 10 \u003d 4.2 so the boy lives on the fifth floor.

ANSWER: 5th floor

Apartments, floors, entrances

All entrances of the building have the same number of floors, and all floors have the same number of apartments. Moreover, the number of floors in a building is greater than the number of apartments on a floor, the number of apartments on a floor is greater than the number of entrances, and the number of entrances is more than one. How many floors are there in the house if there are X apartments in total?

This type of problem is based on the following condition: if in the house there are E - floors, P - entrances and K - apartments on the floor, then the total number of apartments in the house should be equal to E * P * K \u003d X. so we need to represent X as a product of three numbers that are not equal to 1 (by the problem statement). To do this, we will decompose the number X into prime factors. Having made the expansion and taking into account the conditions of the problem, we make a selection of the correspondence of the numbers and those conditions that are indicated in the problem.

EXAMPLE

DECISION

We represent the number 105 as a product of prime factors

105 \u003d 5 * 7 * 3, now let's return to the condition of the problem: since the number of floors is the largest, it is equal to 7, the number of apartments on the floor is 5, and the number of entrances is 3.

ANSWER: entrances - 7, apartments on the floor - 5, entrances - 3.

Exchange

For a gold coins receive from silver and copper;

For x silver coins, get gold and 1 copper.

Nikolai only had silver coins. After the exchange office, he had fewer silver coins, no gold coins appeared, but C copper coins appeared. By how much did Nikolai's number of silver coins decrease?

There are two exchange schemes in the punkte exchange:

EXAMPLE

IN an exchange office can perform one of two operations:

DECISION

5 gold \u003d 4 silver + 1 copper

10 silver \u003d 7 gold + 1 copper

since no gold coins appeared, we need an exchange scheme without gold coins. Therefore, the number of gold coins must be equal in both cases. We need to find the least common multiple of 5 and 7, and bring our gold in both cases to it:

35 gold \u003d 28 silver + 7 copper

50 silver \u003d 35 gold + 5 copper

in the end we get

50 silver \u003d 28 silver + 12 copper

We found an exchange scheme bypassing gold coins, now we need, knowing the number of copper coins, to find how many times such an operation was performed

N=60/12=5

As a result, we get

250 silver \u003d 140 silver + 60 copper

Substituting and receiving the final exchange, we will find how much silver was exchanged. This means that the quantity decreased by 250-140 \u003d 110

ANSWER for 110 coins

6. THE GLOBE

On the surface of the globe, a marker has drawn x parallels and a meridian. How many parts did the drawn lines divide the surface of the globe? (the meridian is a circular arc connecting the North and South poles, and parallel is the boundary of the section of the globe by a plane parallel to the equatorial plane).

DECISION:

Since the parallel is the boundary of the section of the globe with a plane, then one will split the globe into 2 parts, two into three parts, x into x + 1 parts

The meridian is an arc of a circle (more precisely, a semicircle) and at the meridians the surface is divided into parts, therefore, all will be (x + 1) * parts.

EXAMPLE

Following similar reasoning, we get:

(30 + 1) * 24 \u003d 744 (parts)

ANSWER: on part 744

7. SAWS

Cross lines of red, yellow and green are marked on the stick. If you cut the stick along the red lines, you get A pieces, if you cut the yellow ones - B pieces, and if you cut the green ones - C pieces. How many pieces would you get if you sawed the stick along the lines of all three colors?

DECISION

For the solution, take into account that the number of pieces is 1 more than the number of cuts. Now you need to find how many lines are marked on the stick. We get red (A-1), yellow - (B-1), green - (C-1). Having found the number of lines of each color and summing them up, we get the total number of lines: (A-1) + (B-1) + (C-1). We add one to the resulting number (since the number of pieces is one more than the number of cuts), we get the number of pieces if we cut along all the lines.

EXAMPLE

Cross lines of red, yellow and green are marked on the stick. If you cut the stick along the red lines, you get 7 pieces, if you cut the yellow ones - 13 pieces, and if you cut the green ones - 5 pieces. How many pieces would you get if you sawed the stick along the lines of all three colors?

DECISION

Find the number of lines

Red: 7 - 1 \u003d 6

Yellow: 13-1 \u003d 12

Greens: 5-1 \u003d 4

Total number of lines: 6 + 12 + 4 \u003d 22

Then the number of pieces: 22 + 1 \u003d 23

ANSWER: 23 pieces

8. COLUMNS AND ROWS

IN each cell of the table was set natural number so that the sum of all the numbers in the first column is C1, in the second - C2, in the third - C3, and the sum of the numbers in each row is greater than Y1, but less than Y2. How many rows are there in the table?

DECISION

Since the numbers in the cells of the table do not change, the sum of all the numbers in the table is: С \u003d С1 + С2 + С3.

Now let's pay attention to the fact that the table consists of natural numbers, which means that the sum of the numbers along the lines must be integers and be in the range from (U1 + 1) to (U2-1) (since the sum of the lines is strictly limited). Now we can estimate the number of lines:

С / (У1 + 1) - maximum amount

С / (У2-1) - minimum quantity

EXAMPLE

IN the table has three columns and several rows. IN

DECISION

Find the sum of the table

C \u003d 85 + 77 + 71 \u003d 233

Determine the boundaries of the sum of rows

12 + 1 \u003d 13 - minimum

15-1 \u003d 14 - maximum

Let's estimate the number of rows in the table

233/13 \u003d 17.92 maximum

233/14 \u003d 16.64 minimum

Within these limits, only one integer is enclosed - 17

ANSWER: 17

9. FILLING ON RING

and G. Distance between A and B - 35 km, between A and in - 20 km, between B and G - 20 km, between G and A and V.

DECISION

After carefully reading the problem, we will notice that practically the circle is divided into three arcs AB, VG and AG. Based on this, we will find the length of the entire circle (annular). For this problem, it is equal to 20 + 20 + 30 \u003d 70 (km).

Now, having placed all points on the circle and signed the lengths of the corresponding arcs, it is easy to determine the desired distance. In this problem, BV \u003d AB-AB, that is, BV \u003d 35-20 \u003d 15

ANSWER: 15 km

10. COMBINATIONS

DECISION

To solve this type of problem, remember what a factorial is.

Factorial numberN! is called the product of consecutive numbers from 1 toN, that is, 4! \u003d 1 * 2 * 3 * 4.

Now let's get back to the problem. Find the total number of cubes: 3 + 1 + 1 \u003d 5. Since we have three cubes of the same color, the total number of cubes can be found using the formula 5! / 3! We get (5 * 4 * 3 * 2 * 1) / (1 * 2 * 3) \u003d 5 * 4 \u003d 20

ANSWER: 20 ways of placing

11 ... WELLS

The owner agreed with the workers that they would dig a well for him on the following conditions: for the first meter he would pay them X rubles, and for each next meter he would pay U rubles more than the previous one. How many rubles will the owner have to pay to the workers if they dig a well deepN meters?

DECISION:

Since the owner increases the price for each meter, he will pay for the second (X + Y), for the third - (X + 2Y), for the fourth (X + 3Y), etc. It is not difficult to see that this payment system resembles an arithmetic progression, where a1 \u003d X,d= Y, n= N... Then

Payment for work is nothing more than the sum of this progression:

S= ( (2a₁ + d (n-1)) / 2) n

EXAMPLE:

DECISION

Based on the above, we obtaina1=4200

d \u003d 1300

n \u003d 11

substituting this data into our formula, we get

S \u003d ((2 * 4200 + 1300 (11-1) / 2) * 11 \u003d ((8400 + 13000) / 2) * 11 \u003d 10700 * 11 \u003d 117700

ANSWER: 117700

12 . POSTS AND WIRES

X pillars are interconnected by wires, so that exactly Y wires depart from each. How many wires are there between the posts?

DECISION

Let's find how many gaps between the posts. Between two there is one gap, between three - two, between four - 3, between X - (X-1).

At each gap of Y wires, then (X-1) * Y is the total of wires between the posts.

EXAMPLE

Ten pillars are interconnected by wires, so that exactly 6 wires depart from each. How many wires are there between the posts?

DECISION

Returning to the previous notation, we get:

X \u003d 9 Y \u003d 6

Then we get (9-1) * 6 \u003d 8 * 6 \u003d 48

ANSWER: 48

13. SAWING BOARDS AND LOGS

There were several logs. We made X cuts and it turned out At the block. How many logs have you sawed?

DECISION

When solving, we will make one remark: some problems do not always have a mathematical solution.

Now to the task. When deciding, it is necessary to take into account that there are more than one logs and when cutting each log it turns out \u003d 1 piece.

It is more convenient to solve this type of problem by the selection method:

Let there be two logs, then the pieces will be 13 + 2 \u003d 15

Let's take three we get 13 + 3 \u003d 16

And here you can see the dependence that the number of cuts and pieces increases equally, that is, the number of logs that need to be cut is equal to Y-X

EXAMPLE

There were several logs. We made 13 cuts and got 20 shanks. How many logs have you sawed?

DECISION

Returning to our reasoning, we can select, or you can simply 20-13 \u003d 7 means only 7 logs

Answer 7

14 ... LOST PAGES

Several pages in a row fell out of the book. The first of the dropped pages has the number X, and the number of the last is written in the same digits in some other order. How many pages are missing from the book?

DECISION

The page numbers that have dropped out start with an odd number and must end with an even number. Therefore, knowing that the number of the last dropped out is written in the same digits as the first dropped, we know its last figure. By rearranging the remaining digits and, taking into account that the page numbering must be greater than the first dropped out, we get its number. Knowing the page numbers, you can calculate how many fell out, while taking into account that page X also fell out. So from the resulting number, we must clean out the number (X-1)

EXAMPLE

Several pages in a row fell out of the book. The first of the dropped pages has the number 387, and the number of the last one is recorded in the same digits in some other order. How many pages are missing from the book?

DECISION

Based on our reasoning, we find that the number of the last dropped out page should end with the digit 8. So we have only two options for numbers - 378 and 738. 378 does not suit us because it is less than the number of the first dropped page, so the last dropped out is 738

738-(387-1)=352

ANSWER: 352

The following should be added: sometimes it is asked to indicate the number of sheets, then the number of pages should be divided in half.

15. FINAL GRADE

At the end of the quarter, Vovochka wrote down his current singing marks in a row in a row and put a multiplication sign between some of them. The products of the resulting numbers turned out to be equal to X. What mark does Vovochka get in singing quarters?

DECISION

When solving this type of problems, it is necessary to take into account that its estimates should be 2,3,4 and 5. Therefore, we need to decompose the number X into factors 2,3,4 and 5. Moreover, the remainder of the decomposition should also consist of these numbers.

EXAMPLE 1

At the end of the quarter, Vovochka wrote down his current singing marks in a row in a row and put a multiplication sign between some of them. The products of the resulting numbers turned out to be equal to 2007. What mark does Vovochka get in singing quarters?

DECISION

Factor 2007

We get 2007 \u003d 3 * 3 * 223

So his grades: 3 3 2 2 3 now find the arithmetic mean of his scores for this set is 2.6 hence his score is three (more than 2.5)

ANSWER 3

EXAMPLE 2

At the end of the quarter, Little Johnny wrote out all his marks in a row 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 690. What mark does Vovochka get in a quarter in this subject, if the teacher puts only marks 2, 3, 4 and 5 and the final mark in a quarter is the arithmetic average of all current marks, rounded according to the rounding rules? (For example: 2.4 rounds to two; 3.5 rounds up to 4; and 4.8 rounds up to 5.)

DECISION

Let us factor 690 so that the remainder of the decomposition consists of the numbers 2 3 4 5

690=3*5*2*23

Hence his grades: 3 5 2 2 3

Find the arithmetic mean of these numbers: (3 + 5 + 2 + 2 + 3) / 5 \u003d 3

This will be his assessment.

ANSWER: 3

16 ... MENU

The restaurant menu has X types of salads, U type of first courses, A type of main courses and B type of dessert. How many salad, first, second and dessert dining options can the diners of this restaurant choose?

DECISION

When deciding, we will slightly cut the menu: let there be only salad and then the first options will become (X * Y). Now let's add the second dish, the number of options increases by A times and becomes (X * Y * A). Well, now let's add dessert. The number of options will increase in times

We now get the final answer:

N \u003dX * Y * A * B

EXAMPLE

DECISION
Based on the above, we get:

N \u003d 6 * 3 * 5 * 4 \u003d 360

ANSWER: 360

17 ... DIVIDE WITHOUT REMAINS

In this section, we will consider the tasks with a specific example, for greater clarity.

Since we have a product of consecutive numbers and there are more than 7 of them, then at least one must be divisible by 7. So we have a product, one of the factors of which is divisible by 7, therefore the whole product is also divisible by seven, which means the remainder of the division will be equal to zero, or for the second problem the number of factors must be equal to the divisor.

18. TOURISTS

We will also consider this type of tasks using a specific example.

First, let's define what we need to find: route time \u003d ascent + rest + descent

We know rest, now we need to find the time of ascent and descent

Reading the problem, we see that in both cases (ascent and descent) the time depends as an arithmetic progression, but we do not yet know what altitude the ascent was, although it is not difficult to find it:

H=(95-50)15+1=4

We found the ascent height, now we will find the ascent time as the sum of the arithmetic progression: Tpolling \u003d ((2 * 50 + 15 * (4-1)) * 4) / 2 \u003d 290 minutes

Similarly, we find, considering that now the difference in the progression is -10. We get Tspuska \u003d ((2 * 60-10 (4-1)) * 4) / 2 \u003d 180 minutes.

Knowing all the components, you can calculate the total route time:

Route T \u003d 290 + 180 + 10 \u003d 480 minutes or converting into hours (divide by 60) we get 8 hours.

ANSWER: 8 hours

19.RECTANGLES

There are two types of problems on rectangles: perimeters and squares

To solve such a plan of problems, it is not difficult to prove that when we split any rectangle with two straight cuts, we will get four rectangles for which the following relations will always be satisfied:

P1 + P2 \u003d P3 + P4

S1 * S2 \u003d S3 * S4,

where R – perimeter , S - square

Based on these relationships, we can easily solve the following problems

19.1 Perimeters

DECISION

Based on the above, we obtain

24 + 16 \u003d 28 + X

X \u003d (24 + 16) -28 \u003d 12

ANSWER: 12

19.2 AREA

The rectangle is split into four small rectangles by two straight cuts. The areas of three of them, starting from the upper left and further clockwise, are equal to 18, 12 and 20. Find the area of \u200b\u200bthe fourth rectangle.

DECISION

For the resulting rectangles, the following should be performed:

18 * 20 \u003d 12 * X

Then X \u003d (18 * 20) / 12 \u003d 30

ANSWER: 30

20. THERE-HERE

A snail crawls up the tree to A m in a day, and slides down to B m during the night. The height of a tree is C m. How many days does a snail first crawl to the top of a tree?

DECISION

In one day, a snail can rise to a height (AB) meters. Since she can climb to the height A in one day, then before the last ascent she needs to overcome the height (C-A). Based on this, we get that it will rise (C-A) \\ (A-B) +1 (we add one unit since it rises to height A in one day).

EXAMPLE

DECISION

Returning to our reasoning, we get

(10-4)/(4-3)+1=7

ANSWER in 7 days

It should be noted that in this way it is possible to solve problems of filling something, when something comes in and something flows out.

21. STRAIGHT JUMPS

The grasshopper jumps along the coordinate line in any direction one segment per jump. How many different points on the coordinate line are there where a grasshopper can find itself after making X jumps, starting from the origin?

DECISION

Suppose that the grasshopper makes all the jumps in one direction, then he gets to the point with the X coordinate. Now he jumps forward on (X-1) jumps and one back: he gets to the point with the coordinate (X-2). Considering all his jumps in this way, you can see that he will be at points with coordinates X, (X-2), (X-4), etc. This dependence is nothing more than an arithmetic progression with the differenced\u003d -2 and a1 \u003d X, aan=- X... Then the number of members of this progression is the number of points at which it can be. Let's find them

an \u003d a1 + d (n-1)

X \u003d X + d (n-1)

2X \u003d -2 (n-1)

n \u003d X + 1

EXAMPLE

DECISION

Based on the above findings, we obtain

10+1=11

ANSWER 11 points

TASKS FOR AN INDEPENDENT SOLUTION:

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

2. Cross lines of red, yellow and green are marked on the stick. If you cut the stick along the red lines, you get 15 pieces, if you cut the yellow ones - 5 pieces, and if you cut the green ones - 7 pieces. How many pieces would you get if you sawed the stick along the lines of all three colors?

3. The grasshopper jumps along the coordinate line in any direction one segment at a time. The grasshopper starts jumping from the origin. How many different points on the coordinate line are there that a grasshopper can find itself in after making exactly 11 jumps?

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

5. Sasha invited Petya to visit, saying that he lived in the seventh staircase 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 start with one.)

6. Sasha invited Petya to visit, saying that he lived in the eighth staircase in apartment No. 468, but forgot to say the floor. Approaching the house, Petya discovered that the house was twelve-story. What floor does Sasha live on? (On all floors, the number of apartments is the same; apartment numbers in the building start with one.)

7. 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 start with one.)

8. Sasha invited Petya to visit, saying that he lived in the tenth staircase in apartment number 333, but forgot to say the floor. Approaching the house, Petya discovered that the house was nine stories high. What floor does Sasha live on? (On all floors the number of apartments is the same; apartment numbers in the building start with one.)

9. The coach advised Andrey to spend 15 minutes on the treadmill on the first day of training, and increase the time spent on the treadmill by 7 minutes at each next lesson. How many lessons will Andrey spend on the treadmill for a total of 2 hours 25 minutes if he follows the coach's advice?

10. 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. After taking 30 drops, he drinks 30 drops of the medicine for another 3 days, and then daily reduces the intake by 3 drops. 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)?

11. 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 the previous one. After 15 days of taking, the patient takes a break of 3 days and continues to take the medicine in 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?

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

13. In how many ways can two identical red cubes, three identical green cubes and one blue cubic be placed in a row?

14. A full bucket of water with a volume of 8 liters is poured into a 38 liter tank every hour, starting at 12 noon. But there is a small gap in the bottom of the tank, and 3 liters flows out of it in an hour. At what point in time (in hours) the tank will be full.

15. What is the smallest number of consecutive numbers you need to take so that their product is divisible by 7?

16. As a result of the flood, the foundation pit was filled with water to the level of 2 meters. The construction pump continuously pumps out water, lowering its level by 20 cm per hour. Subsoil waters, on the other hand, raise 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?

17. The restaurant's menu includes 6 types of salads, 3 types of first courses, 5 types of second courses and 4 types of desserts. How many salad, first, second and dessert dining options can the diners of this restaurant choose?

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

19. What is the smallest number of consecutive numbers you need to take so that their product is divisible by 9?

20.

for 2 gold coins get 3 silver and one copper;

for 5 silver coins get 3 gold and one copper.

21. 12 parallels and 22 meridians are drawn on the surface of the globe with a felt-tip pen. How many parts did the drawn lines divide the surface of the globe?

The meridian is a circular arc that connects the North and South Poles. A parallel is a circle in a plane parallel to the equatorial plane.

22. The basket contains 50 mushrooms: mushrooms and milk mushrooms. It is known that among any 28 mushrooms there is at least one mushroom, and among any 24 mushrooms there is at least one milk mushroom. How many lunches are in the basket?

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

24. There are four petrol 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.

25. There are four petrol stations on the ring road: A, B, C 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 measured along the ring road in the shortest direction). Find the distance between B and C.

26. There are 25 students in the class. Several of them went to the cinema, 18 people went to the theater, moreover, to the cinema, and 12 people went to the theater. It is known that the three did not go to the cinema or the theater. How many people from the class went to the movies?

27. By Moore's rule of thumb, the average number of transistors on a chip doubles every year. It is known that in 2005 the average number of transistors on a microcircuit was 520 million. Determine how many millions of transistors were on an average in 2003.

28. 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 there in the eighth row?

29. Cross lines of red, yellow and green are marked on the stick. If you cut the stick along the red lines, you get 5 pieces, if you cut the yellow ones - 7 pieces, and if you cut the green ones - 11 pieces. How many pieces would you get if you sawed the stick along the lines of all three colors?

30. In a home appliance store, refrigerator sales are seasonal. 10 refrigerators were sold in January, and 10 refrigerators were sold in the next three months. Since May, sales have increased by 15 units over the previous month. Since September, the sales volume began to decrease by 15 refrigerators each month compared to the previous month. How many refrigerators did the store sell in a year?

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

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

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

Nikola only had silver coins. After visiting the exchange office, he had fewer silver coins, no gold coins appeared, but 35 copper coins appeared. How much has Nikola's number of silver coins decreased?

32. Sasha invited Petya to visit, saying that he lived in the seventh staircase 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 each floor, the number of apartments is the same; apartment numbers in the building start with one.)

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

34. The grasshopper jumps along the coordinate line in any direction one segment per jump. How many different points on the coordinate line are there where a grasshopper can find itself after making exactly 6 jumps, starting from the origin?

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

36. The basket contains 25 mushrooms: mushrooms and milk mushrooms. It is known that among any 11 mushrooms there is at least one mushroom, and among any 16 mushrooms there is at least one milk mushroom. How many mushrooms are in the basket?

37. The basket contains 30 mushrooms: mushrooms and milk mushrooms. It is known that among any 12 mushrooms there is at least one mushroom, and among any 20 mushrooms there is at least one milk mushroom. How many mushrooms are in the basket?

38. On the globe, a felt-tip pen has drawn 17 parallels (including the equator) and 24 meridians. How many parts do the drawn lines divide the surface of the globe?

39. A snail crawls up the tree by 4 m per day, and during the night it slides by 3 m. The height of the tree is 10 m. How many days will the snail first crawl to the top of the tree?

40. A snail crawls up the tree by 4 m per day, and slides by 1 m during the night. The height of the tree is 13 m. How many days will the snail first crawl to the top of the tree?

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

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

43. The basket contains 45 mushrooms: mushrooms and milk mushrooms. It is known that among any 23 mushrooms there is at least one mushroom, and among any 24 mushrooms there is at least one milk mushroom. How many mushrooms are in the basket?

44. The basket contains 25 mushrooms: mushrooms and milk mushrooms. It is known that among any 11 mushrooms there is at least one mushroom, and among any 16 mushrooms there is at least one milk mushroom. How many mushrooms are in the basket?

45. 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, 0 points were given. How many correct answers did a student who scored 42 points give if he was known to be wrong at least once?

46. Cross lines in red, yellow and green are marked on the stick. If you cut the stick along the red lines, you get 5 pieces, if you cut the yellow ones - 7 pieces, and if you cut the green ones - 11 pieces. How many pieces would you get if you sawed the stick along the lines of all three colors?

47. A snail crawls up the tree by 2 m per day, and during the night it slides by 1 m. The height of the tree is 11 m. How many days will the snail crawl from the base to the top of the tree?

48. A snail crawls up the tree by 4 m per day, and during the night it slides by 2 m. The height of the tree is 14 m. How many days will the snail crawl from the base to the top of the tree?

49. The rectangle is split into four smaller rectangles by two straight cuts. The perimeters of three of them, starting from the upper left and further clockwise, are equal to 24, 28 and 16. Find the perimeter of the fourth rectangle.

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

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

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

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

51. The rectangle is split into four smaller rectangles by two straight cuts. The perimeters of three of them, starting from the upper left and further clockwise, are equal to 24, 28 and 16. Find the perimeter of the fourth rectangle.

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

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

2) For 7 silver coins, get 5 gold and one copper.

Nikolai only had silver coins. After several visits to the exchange office, he had fewer silver coins, no gold coins appeared, but 90 copper coins appeared. By how much did Nikolai's number of silver coins decrease?

53. All entrances of the building have the same number of floors, and each floor has the same number of apartments. At the same time, the number of entrances to a building is less than the number of apartments on a floor, the number of apartments on a floor is less than the number of floors, the number of entrances is more than one, and the number of floors is not more than 24. How many floors are there in a building if there are only 156 apartments?

54. IN the class has 26 students. Several of them listen to rock, 14 people listen to rap, and only three of them listen to both rock and rap. It is known that the four do not listen to either rock or rap. How many people in the class listen to rock?

55. IN the cage contains 35 fish: perches and roach. It is known that among any 21 fish there is at least one roach, and among any 16 fish there is at least one perch. How many roaches are in the cage?

56. On the surface of the globe, a marker has drawn 30 parallels and 24 meridians. How many parts did the drawn lines divide the surface of the globe? (the meridian is the arc of a circle connecting the North and South poles, and the parallel is the boundary of the section of the globe by a plane parallel to the equatorial plane).

57. IN prehistoric exchange office could perform one of two operations:
- for 2 cave lion skins get 5 tiger skins and 1 boar skin;
- For 7 tiger skins get 2 cave lion skins and 1 boar skin.
Un, son of the Bull, had only tiger skins. After several visits to the exchange office, his tiger skins did not increase, the skins of the cave lion did not appear, but 80 boar skins appeared. How much, in the end, did the number of tiger skins in Un, the son of the Bull, decrease?

58. IN military unit 32103 has 3 types of salad, 2 types of first course, 3 types of second course and a choice of compote or tea. How many options for a dinner, which necessarily consists of one salad, one first course, one second course and one drink, can the servicemen of this military unit choose?

59. A snail crawls 5 meters up the tree during the day, and slides down 3 meters during the night. The height of the tree is 17 meters. On what day will the snail first crawl to the top of the tree?

60. In how many ways can three identical yellow cubes, one blue cubes, and one green cubes be placed in a row?

61. The product of sixteen consecutive natural numbers was divided by 11. What can be the remainder of the division?

62. Every minute a bacteria divides into two new bacteria. It is known that bacteria fill the entire volume of a three-liter jar in 4 hours. How many seconds does bacteria take to fill a quarter of the jar?

63. 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, 0 points were given. How many correct answers did the student who scored 75 points give if it is known that he was wrong at least once?

64. A grasshopper jumps on a straight road, the length of one jump is 1 cm. First, he jumps 11 jumps forward, then 3 backwards, then again 11 jumps and then backwards 3 jumps, and so on, how many jumps he will make by the time he first reaches a distance of 100 cm from the beginning.

65. Cross lines of red, yellow and green are marked on the stick. If you cut the stick along the red lines, you get 7 pieces, if you cut the yellow ones - 13 pieces, and if you cut the green ones - 5 pieces. How many pieces would you get if you sawed the stick along the lines of all three colors?

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

67. The rectangle is split into four smaller rectangles by two straight cuts.
The perimeters of three of them, starting from the upper left and further clockwise, are equal to 24, 28 and 16. Find the perimeter of the fourth rectangle.

68. IN an exchange office can perform one of two operations:
1) for 4 gold coins, get 5 silver and one copper;
2) For 7 silver coins, get 5 gold and one copper.
Nikola only had silver coins. After visiting the exchange office, he had less silver coins, no gold coins appeared, but 90 copper coins appeared. How much has the amount of silver coins decreased?

69. A snail crawls up the tree by 4 m during the day, and slides down by 2 m during the night. The height of the tree is 12 m. How many days will the snail crawl from the base to the top of the tree?

70. The list of tasks of the quiz consisted of 32 questions. For each correct answer, the student receives 5 points. 9 was written off for wrong, 0 points were given if there was no answer.
How many correct answers did the student who scored 75 points give if he made at least 2 mistakes?

71. 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, 0 points were given. How many correct answers did a student who scored 42 points give if he was known to be wrong at least once?

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

73. The rectangle is split into four small rectangles by two straight cuts. The areas of three of them, starting from the upper left and further clockwise, are equal to 18, 12 and 20. Find the area of \u200b\u200bthe fourth rectangle.

74. The rectangle is split into four small rectangles by two straight cuts. The areas of three of them, starting from the upper left and further clockwise, are equal to 12, 18 and 30. Find the area of \u200b\u200bthe fourth rectangle.

75. IN the table has three columns and several rows. IN each cell of the table was set by a natural number so that the sum of all the numbers in the first column is 85, in the second - 77, in the third - 71, and the sum of the numbers in each row is more than 12, but less than 15. How many rows are there in the table?

76. The grasshopper jumps along the coordinate line in any direction one segment per jump. How many different points on the coordinate line are there where a grasshopper can find itself after making 10 jumps, starting from the origin?

77. Sasha invited Petya to visit, saying that he lived in the seventh staircase 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 start with one.)

78. IN an exchange office can perform one of two operations:
for 2 gold coins get 3 silver and one copper;
for 7 silver coins get 3 gold and one copper.
Nikolai only had silver coins. After the exchange office, he did not have gold coins, but 20 copper coins appeared. By how much did Nikolai's number of silver coins decrease?

79. The grasshopper jumps along the coordinate line in any direction one segment per jump. How many different points on the coordinate line are there where a grasshopper can find itself after making 11 jumps, starting from the origin?

80. There are four petrol stations on the ring road: A, B, C and G. Distance between A and B - 35 km, between A and in - 20 km, between B and G - 20 km, between G and A - 30 km (all distances are measured along the ring road along the shortest arc). Find the distance (in kilometers) between B and V.

81. IN an exchange office can perform one of two operations:
for 4 gold coins get 5 silver and one copper;
for 7 silver coins get 5 gold and one copper.
Nikolai only had silver coins. After the exchange office, he had fewer silver coins, Gold coins did not appear, but 90 copper coins appeared. By how much has decreased the number of silver coins from Nicholas.

82. A grasshopper jumps along the coordinate line in any direction one segment per jump. How many points are there on the coordinate line where a grasshopper can find itself, having made exactly 8 jumps, starting from the origin?

83. IN an exchange office can perform one of two operations:
for 5 gold coins get 4 silver and one copper;
for 10 silver coins get 7 gold and one copper.
Nikolai only had silver coins. After the exchange office, he had fewer silver coins, no gold coins appeared, but 60 copper coins appeared. By how much did Nikolai's number of silver coins decrease?

84. IN an exchange office can perform one of two operations:
for 5 gold coins get 6 silver and one copper;
for 8 silver coins get 6 gold and one copper.
Nikolai only had silver coins. After the exchange office, he had fewer silver coins, no gold coins appeared, but 55 copper coins appeared. By how much did Nikolai's number of silver coins decrease?

85. All entrances of the building have the same number of floors, and all floors have the same number of apartments. Moreover, the number of floors in a building is greater than the number of apartments on a floor, the number of apartments on a floor is greater than the number of entrances, and the number of entrances is more than one. How many floors are there in the building if there are 105 apartments in total?

86. IN an exchange office can perform one of two operations:
1) for 3 gold coins, get 4 silver and one copper;
2) for 7 silver coins, get 4 gold and one copper.
Nikola only had silver coins. After visiting the exchange office, he had fewer silver coins, no gold coins appeared, but 42 copper coins appeared. How much has Nikola's number of silver coins decreased?

ANSWERS

Yakovleva Natalia Sergeevna
Position: mathematic teacher
Educational institution: MCOU "Buninskaya Secondary School"
Locality: Bunino village, Solntsevsky district, Kursk region
Material name: article
Theme: "Methods for solving tasks No. 20 of the exam in mathematics, basic level"
Date of publication: 05.03.2018
Section: complete education

Single state exam is currently the only

form final certification graduates high school... And getting

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

mathematics. Mathematics is not only important academic subjectbut

and quite complex. Math skills are far

not all children, and their future fate depends on the successful passing of the exam.

High school teachers ask the question over and over again: “How to help

to a student in preparation for the Unified State Exam and pass it successfully? " To

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

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

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

solving task number 20 mathematician basic level FIPI 2018 under

edited by M.V. Yashchenko.

1 There are two stripes on the tape on opposite sides of the middle: blue and

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

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

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

stripes.

Decision:

Let a cm be the distance from the left end of the tape to the blue strip, in cm

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

between the stripes. It is known that if the ribbon is cut along the red stripe, 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 b + c -

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

2 ... The arithmetic mean of 6 distinct natural numbers is 8. On

how much to increase the largest of these numbers so that the average

arithmetic became 1 more.

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

therefore, the sum of these numbers is 8 * 6 \u003d 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 stan numbers is 9 * 6 \u003d 54. To find how much one has increased

from numbers, you need to find the difference 54-48 \u003d 6.

3. The cells of the 6x5 table are colored black and white. Couples adjacent

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

adjacent cells are white.

Decision:

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

horizontals in total will be 5 * 5 \u003d 25 pairs of adjacent cells. Vertically

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

the vertical will be 4 * 6 \u003d 24. In total, 24 + 25 \u003d 49 pairs of neighboring cells are formed. Of

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

26-6 \u003d 17 pairs.

Answer: 17.

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

red. To the left of the red vase there are 15 roses, to the right of the blue vase 12

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

Decision: Let x roses be in a white vase, roses - in blue, z roses - in

red. According to the condition of the problem, there are 22 roses in vases, that is, x + y + z \u003d 22. It is known

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

to the right of the blue vase, that is, there are 12 roses in the white and red vases, which means x + z \u003d 12.

Got:

Add the term-by-term 2nd and 3rd equalities: x + y + x + z \u003d 27 or 22 + x \u003d 27, x \u003d 5.

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

at the same time. First, Masha ate jam, and Bear ate cookies, but in some

moment 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?

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

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

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

Masha, it means that the Bear absorbed food 9 times faster than Masha. Then let x

masha ate 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 \u003d 144 cookies.

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

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

the dropped sheets are recorded in the same numbers, but in a different order.

How many sheets fell out?

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

there 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 out

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 \u003d 85.

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

less than 9. Guessed a natural number, then multiplied by A, added B and

subtracted C. Received 164. What number was conceived?

Decision:Let x be a hidden natural number, then Ax + B-C \u003d 164, Ax \u003d

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

hence, Ax \u003d 166; 165; 164; 163; 162. Of the numbers 6,7,8, only 6 is

Assignment number 20 of the exam in mathematics contains a task for quick wits. The tasks in this section are more intuitive than in task 19 of the USE, but nevertheless they are quite difficult for an ordinary student. So let's move on to considering standard options.

Analysis of typical options for assignments No. 20 of the USE in mathematics of the basic level

The first variant of the task (demo version 2018)

for 2 gold coins get 3 silver and one copper;
for 5 silver coins get 3 gold and one copper.

Execution algorithm:

Enter the legend.
Record task data using legend.
Determine the unknown logically.

Decision:

According to the condition, no gold coins appeared, which means that all gold coins received after the second operation were exchanged by Nikolai using the first operation. Gold coins can only be exchanged for 2 pieces, therefore, there were an even number of second operations.

Let us introduce the notation, let the second operations be 2n (the number is always even).

If we apply the second operation, we get:

All gold coins were exchanged in the first transaction. In one operation, you can exchange 2 gold coins at once, which means that the total number of transactions will be (3 2n) / 2 \u003d 3 n. I.e

3 · 2n gold coins were exchanged for 3 · 3n silver + 3n copper.

Or after conversion:

Let's compare the results of the first and second operations:

5 · 2n silver was exchanged for 3 · 2n gold + 2n copper.

3 2n gold exchanged for 9n silver + 3n copper

5 2n silver traded for 9n silver + 3n copper + 2n copper

10 n silver traded for 9n silver + 5n copper

If, having exchanged 10 n silver coins, we get 9 n silver coins, then Nikolay's number of silver coins decreased by n. From the last expression it can be seen that Nikolai received 5n copper coins, and according to the condition, 50 copper coins appeared, that is, 5n \u003d 50.

Second variant of the task

Masha and the Bear ate 100 cookies and a jar of jam, starting and ending 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?

Execution algorithm:

Compare results.
Find the unknown.

Decision:

Since both Masha and the Bear ate the jam equally, and the Bear ate the jam 3 times faster, Masha ate the jam (her half) 3 times longer than the Bear (the same half).
Then it turns out that the Bear ate cookies 3 times longer than Masha and, moreover, ate them 3 times faster, that is, for one cookie Masha eaten, there were 3 ∙ 3 \u003d 9 cookies eaten by the Bear.
In total, these cookies are 1 + 9 \u003d 10, and there are exactly 100: 10 \u003d 10 such amounts in 100 cookies.
This means that Masha ate 10 cookies, and the Bear 9 ∙ 10 \u003d 90.

The third variant of the task

Masha and the Bear ate 51 cookies and a jar of jam, starting and ending at the same time. At first, Masha ate jam, and the Bear ate cookies, but at some point they changed. The bear eats both that, and another four times faster than Masha. How many cookies did the Bear eat if they ate the same amount of jam?

Execution algorithm:

Determine who ate cookies and how many times longer.
Determine who ate the jam and how many times longer.
Compare results.
Find the unknown.

Decision:

Since both Masha and the Bear ate the jam equally, and the Bear ate jam 4 times faster, Masha ate jam (her half) 4 times longer than the Bear (the same half).
Then it turns out that the Bear ate cookies 4 times longer than Masha and, moreover, ate them 4 times faster, that is, for one cookie eaten by Masha, there were 4 ∙ 4 \u003d 16 cookies eaten by the Bear.
In total, these cookies are 1 + 16 \u003d 17, and there are exactly 51:17 \u003d 3 of such amounts in 51 cookies.
This means that Masha ate 3 cookies, and the Bear 3 ∙ 16 \u003d 48.

The fourth variant of the task

If each of the two factors were increased by 1, their product would increase by 11. In fact, each of the two factors was increased by 2. How much did the product increase?

Execution algorithm:

Enter the legend.
Convert the resulting expression.
Find the unknown.

Decision:

When these factors increase by 1, their product increases by 11, that is,

Now, in a similar way, we calculate how much the product will increase if the factors are increased by 2 and substitute the already known a + b \u003d 10:

Fifth variant of the task

If each of the two factors were increased by 1, their product would increase by 3. In fact, each of the two factors was increased by 5. How much did the product increase?

Execution algorithm:

Enter the legend.
Write down the first condition using a legend.
Convert the resulting expression.
Legend the second condition.
Convert the resulting expression.
Find the unknown.

Decision:

Let the first factor be a and the second b, their product is ab.

As these factors increase by 1, their product increases by 3, that is,

Move the product ab to the left-hand side with the opposite sign and expand the parentheses by multiplying.

Now, in a similar way, we calculate how much the product will increase if the factors are increased by 5 and substitute the already known a + b \u003d 2:

Option twentieth assignment 2017

The rectangle is split into four smaller rectangles by two straight line segments. The perimeters of three of them, starting from the upper left and further clockwise, are equal to 24, 28 and 16. Find the perimeter of the fourth rectangle.

Let's redraw the rectangle in a convenient way for us:

Now let's write the equations using the rectangle perimeter formula:

Option twentieth assignment 2019 (1)

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, 0 points were given. How many correct answers did a student who scored 42 points give if he was known to be wrong at least once?

Algorithm of execution

We make combinations of correct and incorrect answers and determine the number of points in them, for example: 1) 1 right + 1 wrong \u003d 7–10 \u003d –3 points; 2) 2 right + 1 wrong \u003d 2 · 7–10 \u003d 4 points, etc.
Of the points for right answers and points for their combinations, we "gain" 42 points. We count the number of questions that were asked.
The remaining difference between the received number of questions and the given 25 questions is defined as those to which no answer was given.
We check the result.

Decision:

Let's enter the designations: right answer - 1P, wrong answer - 1H.

We set the combinations and determine the number of points that will be awarded in this case:

1p \u003d 7 points

1P + 1H \u003d 7–10 \u003d –3 pts.

2P + 1H \u003d 2 · 7-10 \u003d 4 pts.

3P + 1H \u003d 3 · 7-10 \u003d 11 pts.

Let's summarize the points that can be obtained at the same time: 7+ (–3) + 4 + 11 \u003d 19. This is clearly not enough. And it's guaranteed you can add another 11: 19 + 11 \u003d 30. To "get" up to 42 points, you need to further add 12 points, which are gained by triple occurrence of 4 points. In general, we get:

7 + (- 3) + 4 + 11 + 11 + 34 \u003d 42.

Let's write down the resulting combination of terms in the form of answers:

1P + (1P + 1H) + (2P + 1H) + (3P + 1H) + (3P + 1H) + 3 (2P + 1H) \u003d 1P + 1P + 1H + 2P + 1H + 3P + 1H + 3P + 1H + 6P + 3H \u003d 16P + 7H (answers).

16 + 7 \u003d 23 answers. 25-23 \u003d 2 answers for which 0 points were received, ie these are unanswered questions.

So, according to our count, the correct answers were given 16.

Let's check this:

16 answers 7 b each. + 7 answers for (–10) b. + 2 answers 0 bp each. \u003d 16 · 7–7 · 10 + 2 · 0 \u003d 112–70 + 0 \u003d 42 (points).

Option twentieth assignment 2019 (2)

The table has three columns and several rows. In each cell of the table, a natural number was entered so that the sum of all numbers in the first column is 103, in the second - 97, in the third - 93, and the sum of the numbers in each row is more than 21, but less than 24. How many rows are there in the table?

Algorithm of execution

We find the total for all numbers in the table (adding the sums for each of the 3 columns).
Determine the range of acceptable values \u200b\u200bfor the sums of numbers in each line.
Dividing the total amount first by the smallest sum of numbers in each row, and then by the largest, we get the required number of rows.

Decision:

The total sum of the numbers in the table is: 103 + 97 + 93 \u003d 293.

Since by condition the sums of the numbers in each line are\u003e 21, but<24, то кол-во строк X может быть равным меньше, чем 293:21≈13,95, и больше, чем 293:24≈12,21. Т.е.: 12,21 < X < 13,95. Единственное целое число в полученном диапазоне – 13. Значит, искомое кол-во строк равно 13.

Option twentieth assignment 2019 (3)

The house has a total of eighteen apartments with numbers from 1 to 18. Each apartment is home to at least one and no more than three people. A total of 15 people live in apartments from 1 to 13 inclusive, and a total of 20 people live in apartments from 11 to 18 inclusive. How many people live in this house?

Algorithm of execution

We determine the maximum number of people living in the 11-13th apartments, using data on how many people live in the 1-13th apartments.
We find the minimum number of residents of the 11-13th apartments, taking into account the data on the residents of the 11-18th apartments.
Compares the data obtained in paragraphs 1–2, we get the exact number of residents of these apartments No. 11-13.
Find the number of people living in apartments 1-10 and 14-18.
We calculate the total number of residents of the house.

Decision:

The first 13 apartments (1st to 13th) are home to 15 people. This means that 1 person lives in 11 apartments, plus 2 people live in 2 apartments (11 1 + 2 2 \u003d 15). Consequently, at least 3 and no more than 5 (1 + 2 + 2) people live in the 11-13th (ie 3) apartments.

The second 8 apartments (11th to 18th) are home to 20 people. At the same time, from the 14th to the 18th apartments (i.e. in 5 apartments) more than 5 · 3 \u003d 15 people cannot live. Consequently, no less than 20-15 \u003d 5 people live in apartments 11-13.

Those. on the one hand, no more than 5 people should live in the 11-13th apartments, and on the other, no less than 5. Conclusion: exactly 5 people live in these apartments, because there are no other values \u200b\u200bvalid for both cases.

Then we get: 15–5 \u003d 10 people live in apartments 1–10, 20–5 \u003d 15 people in apartments 14–18. In total, the house is home to: 10 + 5 + 15 \u003d 30 people.

Option twentieth assignment 2019 (4)

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

for 4 gold coins get 5 silver and one copper;
for 7 silver coins get 5 gold and one copper.

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

Algorithm of execution

Determine the number of silver coins that Nikolai needs to make a double exchange so that he does not have gold coins. Double exchange is the exchange of first silver coins for gold and copper, and then gold coins for silver and copper.
Determine the number of different coins that Nikolai will have as a result of 1 double exchange.
We calculate the number of double exchanges that need to be made in order for 45 copper coins to appear.
We find the number of silver coins that Nikolai should have had initially in order to make the required number of exchanges, and which he received as a result of all exchanges.
Determine the desired difference.

Decision:

Nikolay must make the 1st exchange according to the 2nd scheme, since he only has silver coins. In order to ensure that he does not have gold coins as a result, you need to find the minimum multiple of 5 gold that he will receive, and 4 gold, which he can accept in full at one time (without a remainder). This is the number 20.

Accordingly, to get 20 gold coins, Nicholas must have 20: 5 \u003d 4 sets of 7 silver coins. This means that initially he should have 4 7 \u003d 28. And at the same time, Nikolai also receives 1 · 4 \u003d 4 copper coins.

Making an exchange, Nikolai gives 20: 4 \u003d 5 sets of gold medals. In return, he receives 5 5 \u003d 25 silver coins and 1 5 \u003d 5 copper coins.

Thus, as a result of one exchange, Nikolai will have 25 silver coins and 4 + 5 \u003d 9 copper coins. Since in the end Nicholas had 45 copper coins, it means that 45: 9 \u003d 5 double exchanges were made.

If, as a result of 1 double exchange, Nikolai had 25 silver coins, then after 5 such exchanges, he will have 25 · 5 \u003d 125 pieces. And initially he had to have 28 · 5 \u003d 140 silver coins for this. Consequently, Nikolai's number decreased by 140–125 \u003d 15 pieces.

Option twentieth assignment 2019 (5)

Algorithm of execution

We determine the equation for determining the number of apartments in the house in total through the parameters stated in the condition (i.e. through the number of apartments on the floor, etc.).
Factor 357.
We find the correspondence of the obtained multipliers to specific parameters, proceeding from the condition of which of the parameters is greater or less than the others.

Decision:

Because on all floors the same number of apartments (X), on all entrances the same number of floors (Y), then denoting the number of entrances through Z, we can write: 357 \u003d X * Y * Z.

Let's factor 357 into prime factors. We get: 357 \u003d 3 · 7 · 17 · 1. Moreover, this is the only scenario. Because Y\u003e X\u003e Z\u003e 1, then we disregard the unit in the layout and determine that Z \u003d 3, X \u003d 7, Y \u003d 17.

Since the number of floors was designated by Y, the required number is 17.

Option twentieth assignment 2019 (6)

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

Algorithm of execution

We count the number of agreements signed by 7 countries.
Determine the number of agreements signed by the 3 remaining countries.
We find the total number of signed contracts. We divide it by 2, because bilateral agreements.

Decision:

The first 7 countries have signed agreements with 3 countries, i.e. 7 · 3 \u003d 21 signatures were put on these contracts. Similarly, the other 3 countries, when drawing up agreements with 7 countries, put 3 · 7 \u003d 21 signatures. This means that a total of 21 + 21 \u003d 42 signatures have been delivered.

Because all agreements are bilateral, which means that each of them has 2 signatures. Consequently, the number of contracts is half the number of signatures, i.e. 42: 2 \u003d 21 contracts.

Option twentieth assignment 2019 (7)

On the surface of the globe, a felt-tip pen has drawn 13 parallels and 25 meridians. How many parts did the drawn lines divide the surface of the globe?

The meridian is a circular arc that connects the North and South Poles. A parallel is a circle in a plane parallel to the equatorial plane.

Algorithm of execution

We prove that the parallels divide the globe into 13 + 1 parts.
We prove that the meridians divide the globe into 25 parts.
We determine the number of parts into which the globe is divided as a whole, as the product of the found numbers.

Decision:

If any parallel is a circle, then it is a closed line. This means that the 1st parallel divides the globe into 2 parts. Further, the 2nd parallel provides division into 3 parts, the 3rd - into 4, etc. As a result, 13 parallels will divide the globe into 13 + 1 \u003d 14 parts.

The meridian is a circular arc connecting the poles, i.e. it is not a closed line and does not divide the globe into parts. But 2 meridians are already divided, i.e. 2 meridians provide division into 2 parts, then the 3rd meridian adds the 3rd part, the 4th - the 5th part, etc. This means, ultimately, 25 meridians create 25 parts on the globe.

The total number of parts on the globe is: 14 · 25 \u003d 350 parts.

Option twentieth assignment 2019 (8)

The basket contains 30 mushrooms: mushrooms and milk mushrooms. It is known that among any 12 mushrooms there is at least one mushroom, and among any 20 mushrooms there is at least one milk mushroom. How many mushrooms are in the basket?

Algorithm of execution

We determine the number of milk mushrooms among 12 mushrooms and camelina among 20 mushrooms.
We prove that there is a unique true number representing the number of saffron milk caps. We fix it in the answer.

Decision:

If there is at least 1 mushroom among 12 mushrooms, then there are no more than 11. There are no more than 11 mushrooms here. If among 20 mushrooms there is at least 1 mushroom, then there are no more than 19 mushrooms.

This means that if the milk mushrooms cannot be more than 11, then the mushrooms cannot be less than 30–11 \u003d 19 pieces. Those. saffron milk caps on the one hand are not more than 19, and on the other - at least 19. Therefore, there can be only exactly 19 saffron milk caps.

Option twentieth assignment 2019 (9)

If each of the two factors were increased by 1, then their product would increase by 3. How much would the product of these factors increase if each of them was increased by 5?

Algorithm of execution

We introduce the notation for the factors. This will allow expressing the original product as well (before increasing the multipliers).
We draw up an equation for the situation when the factors are increased by 1. Perform the transformations. We get a new expression that displays the relationship between the original factors.
We make an equation for the situation when the factors are increased by 5. We carry out the transformations. We introduce into the equation the expression obtained in Section 2, we find the desired difference.

Decision:

Let the 1st factor be x, the 2nd - y. Then their work is xy.

After the multipliers are increased by 1, we get:

(x + 1) (y + 1) \u003d xy + 3

xy + y + x + 1 \u003d xy +3

After increasing the multipliers by 5 we have:

(x + 5) (y + 5) \u003d xy + N, where N is the desired difference of the products.

We perform transformations:

xy + 5y + 5x + 25 \u003d xy + N

N \u003d xy + 5y + 5x + 25– xy

Because it is already defined above that x + y \u003d 2, then we get:

Option twentieth assignment 2019 (10)

Sasha invited Petya to visit, saying that he lived in the seventh staircase 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; the numbering of apartments in the building starts with one.)

Algorithm of execution

The selection method is used to determine the number of apartments on the site. This should be such a number that the number of the apartment turns out to be greater than the number of apartments in 6 entrances, but less than the number of apartments in 7.
Determine the number of apartments in 6 entrances. We subtract this number from 462 and divide by the number of apartments on the site. So we find out the desired floor number. Note: 1) if an integer is received, then the desired floor number is 1 more than the calculated value; 2) if received a fractional number, then the floor number will be the rounded up result.

Decision:

We are looking for the number of apartments on the site, checking number by number.

Suppose that this number is 3. Then we get that there are 7 6 3 \u003d 126 apartments in 7 entrances on 6 floors,

and in 7 entrances on 7 floors there are 7 · 7 · 3 \u003d 147 apartments.

Apartment No. 462 definitely does not fall into the range of apartments No. 126-147.

Similarly, checking the numbers 4, 5, etc., we arrive at the number 10. Let us prove that it is precisely this number that fits:

there are 7 6 10 \u003d 420 apartments in 7 entrances on 6 floors,

in 7 entrances on 7 floors: 7 7 10 \u003d 490 apartments. Since 420<462<490, то условие задания выполнено.

In order to get to apartment no. 462, you need to walk past 462–420 \u003d 42 apartments. Because there are 10 apartments on each site, then 42: 10 \u003d 4.2 floors must be overcome for this. 4.2 means that you need to go through 4 floors completely and go up to the 5th. Thus, the desired floor is the 5th.

Problem number 5922.

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

Since the payment of each next meter differs from the payment of the previous one by the same number, in front of 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 the arithmetic progression is found by the formula:

Let's substitute these tasks into this formula.

Answer: 89100.

Problem number 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.

Nikolai only had silver coins. After several visits to the exchange office, he had fewer silver coins, no gold coins appeared, but 100 copper coins appeared. How much decreased the number of silver coins from Nikolai?

Problem number 5960.

The grasshopper jumps along the coordinate line in any direction one segment per jump. How many different points on the coordinate line are there, where a grasshopper can find itself after making exactly 5 jumps, starting from the origin?

If the grasshopper makes five jumps in one direction (to the right or to the left), then it will be 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 (5 jumps in total), he will be at the point with coordinate -3. Similarly, if the grasshopper makes 1 jump to the left and 4 jumps to the right (5 jumps in total), it will be at the point with coordinate 3:

If the grasshopper makes 2 jumps to the right and 3 jumps to the left (5 jumps in total), it will be at the point with coordinate -1. Similarly, if the grasshopper makes 2 jumps to the left and 3 jumps to the right (5 jumps in total), then it will be 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 of coordinates, that is, it can only get to points with odd coordinates:

There are 6 of these points.

If the number of jumps were even, then the grasshopper would be able to return to the origin and all points on the coordinate line, in which it could fall, would have even coordinates.

Answer: 6

Problem # 5990

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

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

The problem asks exactly how much daysthe snail will crawl to the top of the tree.

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

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

Answer: 8 days.

Task number 6010.

All entrances of the building have the same number of floors, and each floor has the same number of apartments. Moreover, the number of floors in a building is greater than the number of apartments on a floor, the number of apartments on a floor is greater than the number of entrances, and the number of entrances is more than one. How many floors are there in the 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 on a 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 a building is more than the number of apartments on a floor, the number of apartments on a floor is more than the number of entrances, and the number of entrances is more than one."

That is, () is the largest number.

Let's expand 105 into prime factors:

Taking into account condition (1),.

Answer: 7.

Problem number 6036.

As among any 12 mushrooms, there is at least one mushroom(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.

As among any 20 mushrooms, at least one lump(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 saffron milk caps is greater than or equal to 19 , and on the other - less than or equal to 19 .

Therefore, the number of saffron milk caps equally 19.

Answer: 19.

Problem number 6047.

Sasha invited Petya to visit, saying that he lived in the seventh entrance in apartment number 333, but forgot to say the floor. Approaching the house, Petya discovered that the house was nine stories high. What floor does Sasha live on? (On each floor, the number of apartments is the same; apartment numbers in the building start with one.)

Let it be on every floor of the apartments.

Then the number of apartments in the first six entrances is

Let's find the maximum natural value satisfying the inequality (is 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.

Consequently, 333 apartments are located on the second floor.

Answer: 2

Problem number 6060.

On the surface of the globe, 17 parallels and 24 meridians are drawn with a felt-tip pen. How many parts do the drawn lines divide the surface of the globe? The meridian is a circular arc that connects the North and South Poles. parallel is a circle in a plane parallel to the equatorial plane.

Imagine a watermelon that we cut into pieces.

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

Now we are going to cut each slice.

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

If we make 2 cross cuts (parallels), then we 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 the pieces.

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

Answer: 432.

Problem number 6069

Cross lines of red, yellow and green are marked on the stick. If you cut the stick along the red lines, you get 5 pieces, if you cut the yellow ones - 7 pieces, and if you cut the green ones - 11 pieces. How many pieces would you get if you sawed the 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 general: if you make cuts, you get a piece.

Conversely: to get the 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 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 number 9626.

There are four petrol 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 this arrangement, the distance between G and A cannot be 35 km.

Let's try this:

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

Consider this option:

This variant satisfies the condition of the problem.

Answer: 10.

Problem number 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 in the absence of an answer, 0 points were given. How many correct answers did a student with 56 points give if it is known that he was wrong at least once?

Have the student give correct and incorrect answers (). Since there were possibly more questions that he did not answer, we get the inequality:

In addition, by condition,

Since the correct answer adds 7 points, and the wrong one subtracts 9, and eventually the student scored 56 points, we get the equation:

This equation must be solved in whole numbers.

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

Let, then.

In this case, all conditions are met.

Problem number 10056.

The rectangle is split into four small rectangles by two straight cuts. The areas of three of them, starting from the upper left and further clockwise, are equal to 15, 18, 24. Find the area of \u200b\u200bthe fourth rectangle.

The area of \u200b\u200ba 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).

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 number 10071.

The rectangle is split into four small rectangles by two straight cuts. The perimeters of three of them, starting with the upper left and then hourly hand, 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's designate the sides of the rectangles as shown in the figure and express the perimeters of the rectangles through the indicated variable. We get:

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

Let us subtract the second from the third equation and add the third. We get:

Simplifying the right and left sides, we get:

So, .

Answer: 18.

Problem number 10086.

The table has three columns and several rows. A natural number was placed in each cell of the table 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 more than 13, but less than 16. How many rows are there in the table?

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

Let the number of rows in the table be.

By the condition of the problem, the sum of the 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 equal, 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 equal, 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 thought of a natural number, then multiplied it by A then added to the resulting product B and subtracted C. It turned out 165. What number was guess?

Integers A, B and Ccan be equal to 5, 6 or 7.

Let the unknown natural number be.

We get:;

Let's consider various options.

Let A \u003d 5. Then B \u003d 6 and C \u003d 7, or B \u003d 7 and C \u003d 6, or B \u003d 7 and C \u003d 7, or B \u003d 6 and C \u003d 6.

Let's check:; (one)

165 is divided 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 equal, then equality (1) is impossible. Therefore, the difference is 0 and

Let A \u003d 6. Then B \u003d 5 and C \u003d 7, or B \u003d 7 and C \u003d 5, or B \u003d 7 and C \u003d 7, or B \u003d 5 and C \u003d 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 + an even number) cannot be an even number.

Let A \u003d 7. Then B \u003d 5 and C \u003d 6, or B \u003d 6 and C \u003d 5, or B \u003d 6 and C \u003d 6, or B \u003d 5 and C \u003d 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. When divided by 7, the number 165 gives a remainder of 4. Therefore, it is also not divisible by 7, and equality (3) is impossible.

Answer: 33

Several consecutive sheets 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 more than 352, which means it can be either 532 or 523.

Each dropped sheet contains 2 pages. Consequently, 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 sheets.

Answer: 85

Masha and the Bear ate 160 cookies and a jar of jam, starting and ending 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 the jam equally, and the bear ate three times as much jam per unit of time, then he ate jam three times less time 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. Consequently, the bear ate cookies three times longer than Masha. But the Bear, moreover, ate three times more cookies per unit of time than Masha, therefore, in the end, he ate 9 times more cookies than Masha.

Now it's easy to write the 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 of roses: orange, white and blue. To the left of the orange vase 15 roses, to the right of the blue vase 12 roses. There are 22 roses in vases. how many roses are in an orange vase?

Since 15 + 12 \u003d 27, and 27\u003e 22, therefore, the number of flowers in one vase was counted twice. And this is a white vase, as it must be the vase that stands to the right of the blue and to the left of the orange. This means that the vases are in this order:

From here we get the system:

Subtracting the first from the third equation, we get O \u003d 7.

Answer: 7

Ten pillars are interconnected by wires so that exactly 8 wires leave each pillar. how many wires are there between these ten posts?

Decision

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

That is, despite the fact that 2 wires leave the posts, only one wire will be stretched between the posts. This means that the number of drawn wires is two times less than the number of outgoing ones.

We get: - the number of outgoing wires.

The number of wires pulled.

Answer: 40

Of the ten countries, seven signed a friendship agreement 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 common agreement. Each contract has two signatures. That is, the number of signed contracts is half the number of signatures.

Let's find the number of signatures:

Let's find the number of signed contracts:

Answer: 21

Three rays emanating from one point split the plane into three different angles, measured in an integer number of degrees. The largest angle is 3 times the smallest. How many values \u200b\u200bcan the mean angle take?

Let the smallest angle be, then the largest angle is. Since the sum of all angles is equal, 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 \u200b\u200bin the range from 52 to 71 degrees, that is, all possible values.

Answer: 20

Misha, Kolya and Lesha play table tennis: the player who lost 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?

Decision

It should be explained how the tournament works: the tournament consists of a fixed number of games; the player who loses in this game gives way to a player who did not participate in this game. According to the results of the next game, the player who did not take part in it takes the place of the loser. Consequently, 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 Alex played the remaining 13.

Answer: 13

At the end of the quarter, Petya wrote out all his marks in a row 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 only puts marks 2, 3, 4 or 5 and the final mark in a quarter is the arithmetic average of all current marks, rounded according to the rounding rules? (For example, 3.2 is rounded to 3; 4.5 to 5; 2.8 to 3)

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

Got that

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

Answer: 3

The arithmetic mean of 6 different natural numbers is 8. How much do you need to increase the largest of these numbers to make their arithmetic mean 1 more?

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

The arithmetic mean became 1 more, that is, it became equal to 9. If one of the numbers was increased by, then the sum increased by and became equal.

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

We get equality:

Secondary general education

UMK line G.K. Muravin. Algebra and the beginnings of mathematical analysis (10-11) (in-depth)

UMK Merzlyak line. Algebra and the beginnings of analysis (10-11) (U)

Maths

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

We analyze tasks and solve examples with a teacher

The examination work at the 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 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 (a complete record of the decision with justification of the actions performed).

Panova Svetlana Anatolyevna, teacher of mathematics of the highest category of school, work experience 20 years:

“In order to get a school certificate, a graduate must pass two compulsory examinations in the form of the Unified State Exam, one of which is mathematics. In accordance with the Concept for the Development of Mathematical Education in the Russian Federation, the Unified State Exam in Mathematics is divided into two levels: basic and specialized. Today we will consider options for the profile level. "

Task number 1 - tests the USE participants' ability to apply the skills acquired in the course of 5-9 grades in elementary mathematics in practical activities. The participant must have computational skills, be able to work with rational numbers, be able to round decimal fractions, be able to convert one unit of measurement to another.

Example 1. In the apartment where Peter lives, a cold water meter (meter) was installed. On May 1, the meter showed a consumption of 172 cubic meters. m of water, and on June 1 - 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.

Decision:

1) Find the amount of water spent per month:

177 - 172 \u003d 5 (cubic meters)

2) Let's find how much money will be paid for the water spent:

34.17 5 \u003d 170.85 (rub)

Answer: 170,85.

Task number 2-is one of the simplest exam tasks. Most graduates successfully cope with it, which indicates that they have mastered the definition of the concept of function. Type of task number 2 according to the requirements codifier is a task for using the acquired knowledge and skills in practical activities and everyday life... Task number 2 consists of the description using functions of various real relationships between the quantities and the interpretation of 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 in various ways of defining a function and describe the behavior and properties of a function by its graph. It is also necessary to be able to find the highest or the lowest value on the graph of the function and to plot the graphs of the studied functions. The mistakes made are random in reading the problem statement, reading the diagram.

# ADVERTISING_INSERT #

Example 2. The figure shows the change in the market value of one share of a mining company in the first half of April 2017. On April 7, the businessman acquired 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 rest. How much did the businessman lose as a result of these operations?

Decision:

2) 1000 3/4 \u003d 750 (shares) - make up 3/4 of all purchased shares.

6) 247500 + 77500 \u003d 325000 (rubles) - the businessman received after the sale of 1000 shares.

7) 340,000 - 325,000 \u003d 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 \u200b\u200ba figure on checkered paper, the ability to calculate the degree measures of angles, calculate the perimeters, etc.

Example 3. Find the area of \u200b\u200ba rectangle depicted on checkered paper with a cell size of 1 cm by 1 cm (see figure). Give your answer in square centimeters.

Decision: To calculate the area of \u200b\u200ba given shape, you can use the Pick formula:

To calculate the area of \u200b\u200bthis rectangle, we use the Peak formula:

S \u003d B +	D
	2

where B \u003d 10, G \u003d 6, therefore

S = 18 +	6
	2

Answer: 20.

See also: Unified State Exam in Physics: Solving Oscillation 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 points marked on the circle. Determine which polygons there are more: those with all the vertices are red, or those with one of the vertices blue. In your answer, indicate how many of some are more than others.

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

in which all vertices are red.

3) One pentagon with all vertices red.

4) 10 + 5 + 1 \u003d 16 polygons with all vertices red.

whose vertices are red or with one blue vertex.

8) One hexagon, with red peaks with one blue peak.

9) 20 + 15 + 6 + 1 \u003d 42 polygons, in which all vertices are red or with one blue vertex.

10) 42 - 16 \u003d 26 polygons using the blue point.

11) 26 - 16 \u003d 10 polygons - how many polygons with one of the vertices - a blue point, more than polygons with all vertices 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 the equation 2 3 + x \u003d 0.4 5 3 + x .

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

2 3 + x	\u003d 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 on planimetry for finding geometric quantities (lengths, angles, areas), modeling real situations in the language of geometry. Research of the constructed models using geometric concepts and theorems. The source of difficulties is, as a rule, ignorance or incorrect application of the necessary planimetry theorems.

Area of \u200b\u200ba triangle ABC is equal to 129. DE - the middle line parallel to the side AB... Find the area of \u200b\u200bthe trapezoid ABED.

Decision. Triangle CDE like a triangle CAB in two corners, since the apex angle C general, angle CDE equal to the angle CAB as the corresponding angles at DE || AB secant AC... As DE - the middle line of the triangle by condition, then by the property of the middle line | DE = (1/2)AB... This means that the coefficient of similarity is 0.5. The areas of such figures are related as the square of the coefficient of similarity, therefore

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 knowledge of the concept of a derivative is required.

Example 7. Go to function graph 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).

Decision. 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 | · (-one)

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 straight line y = 4x - 13, where k 1 \u003d 4, according to the formula:

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

Answer: –0,25.

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

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

Decision. 1) V cube \u003d a 3 (where a Is the length of the edge of the cube), therefore

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.

{!LANG-e6f00da1f9c083a2f8c1dcb806449042!}{!LANG-be5576e76fbe7f31317d3691ff16bff9!}

{!LANG-b703ba43309da018952f1743c267551b!}

{!LANG-9aa57c5c8975cc5e0d50016d387b7e1d!}

{!LANG-6ea6e277832deb48cf70b299ba0f62e1!}

{!LANG-e5a3df20b4ba69a04b20b36ba1e08a86!}

{!LANG-f67412a21f9a2752621254488d19708f!}

{!LANG-ae8b4735be6ed96b93b3ee44e1fcc261!}

{!LANG-643f91d52e3700f228eda6f886fae814!}{!LANG-ed62e0c302788e9f2707635ff9578ce9!}

{!LANG-46d8ba38cf5a13bd4d2a41509bbbe61e!}	< α < π.
4

Decision.{!LANG-08fdca131e8ffaf6c8ae3627a3cce6ed!}

{!LANG-112f04e261008cff8e439385a03ee840!}	1	– 1 =	1	– 1 =	10	– 1 =	5	– 1 = 1	1	– 1 =	1	= 0,25.
	{!LANG-40e21cf6b84dc3b505e791bd16f3c0c0!}		0,8		8		4		4		4

{!LANG-e7dc93ca61068960e54b0f985545aa6b!}

{!LANG-6cc7827c5e938c71f416e4f334866a03!}

{!LANG-46d8ba38cf5a13bd4d2a41509bbbe61e!}	< α < π,
4

{!LANG-51547604c12d150a7dfec3cfcf1c638e!}< 0, поэтому tgα = –0,5.

Answer: –0,5.

{!LANG-b251ea7075402cd53b0c52e05aa4e216!} {!LANG-3cbec0303bc66bceb9a79601c0e765b5!}{!LANG-1efea8a954d00a9cdf7ebee3b850c5e8!}

{!LANG-45fb9f68ee62fc859eaa8cc2358bc20a!} {!LANG-69b64623f86def16ce17d454b8be41ae!}{!LANG-8c0aefdb4cc56e5fb820af340a6f47e5!} {!LANG-e734a88a1110fa3d657454b2dd348822!}{!LANG-e8b3ab705184bc2eadd1d8da5527bd12!} {!LANG-132bf2be3e39bd19ca8b930e92e955ad!} = {!LANG-13c4292265ca98e8ffeb2ad69b8259f3!}{!LANG-3f8c0c84a9a93dc776460bb758f52f87!}
Decision.{!LANG-12e141d60a58f53b13e995eeb1b19615!}

{!LANG-13c4292265ca98e8ffeb2ad69b8259f3!}{!LANG-097a26a1c33bdbe52ccc7676016fd58d!}

{!LANG-5c7b3b7c666a171e80fc3155a8c2d4a5!}

{!LANG-5b6e1c120979f2b0bd69193eef0e8863!}

{!LANG-467a4d944d6b2cf5b73fac36f22898f2!}

{!LANG-d389180e0f8082d64adb583be5ff9743!}

{!LANG-9c38976700075931ee6651d0a1d42dc6!}< 90°. Получили, что наименьший угол α равен 30°, тогда наименьший угол 2α = 60°.

{!LANG-e19f1223e32763c69c5223535ede0c4a!}{!LANG-38b31a8303b4729bfd6e3b223c293de3!}

{!LANG-934c1eaeef393d242b56c6e0d4897ac0!}{!LANG-a734992d167f92f595294e769735ccb9!}

Decision:{!LANG-05c7a0562ea72bd9585ee3d49dba7cf1!} a{!LANG-7e0bc014ade2ee9766272843bc879cac!} d{!LANG-61ab199507e55fd0053af8d8174493a2!} n{!LANG-0bafd31e7810efc7c5ff0373dd3273b3!} S{!LANG-bc85b91efa4d4e1f1c96a49ca48bcbbf!} a{!LANG-c2924b939cb1f569eb461568326e6035!}

560 = (5 + a{!LANG-2b87be898436c865d8ede4fbc0a142e0!}

5 + a 16 = 560: 8,

5 + a 16 = 70,

a 16 = 70 – 5

a 16 = 65.

Answer: 65.

{!LANG-10be886379549b0f7706bc4030bb8be5!}{!LANG-f8fd7c86f18265d03613626066303700!}

{!LANG-3d7e205f9a39c1479508f9138c2403fb!} y{!LANG-e98ded2b661a806f9f49ef960997ed1f!} x + 9) – 10x + 1.

Decision:{!LANG-b8eb568d5cfe1393aa322596027b1a40!} x + 9 > 0, x{!LANG-f54e7f16f8a781960c136d4acdbaaf4a!}

{!LANG-f4b245e293636757c778335ae7a61c71!}

{!LANG-785808428b2a57526ce81adc6fe27b3e!}

{!LANG-e473623a5a905e33e61a7a38ad45d969!} x = –8.

{!LANG-5c3702d740dde37b1a46f9be3857e5bd!} {!LANG-9da319b0fbb6fd84ac1f9fde272d480d!}

{!LANG-f12738d5013c649e8060d6ef3885723e!}{!LANG-6f228f711630b0dbee15c149fba3e760!}

{!LANG-8946118a48ea94cdc87453648df86aa4!} x{!LANG-a37f7df106a4de70d43a2312332e3b9a!} x) + 2 = 0

{!LANG-622dc5a91502472d3d8399783e645a10!} {!LANG-45673f1fe6004aaa13137d1473e60bdd!}.

Decision:{!LANG-8d4e6fdfbc85462432d29b7ea5132f3f!} x) = {!LANG-b7269fa2508548e4032c455818f1e321!}{!LANG-abc353e1dfb37645e0233d87550c8372!} {!LANG-b7269fa2508548e4032c455818f1e321!} 2 – 5{!LANG-b7269fa2508548e4032c455818f1e321!} + 2 = 0,

{!LANG-82caf24adb026417cc90780a8dfb47fd!} x) =	2	⇔	{!LANG-22461111f442acde3c09710c17cd883e!} x = 9	⇔	{!LANG-2f1795e590b4e611b669b58449a2a508!} x =	4,5	{!LANG-d2531f73ab43b90f0e16206524f9db8e!} x\| ≤ 1,

{!LANG-82caf24adb026417cc90780a8dfb47fd!} x) =	1		{!LANG-22461111f442acde3c09710c17cd883e!} x = √3		{!LANG-2f1795e590b4e611b669b58449a2a508!} x =	√3
	2					2

{!LANG-4b62c273c3e028f43413077f7dce0597!} x =	√3
	2

x =	π	{!LANG-27603f0974c99efa13cdcc2775bd1f13!} k
	6

x = –	π	{!LANG-27603f0974c99efa13cdcc2775bd1f13!} k, k ∈ {!LANG-41ff0912a07fdc52799ff27b38e7f140!}
	6

{!LANG-634ae70a41210397d9b6035ee9873d62!}

{!LANG-8d8d2819f520b12d3320a841c53f0d3e!}

{!LANG-1d630a76546ab351644a9cbbfc1e3a54!}	{!LANG-daad04b58fc0326a33ac665c82486ec8!}	{!LANG-e41b4bb03b2c0e4d9edb59ba28fc8a25!}	.
6		6

Answer:{!LANG-4336b0bdd12cbc9db870dbca750f2d48!}	π	{!LANG-27603f0974c99efa13cdcc2775bd1f13!} k; –	π	{!LANG-27603f0974c99efa13cdcc2775bd1f13!} k, k ∈ {!LANG-41ff0912a07fdc52799ff27b38e7f140!}{!LANG-c10ca531b5d4af171c6f8a210bb67eb4!}	{!LANG-1d630a76546ab351644a9cbbfc1e3a54!}	;	{!LANG-e41b4bb03b2c0e4d9edb59ba28fc8a25!}	.
	6		6		6		6

{!LANG-d08bcb2268056cb5d124414001e40b4d!}{!LANG-0473e7529be80bdb6e21789d9cf5e002!}

{!LANG-e29c987c99e155d99aced1a6197bbb96!}

{!LANG-666c3888a55985da2bc2cc73f0551cc5!}

{!LANG-fc2c60365c6ab85e432af754ecdebb78!}

Decision:{!LANG-01ba2af1b096916b8ac320b2a2b06f91!}

{!LANG-74cac91df02b2fbfd8bd97c931fcfa9a!}

= = √980 = = 2√245

= = √788 = = 2√197.

{!LANG-324477afb5b1b1f9deb02fc0b74eec51!} {!LANG-096596c72fe4b92333fdbeb8b244f2ce!}{!LANG-0515f2b65d787f37aca585cb3f87ce8f!}

{!LANG-5572f97135d47b9a5260fd420f7770b3!}

{!LANG-5107261ccae5ad01d38c29a05d6a2266!}

{!LANG-c5724e3c586a3ec7385806a7e49da6a1!}	{!LANG-4c9b39b489fefa5458e3aa4ee5c00ad5!}	{!LANG-4931c9de29745d519c88c51da1b7f5ee!}	28	{!LANG-1184c0cdd68cd16d52dda7c55a100fc6!}
	{!LANG-feadb2ab179825bdf6be37f87719686c!}		8 – 6

{!LANG-c9435b67532b7db09186c83e063e0bbe!}{!LANG-056e1b6c6daac0f9d20d3fc5752ef915!}

{!LANG-1d8367507e2f1cfbda39e5041b4a3196!}{!LANG-1b9cf6b8c76b936903a70b88ba735ef7!} x 2 – 3x{!LANG-341568310f7f9ed6b12d86df4195a355!} x + 1) ≤ 3x – x 2 .

Decision:{!LANG-658c3e2371dcc0781fdf7b5a2314023a!}

{!LANG-d79e42588274b48bf0cbe59f1da09c3f!} x 2 – 3x{!LANG-88738f617c8502ada8325160bacab5df!} x{!LANG-cd0e323655713d49bb23c2082db2035b!} x{!LANG-14fe48ce8ff14542d10204fb28227388!}

{!LANG-a8f668fd2e0c524ec4794ffe642c3f19!} x 2 – 3x{!LANG-750e21166398633f439754d18addfda6!} x{!LANG-ccd71658b75b35b423c9ff6a058f1763!} x 2 – 3x{!LANG-c9db67539d1ac47c12ad75680fb5b4de!} x + 1) ≤ 3x – x{!LANG-eef79b9e58eb6dcb8235acae7ae8474e!} x 2 – 3x{!LANG-9836f203108900e52338681c89da2f22!} x + 1) ≤ –1, x + 1 ≤ 2 –1 , x{!LANG-932cb80d78127c2a5673eba7a33b9ab3!} {!LANG-392b105058e992a95c797aaff50a0f71!}{!LANG-0ac61642c8e9bf64f5b755ace8bc93fb!} x ∈ (–1; –0,5].

{!LANG-4af85f5f05562c4ce3707af60b0b548b!} x 2 – 3x < 0, при этом x{!LANG-676b79154378750425260000ebce5465!} x – x{!LANG-015cf31a31ceb038448ab72bc42a80a2!} x + 1) ≤ 3x – x{!LANG-1a8f3bdb95f3789d2930bfdb44ca5747!} x – x{!LANG-3fe8d2f4b673263e6b712b1577e27f17!} x + 1) ≤ 1, x + 1 ≤ 2, x{!LANG-c99843d323ee48b1f9303aa4a74a1209!} x ∈ (0; 1].

{!LANG-6c503f60ef0043d47437bd94fb39441a!} x ∈ (–1; –0.5] ∪ ∪ {3}.

Answer: (–1; –0.5] ∪ ∪ {3}.

{!LANG-37d56b1e43e797285f407d2256c044dd!}{!LANG-a51f555881fca8f1bb1f56dc9ca959ff!}

{!LANG-3e1a7e06dd77322bbda52d6bce94564a!}

Decision:{!LANG-95dc07cc33bf6199b22a28e7d583def1!}

{!LANG-05724a6a6d65bf1bc1daea52fccb0311!}

{!LANG-153b688417071d4a0d96c02da8315c57!} x{!LANG-4afa54fcb333260b6a043859d6d202ad!} x{!LANG-b999ee3bbbec86803db9029aad763e50!} x{!LANG-a16646e804c74181b8c140fa3e73bda8!}

{!LANG-c8834e15bb34c6dbfe159f63c6ef7a61!}

{!LANG-f1e6f50460c8cb6fc7488d46348f6c77!}

{!LANG-78e593564bebf208079dde403096a92a!}

2x	=	4 – 2x
2x(√3 + 1)	=	4

1	=	2 – x
√3 + 1	=	2

√3 – 1 = 2 – x

x = 3 – √3

{!LANG-bffe40ce603a643628895e876d487b6a!}

2) S{!LANG-31f4182c067d4cd793c1050e7d677179!}

S{!LANG-9375e838f9d9c935adb1893d6f9420fa!}

Answer: 24 – 12√3.

{!LANG-627db2ea237d149b49f67174d468162e!}{!LANG-1393b3712d49b114907348498850d401!} {!LANG-81c1a2731d128ffffeec8b277181a2a9!}{!LANG-918960bf67f5f8525e763fb276a5b433!}

{!LANG-7e999753325a58ef015a83da2f45e70d!}{!LANG-cbcdf74170cff940b72cb9d2a9ee8eee!} x{!LANG-aae0f0af382c45145636187e84788cae!} x - {!LANG-c72dd16739a9448cb1e8818cd573ad16!}{!LANG-93d323a484bc33efd3cc1a68e52ae1bb!} x{!LANG-d92e567abe36fa0a97655fc755780a81!}

Decision:{!LANG-b94f0b8b026b09f79852eb2e12bfadd9!} x{!LANG-b2d6a31da1aa1038fb347f3c4b7f47a1!} {!LANG-5194fd2178d1fcc0029a7b6ac5ee9f9c!} + (24,2 + {!LANG-5194fd2178d1fcc0029a7b6ac5ee9f9c!}{!LANG-a85ef7acda471898e1fb9aa150975999!} x{!LANG-a35d3360143350ae4ff5f04c0fe269d6!} {!LANG-5194fd2178d1fcc0029a7b6ac5ee9f9c!}{!LANG-8047810f63c5fba5b7356c583a3320d6!} x) + (26,62 + 2,1x{!LANG-cfa0a6d0e506ba4be01e96215d766717!} x{!LANG-d0087c35ca5f1cef0dfef812e6025a69!}

(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

{!LANG-580845e4cbffdeda38dc1e6342d05b42!}

Answer: 24.

{!LANG-6f855af9cfb404b7fc72fbcde7f17ab2!}{!LANG-8d359a904bf4eb78fbec4d6d0e0fca16!} {!LANG-dec5ecb573d3f81dce25056258d51e26!}{!LANG-02510d9d893ccfa4ae4e477045b115a5!} {!LANG-4382a2d4b478a51003b7593f3fea9288!}{!LANG-9a589f8de007b8e9b9e307a13ff6682d!}

{!LANG-c2d3c0c86c812a76b810e132bbadb627!} a{!LANG-cbb333df63a9bb848e88e0e70745b5d7!}

	x 2 + y 2 ≤ 2{!LANG-2cb289f5d1dccd216d3555488cd25a28!} – a 2 + 1

	y + a ≤ \|x\| – a

{!LANG-d5cd1078d64db1aa082346d811bf8299!}

Decision:{!LANG-d7ab205d14dfa47d55d0febc72450a45!}

	x 2 + (y– a) 2 ≤ 1

	y ≤ \|x\| – a

{!LANG-988a1e65975acb1dc5cf13b902b300fd!} a{!LANG-68f86bac884d6a1208c74fc3961e4e33!} y = | x| – a, {!LANG-d7f06ad81de47ce522f47b68b66242f4!}
y = | x| {!LANG-b3aa51a4a7a082b85807522847d43089!} a{!LANG-dedcf6c6554e3d7da494b5842fd818c1!}

{!LANG-730fac31cae6cbe3465c966072eceeeb!}

{!LANG-119e0ff155525621be0e4c60c225897b!} {!LANG-8cdf36886a290da85338c96f823422da!}{!LANG-9858aaa6f09ec981ba627ddab85c48fa!} {!LANG-132bf2be3e39bd19ca8b930e92e955ad!}{!LANG-7c8ed0bf11d557c5e05ef7afe8123de0!} a{!LANG-62b75e75faf84be33196bb133e5359d5!} R{!LANG-54a9704319c19bd6948debc17ffbe1df!} a{!LANG-0730167a6d12e0c37ce6acdd7bd8f0e4!} {!LANG-cbd4cfee378c8c5676508991c8aa1cc6!}{!LANG-73365e38bf2b65234d77914e34d9fae0!} {!LANG-126c37b54d4f1367d46ec571c56697dc!}{!LANG-854d9ae696eda2db873839c77e3866dd!}

{!LANG-af9b5b70f604acd24883582fa253306f!}= 2a = √2, a =	√2	.
	2

Answer: a =	√2	.
	2

{!LANG-97bd5f6f2b739c816884669f8b0847d1!}{!LANG-34d9d1873c7afee6a4ce16a047acf68e!}

{!LANG-5132beec13c9bb82fb30320aed5dea2c!} {!LANG-27c54d4499b00eb2caff50c910ec2194!}{!LANG-50a9b26139ff33386a53ac722694026b!} {!LANG-46e29e7e0aa8e6033dbbc285732f3cab!}{!LANG-b250adfac245171497e92ad4c7ef17c8!} {!LANG-43bbbdc24ea1e5c35d1d1ec7d836dde1!}{!LANG-67113fd7e03426666b1c704320b03d9b!} {!LANG-ffc04eebe98ae478ecf3473031d50006!} + 1 = 2n 2 – 21n – 23.

{!LANG-0fd4cb4545c8c8fd42b2668958602aff!} {!LANG-46e29e7e0aa8e6033dbbc285732f3cab!}{!LANG-5c0c09efd4706f17cbd5cf42b2eac136!}

{!LANG-8915e613c42fbfc299c09daf5a82145f!} {!LANG-ffc04eebe98ae478ecf3473031d50006!}.

{!LANG-2704bbeefe469e59b323358e4eba84f6!} {!LANG-46e29e7e0aa8e6033dbbc285732f3cab!}{!LANG-afecf6c4d302dd1d215e3c561ca6aa66!} {!LANG-ffc04eebe98ae478ecf3473031d50006!}{!LANG-02aca2ac1ecd9b65190eeb75a84b8e03!}

Decision{!LANG-d982c69935e60d16c851ff8da2f297fb!} {!LANG-8a1620fcd0314fad3404d07fee591423!} = {!LANG-ffc04eebe98ae478ecf3473031d50006!} – {!LANG-ffc04eebe98ae478ecf3473031d50006!}{!LANG-8ed6acddb5279797e9cef54b541d6f6d!}

{!LANG-ffc04eebe98ae478ecf3473031d50006!} = S (n – 1) + 1 = 2(n – 1) 2 – 21(n – 1) – 23 = 2n 2 – 25n,

{!LANG-ffc04eebe98ae478ecf3473031d50006!} – 1 = S (n – 2) + 1 = 2(n – 1) 2 – 21(n – 2) – 23 = 2n 2 – 25n+ 27

{!LANG-7b72f13338cae6352e3dfecd652917da!} {!LANG-8a1620fcd0314fad3404d07fee591423!} = 2n 2 – 25n – (2n 2 – 29n + 27) = 4n – 27.

{!LANG-3afea6527ea026150c4f6e158696024a!} {!LANG-ffc04eebe98ae478ecf3473031d50006!} = 2n 2 – 25n{!LANG-eaa7ff7987b7e58ff43cb78961a3a440!} S(x) = | 2x 2 – 25{!LANG-7170d2ce565f93ce8db884d28632748d!}{!LANG-1c03a3d907bafc5819df5bfe62b9c41e!}

{!LANG-45aeab9a67f945c33ae87c382aabdcf1!} x= 1, x{!LANG-dd571a09e2ff71ae37e79fda04727a65!} x{!LANG-0031ef932b8f8f6bcb7aa43833d64f73!} S(1) = |S 1 | = |2 – 25| = 23, S(12) = |S{!LANG-3ba17f58022a271d63c5106308f21d3a!} S(13) = |S{!LANG-cb97d023767e0c18cb972af506b91faf!}

{!LANG-1976b91b47ba841b5c8cf4a5845b0b1d!} {!LANG-27c54d4499b00eb2caff50c910ec2194!}{!LANG-919b8cc1950841a5e6f3e546000168b3!} n{!LANG-e24403f946b112e819dca0c0ca392cab!} {!LANG-ffc04eebe98ae478ecf3473031d50006!} = 2n 2 – 25n = n(2n{!LANG-76f926e431f6c7a074d2d0ca189a899f!} n = 2n{!LANG-dfa1905eb8c42d1c2b5331d7ea16c3d4!} {!LANG-46e29e7e0aa8e6033dbbc285732f3cab!}= 25.

{!LANG-f6ce803915b172b236e2d3c7651002b0!}

S{!LANG-11bde3b61191f8ed4b731eeec27ff854!} S{!LANG-d598b9da6dbf732934c757337f6ac557!} S{!LANG-52209e7ad19b8ab6ba853a07137efcc3!} S{!LANG-b6c8914ea9ac40a2a18feb984bd54cc5!} S{!LANG-bf02215a95b783da8ff22ba3ca19775e!} S{!LANG-6f163f875869703deca849971cbf1fde!} S{!LANG-1dfa6a6c758466a782c457b4724dd040!} S{!LANG-bef5f9e6a6ea88e1d83f1f272aca6eb1!} S{!LANG-ddb8991d5b574cb7ba3db857e503e34c!} S{!LANG-be1f172af249d7df216c683006736021!} S{!LANG-15abe2167bc324014f649081d586ec72!} S{!LANG-746fad148a25eae524df3cc50f427f0c!}

{!LANG-f797bac5e55b2879c8c56cfd3d25d73e!} {!LANG-46e29e7e0aa8e6033dbbc285732f3cab!} {!LANG-8acf24a53857a0ffddf099029ddb7b26!}{!LANG-98bc3f46206e4ee995ed49d5af0c98d3!}

Answer:{!LANG-e9b90312998da2b8c8c7663f3b8d6753!} {!LANG-8a1620fcd0314fad3404d07fee591423!} = 4n{!LANG-f19f7eb9fff71d90dd5dbebe04e336fe!}

________________

{!LANG-70cef5efc90003daf30be5db59caf13d!} {!LANG-12512fc7838472f3902657a7df783309!}{!LANG-25d21e2ae48468ff1d1b3b576ef12f9c!} {!LANG-281646c3fb0c8c2725786a9c36716ef4!}{!LANG-9a819597f2e00689b635e9ccf1e19c9b!} {!LANG-bc944644a98ce44892a370f8b49a82b9!}{!LANG-4e4c214ff6c4aad612f38c22e08ada58!} {!LANG-1eae3ddc5cf3806478df8cfd41b3e1bc!}{!LANG-36902979180c9381d4523a5e769ee157!} {!LANG-6fb709807954cb90a5dadcf9acee3b50!}{!LANG-4656e0357b9e7064ebedcfd91c5db501!} {!LANG-a6c174181f8bb1448f55a32212bf1b00!}{!LANG-7d38d3270430ee255595ff91ff1a7d91!} {!LANG-ed5fa99282dd201d0987f203dbd1d7d8!}{!LANG-71b584efbf94292f09119909ddb2ca37!}