Solution 20 of the exam task, basic level. Sasha invited Petya to visit, saying that he lived

Task 20 Basic level of the Unified State Exam

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

12) There are 50 mushrooms in the basket: saffron milk caps and milk mushrooms. It is known that among any 28 mushrooms there is at least one saffron milk cap, and among any 24 mushrooms there is at least one milk mushroom. How many milk mushrooms are there in the basket? ( (50-28)+1=23 - there must be saffron milk caps. (50-24)+1=27 - there must be milk mushrooms. Answer: milk mushrooms in a basket 27 .)

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

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

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

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

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

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

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

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

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

22) The trainer advised Andrey to spend 22 minutes on the treadmill on the first day of classes, and at each subsequent lesson to increase the time spent on the treadmill by 4 minutes until it reaches 60 minutes, and then continue to train for 60 minutes every day. In how many sessions, starting from the first, will Andrey spend a total of 4 hours and 48 minutes on the treadmill? ( 8 )

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

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

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

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

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

A meridian is an arc of a circle connecting the North and South Poles. A parallel is a circle lying in a plane parallel to the plane of the equator. (13 22=286)

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

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

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

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

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

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

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

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

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

Problem No. 5922.

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

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

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

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

Let's substitute these problems into this formula.

Answer: 89100.

Problem No. 5943.

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

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

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

Problem No. 5960.

The grasshopper jumps along the coordinate line in any direction for a unit segment per jump. How many different points are there on the coordinate line at which the grasshopper can end up after making exactly 5 jumps, starting from the origin?

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

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

If the grasshopper makes 2 jumps to the right and 3 jumps to the left (5 jumps in total), it will end up at the point with coordinate -1. Similarly, if the grasshopper makes 2 jumps to the left and 3 jumps to the right (5 jumps in total), it will end up at 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 only 6 of these points.

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

Answer: 6

Problem No. 5990

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

Note that in this problem we should distinguish between the concept of “day” and the concept of “day”.

The problem asks exactly how long days the snail will crawl to the top of the tree.

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

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

Answer: 8 days.

Problem No. 6010.

All entrances of the house have the same number of floors, and each floor has the same number of apartments. In this case, the number of floors in the house is greater than the number of apartments on the floor, the number of apartments on the floor is greater than the number of entrances, and the number of entrances is more than one. How many floors are in the building if there are 105 apartments in total?

To find the number of apartments in a house, you need to multiply the number of apartments on the 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 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.”

That is, ( ) is the largest number.

Let's factor 105 into prime factors:

Taking into account condition (1), .

Answer: 7.

Problem No. 6036.

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

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

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

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

Then we found that, on the one hand, the number of saffron milk caps is greater than or equal to 19 , and on the other hand - less than or equal to 19 .

Therefore, the number of saffron milk caps equals 19.

Answer: 19.

Problem No. 6047.

Sasha invited Petya to visit, saying that he lived in the seventh entrance in apartment No. 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 begin with one.)

Let there be apartments on each floor.

Then the number of apartments in the first six entrances is equal to

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

From here

The number of the last apartment in the sixth entrance is

The seventh entrance starts from apartment 325.

Therefore, apartment 333 is on the second floor.

Answer: 2

Problem No. 6060.

On the surface of the globe, 17 parallels and 24 meridians were drawn with a felt-tip pen. How many parts do the drawn lines divide the surface of the globe into? A meridian is an arc of a circle connecting the North and South Poles. parallel is a circle lying in a plane parallel to the plane of the equator.

Let's imagine a watermelon that we cut into pieces.

By making two cuts from the top to the bottom (drawing two meridians), we will cut the watermelon into two slices. Therefore, by making 24 cuts (24 meridians), we will cut the watermelon into 24 slices.

Now we will cut each slice.

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

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

This means that by making 17 cuts we will cut one slice into 18 parts.

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

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

Answer: 432.

Problem No. 6069

The stick is marked with transverse lines of red, yellow and green. If you cut a stick along the red lines, you will get 5 pieces, if along the yellow lines, 7 pieces, and if along the green lines, 11 pieces. How many pieces will you get if you cut a stick along the lines of all three colors?

If you make 1 cut, you will get 2 pieces.

If you make 2 cuts, you will get 3 pieces.

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

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

Let's find the total number of lines along which the stick was cut.

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

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

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

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

Answer: 21.

Problem No. 9626.

There are four gas stations on the ring road: A, B, B, and D. The distance between A and B is 50 km, between A and B is 40 km, between C and D is 25 km, between G 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 equal to 35 km.

Let's try this:

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

Let's consider this option:

This option satisfies the conditions of the problem.

Answer: 10.

Problem No. 10041.

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

Let the student give correct and incorrect answers ( ). Since there were possibly other questions that he answered, we get the inequality:

Moreover, according to the condition,

Since the correct answer adds 7 points and the wrong answer subtracts 9, and the student ends up with 56 points, the equation is:

This equation must be solved in whole numbers.

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

Let it be then.

In this case, all conditions are met.

Problem No. 10056.

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

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

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

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

By the property of proportion we get

From here.

Problem No. 10071.

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

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

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

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

Let's 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 No. 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 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 the numbers in the table: .

Let the number of rows in the table be .

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

Since the sum of 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 to , 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 to , and this number satisfies the inequality.

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

The only natural that satisfies this system is

Answer: 17.

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

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

Let the unknown natural number be equal to .

We get: ;

Let's consider various options.

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

Let's check: ; (1)

165 is divisible by 5.

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

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

Let's check: ; (2)

The difference between 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=7. Then B=5 and C=6, or B=6 and C=5, or B=6 and C=6, or B=5 and C=5.

Let's check: ; (3)

The difference between numbers B and C is either equal to or equal to 0 if these numbers are equal. The number 165 when divided by 7 leaves a remainder of 4. Consequently, 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 with the same numbers, but in a different order. How many sheets fell out?

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

Each dropped sheet contains 2 pages. Therefore, there is an even number of pages. 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 the result is sheets.

Answer: 85

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

If Masha and the Bear ate jam equally, and the bear ate three times as much jam per unit time, then he ate jam in three times less time than Masha. In other words, Masha ate jam three times longer than 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 create an equation. Let Masha eat the cookies, then the Bear ate the cookies. Together they ate the cookies. we get the equation:

Answer: 144

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

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

From here we get the system:

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

Answer: 7

Ten pillars are connected to each other by wires so that exactly 8 wires come from each pillar. How many wires are there between these ten poles?

Solution

Let's simulate the situation. Let us have two pillars, and they are connected to each other by wires so that exactly 1 wire comes from each pillar. Then it turns out that there are 2 wires coming from the poles. But we have this situation:

That is, even though there are 2 wires coming from the poles, only one wire will be stretched between the poles. This means that the number of extended wires is two times less than the number of outgoing ones.

We get: - the number of outgoing wires.

Number of wires pulled.

Answer: 40

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

This task is similar to the previous one: two countries sign one general treaty. Each agreement has two signatures. That is, the number of signed agreements is half as large as 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 divide the plane into three different angles, measured in an integer number of degrees. The largest angle is 3 times the smallest. How many values can the average angle take?

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

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

We obtain a system of inequalities:

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

Answer: 20

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

Solution

It should be explained how the tournament is structured: the tournament consists of a fixed number of games; the loser in a given game gives way to a player who did not participate in this game. At the end 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 in total.

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

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

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

Answer: 13

At the end of the quarter, Petya wrote down all his grades 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 equal to 3495. What mark does Petya get in a quarter in this subject if the teacher gives only marks 2, 3, 4 or 5 and the final mark in a quarter is the arithmetic mean 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 factor 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, therefore the number is divisible by 3.

Got that

Therefore, Petit’s estimates are 3, 5, 2, 3, 3. Let’s find the arithmetic mean:

Answer: 3

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

The arithmetic mean is equal to the sum of all numbers divided by their number. Let the sum of all numbers be equal. According to the conditions 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 to .

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

We get equality:

Yakovleva Natalya Sergeevna
Job title: mathematic teacher
Educational institution: MCOU "Buninskaya Secondary School"
Locality: Bunino village, Solntsevsky district, Kursk region
Name of material: article
Subject:"Methods for solving tasks No. 20 of the Unified State Examination in mathematics, basic level"
Publication date: 05.03.2018
Chapter: complete education

The Unified State Exam is currently the only

form of final certification of high school graduates. And receiving

a certificate of secondary education is not possible without successfully passing the Unified State Examination

mathematics. Mathematics is not only an important academic subject, but

and quite complex. They have far superior mathematical abilities

Not all children, but their future fate depends on successfully passing the exam.

Graduation teachers ask the question again and again: “How to help

a student in preparation for the Unified State Exam and successfully pass it?” In order to

The graduate has received a certificate; it is enough to pass basic level mathematics. A

success in passing the exam is directly related to the teacher’s command of

methods for solving various problems. I offer you examples

solutions to task No. 20 mathematics basic level FIPI 2018 under

edited by M.V. Yashchenko.

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

red. If you cut the tape 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.

Solution:

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

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

between the stripes. It is known that if the ribbon is cut along the red stripe, then

one part is 5 cm longer than the other, that is, a + c – b = 5. If you cut along

blue stripe, then one part will be 15 cm longer than the other, which means in +c –

a=15. Let's add the two equalities term by term: a+c-b+c+c-a=20, 2c=20, c=10.

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

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

the arithmetic one increased by 1.

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

This means that the sum of these numbers is 8*6=48. Arithmetic mean of numbers

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

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

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

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

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

neighboring cells are white.

Solution:

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

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

4 pairs of neighboring cells are formed, that is, only pairs of neighboring cells

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

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

26-6 = 17 pairs.

Answer: 17.

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

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

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

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

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

that to the left of the red vase, that is, there are 15 roses in the blue and white, which means x + y = 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= 12.

Got:

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

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

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

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

How many cookies did the Bear eat if they ate the same amount of jam?

Solution: 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

different, and according to the conditions of the problem, the Bear eats both 3 times faster than

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

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

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

16*9=144 cookies.

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

pages before dropped sheets 352. First page number after

the dropped sheets are written down with the same numbers, but in a different order.

How many sheets fell out?

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

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

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

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

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

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

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

subtract C. We get 164. What number was intended?

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

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

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

Task No. 20 of the Unified State Examination in mathematics contains a task of intelligence. The tasks in this section are more intuitive than in task 19 of the Unified State Exam, but nevertheless they are quite complex for an ordinary student. So, let's move on to considering typical options.

Analysis of typical options for tasks No. 20 of the Unified State Examination in basic level mathematics

First version of the task (demo version 2018)

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

Execution algorithm:

Enter symbols.
Write down these tasks using symbols.
Determine the unknown using logical reasoning.

Solution:

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

Let us introduce the notation, let there be 2n second operations (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 operations will be (3 · 2n)/2 = 3 n. That is

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

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

5 · 2n silver exchanged for 9n silver + 3n copper+2n copper

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

If, after exchanging 10n silver coins, we get 9n silver coins, then the number of silver coins Nicholas has decreased by n. From the last expression it is clear that Nikolai received 5n copper coins, and according to the condition, 50 copper coins appeared, that is, 5n = 50.

Second version of the task

Masha and the Bear ate 100 cookies and a jar of jam, starting and finishing at the same time. At first Masha ate jam, and Bear ate cookies, but at some point they switched. 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 the results.
Find the unknown.

Solution:

Since both Masha and the Bear ate the jam equally, and the Bear ate the jam 3 times faster, then 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 also ate them 3 times faster, that is, for one cookie eaten by Masha there were 3∙3=9 cookies eaten by the Bear.
The total of these cookies is 1+9=10 and there are exactly 100:10 = 10 such amounts in 100 cookies.
This means that Masha ate 10 cookies, and Bear ate 9∙10=90.

Third version of the task

Masha and the Bear ate 51 cookies and a jar of jam, starting and finishing at the same time. At first Masha ate jam, and Bear ate cookies, but at some point they switched. The bear eats both 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 the cookies and for how many times longer.
Determine who ate the jam and for how many times longer.
Compare the results.
Find the unknown.

Solution:

Since both Masha and the Bear ate the jam equally, and at the same time the Bear ate the jam 4 times faster, then Masha ate the 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 also ate them 4 times faster, that is, for one cookie eaten by Masha there were 4∙4 = 16 cookies eaten by the Bear.
The total of these cookies is 1+16=17 and there are exactly 51:17 = 3 such sums in 51 cookies.
This means that Masha ate 3 cookies, and Bear ate 3∙16=48.

Fourth version 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 symbols.
Convert the resulting expression.
Find the unknown.

Solution:

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

Now let’s similarly calculate how much the product will increase if the factors are increased by 2 and substitute what we already know a + b = 10:

Fifth version 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 symbols.
Write down the first condition using symbols.
Convert the resulting expression.
Write down the second condition using symbols.
Convert the resulting expression.
Find the unknown.

Solution:

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

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

Let's move the product ab to the left side with the opposite sign and open the brackets by multiplying.

Now let’s similarly calculate how much the product will increase if the factors are increased by 5 and substitute what we already know a + b = 2:

Option for the twentieth task 2017

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

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

Now let's create equations using the formula for the perimeter of a rectangle:

Option for the twentieth task of 2019 (1)

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

Execution algorithm

We make combinations of correct and incorrect answers and determine the number of points in them, for example: 1) 1 right + 1 wrong = 7–10 = –3 points; 2) 2 right + 1 wrong = 2 7–10 = 4 points, etc.
From the points for correct answers and points for their combinations, we “score” 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 that were not answered.
We check the obtained result.

Solution:

Let us introduce the following notations: correct answer – 1P, incorrect answer – 1H.

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

1P=7 points

1P+1N=7–10=–3 b.

2P+1N=2·7–10=4 b.

3P+1N=3·7–10=11 b.

Let's sum up the points that you can get: 7+ (–3)+4+11=19. This is clearly not enough. And you are guaranteed to add 11 more: 19+11=30. To “get” to 42 points, you need to further add 12 points, which are gained by triple entry of 4 points. In general we get:

7+(–3)+4+11+11+3·4=42.

Let us write the resulting combination of terms in the form of answers:

1P+(1P+1N)+(2P+1N)+(3P+1N)+(3P+1N)+3 (2P+1N)=1P+1P+1N+2P+1N+3P+1N+3P+ 1N+6P+3N=16P+7N (answers).

16+7=23 answers. 25–23 = 2 answers for which 0 points were received, i.e. these are unanswered questions.

So, according to our calculations, 16 correct answers were given.

Let's check this:

16 answers, 7 points each. + 7 answers for (–10) b. + 2 answers 0 points each. = 16·7–7·10+2·0=112–70+0=42 (points).

Option for the twentieth task of 2019 (2)

The table has three columns and several rows. A natural number was written in each cell of the table 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?

Execution algorithm

Find the total sum for all numbers in the table (by adding the sums for each of the 3 columns).
We determine the range of acceptable values for the sums of numbers in each line.
By dividing the total amount first by the smallest sum of numbers in each line, and then by the largest, we get the required number of lines.

Solution:

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

Since by condition the sum of the numbers in each line is >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 for the twentieth task of 2019 (3)

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

Execution algorithm

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

Solution:

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=15). Consequently, at least 3 and no more than 5 (1+2+2) people live in apartments 11–13 (i.e. 3).

The second 8 apartments (11th to 18th) house 20 people. At the same time, from the 14th to the 18th apartments (i.e., 5 apartments) no more than 5·3=15 people can live. And therefore, no less than 20–15 = 5 people live in apartments 11-13.

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

Then we get: 15–5=10 people live in apartments 1–10, 20–5=15 people live in apartments 14–18. Total number of people living in the house: 10+5+15=30 people.

Option for the twentieth task of 2019 (4)

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

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

Execution algorithm

We determine the number of silver coins that Nikolay needs to make a double exchange so that he does not have gold coins. A double exchange is the exchange of first silver coins for gold and copper, and then gold for silver and copper.
We 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.
We determine the desired difference.

Solution:

Nikolay must make the 1st exchange according to the 2nd scheme, because he only has silver coins. In order for him to end up with no gold coins, he needs to find the minimum multiple of the 5 gold coins that he will receive and the 4 gold coins that he can accept in full (without remainder) at one time. This is the number 20.

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

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

Thus, as a result of one exchange, Nikolai will have 25 silver coins and 4+5=9 copper coins. Since Nicholas ended up with 45 copper coins, it means that 45:9 = 5 double exchanges were made.

If, as a result of 1 double exchange, Nikolai ended up with 25 silver coins, then after 5 such exchanges he will have 25·5=125 pieces. And initially he had to have 28·5=140 silver coins for this. Consequently, their number in Nikolai decreased by 140–125 = 15 pieces.

Option for the twentieth task of 2019 (5)

All entrances of the house have the same number of floors, and all floors have the same number of apartments. In this case, the number of floors in the house is greater than the number of apartments on the floor, the number of apartments on the floor is greater than the number of entrances, and the number of entrances is more than one. How many floors are there in the building if there are 357 apartments in total?

Execution algorithm

We define an equation for determining the number of apartments in a building using the parameters stated in the condition (i.e., through the number of apartments on the floor, etc.).
Let's factor 357.
We find the correspondence of the resulting multipliers to specific parameters, based on the condition of which of the parameters is greater or less than the others.

Solution:

Because on all floors there is the same number of apartments (X), on all entrances there is the same number of floors (Y), then denoting the number of entrances by Z, we can write: 357 = X·Y·Z.

Let's factor 357 into prime factors. We get: 357=3·7·17·1. Moreover, this is the only option for the layout. Because Y>X>Z>1, then we do not take into account the unit in the layout and determine that Z=3, X=7, Y=17.

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

Option for the twentieth task of 2019 (6)

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

Execution algorithm

We count the number of agreements signed by 7 countries.
We 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.

Solution:

The first 7 countries signed agreements with 3 countries, i.e. These contracts have 7·3=21 signatures. Similarly, the remaining 3 countries, when drawing up agreements with 7 countries, put 3·7=21 signatures. This means that there are 21+21=42 signatures in total.

Because All contracts are bilateral, which means that each of them has 2 signatures. Consequently, there are half as many contracts as there are signatures, i.e. 42:2=21 agreements.

Option for the twentieth task of 2019 (7)

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

A meridian is an arc of a circle connecting the North and South Poles. A parallel is a circle lying in a plane parallel to the plane of the equator.

Execution algorithm

We prove that 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 a product of the numbers found.

Solution:

If every 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=14 parts.

A meridian is an arc of a circle connecting the poles, i.e. It is not a closed line and does not divide the globe into parts. But 2 meridians are already dividing, 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=350 parts.

Option for the twentieth task of 2019 (8)

Execution algorithm

We determine the number of milk mushrooms among 12 mushrooms and saffron milk caps among 20 mushrooms.
We prove that there is only one correct number representing the number of saffron milk caps. We record it in the answer.

Solution:

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

This means that if there cannot be more than 11 milk mushrooms, then there cannot be less than 30 – 11 = 19 mushrooms. Those. there are no more than 19 saffron milk caps on one side, and no less than 19 on the other. Therefore, there can only be exactly 19 saffron milk caps.

Option for the twentieth task of 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?

Execution algorithm

We introduce notation for the factors. This will allow us to express the original product (before increasing the factors).
We compose an equation for the situation when the factors are increased by 1. We carry out the transformations. We obtain a new expression that displays the relationship between the original factors.
We create an equation for the situation when the factors are increased by 5. We carry out the transformations. We enter the expression obtained in step 2 into the equation and find the desired difference.

Solution:

Let the 1st factor be equal to x, the 2nd – y. Then their product is xy.

After the multipliers are increased by 1, we get:

(x+1)(y+1)=xy+3

xy +y+x+1= xy +3

After increasing the multipliers by 5 we have:

(x+5)(y+5)=xy+N, where N is the desired difference in products.

We carry out the transformations:

xy+5y+5x+25=xy+N

N= xy +5y+5x+25– xy

Because It has already been determined above that x + y = 2, then we get:

Option for the twentieth task of 2019 (10)

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

Execution algorithm

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

Solution:

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

Let's assume that this number is 3. Then we get that in 7 entrances on 6 floors there are 7 6 3 = 126 apartments,

and in 7 entrances on 7 floors there are 7·7·3=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’s prove that it is exactly the right one:

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

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

In order to get to apartment No. 462, you need to pass by 462–420 = 42 apartments. Because On each site there are 10 apartments, then 42:10 = 4.2 floors need to be overcome. 4.2 means that you need to go through 4 floors completely and go up to the 5th. Thus, the required floor is the 5th.

Mysikova Yulia

The Unified State Exam in basic level mathematics consists of 20 tasks. Task 20 tests logical problem solving skills. The student must be able to apply his knowledge to solve problems in practice, including arithmetic and geometric progression. This work examines in detail how to solve task 20 of the Unified State Exam in basic level mathematics, as well as examples and methods of solutions based on detailed tasks.

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Tasks for ingenuity of the Unified State Examination in basic level mathematics. Assignments No. 20 Yulia Aleksandrovna Mysikova, student 11 “A” socio-economic class Municipal educational institution “Secondary school No. 45”

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

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

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

Mushrooms in a basket Solution. From the condition that among any 27 mushrooms there is at least one milk cap, it follows that the number of mushrooms is no more than 26. From the second condition that among any 25 mushrooms there is at least one mushroom, it follows that the number of mushrooms is no more than 24. Since there are 50 mushrooms in total, then there are 24 saffron milk caps, and 26 milk mushrooms. Answer: 24. There are 50 mushrooms in the basket: saffron milk caps and milk mushrooms. It is known that among any 27 mushrooms there is at least one saffron milk cap, and among any 25 mushrooms there is at least one milk mushroom. How many saffron milk caps are in the basket?

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

On the treadmill The trainer advised Andrey to spend 15 minutes on the treadmill on the first day of classes, and at each subsequent lesson to increase the time spent on the treadmill by 7 minutes. In how many sessions will Andrey spend a total of 2 hours and 25 minutes on the treadmill if he follows the trainer’s advice? Solution. 1 way. We note that we need to find the sum of the arithmetic progression with the first term 15 and the difference equal to 7. Using the formula for the sum of the first n terms of the progression S n =(2a 1 +(n-1)d)*n/2 we have 145=(2*15+ (n–1)*7)*n/2, 290=(30+(n–1)*7)*n, 290=(30+7n–7)*n, 290=(23+7n)*n , 290=23n+7n 2 , 7n 2 +23n-290=0, n=5 . Answer: 5. Method 2. More labor intensive. 1-15-15 2-22-37 3-29-66 4-36-102 5-43-145. Answer: 5.

Changing coins Task 20. At the exchange office you can perform one of two operations: for 2 gold coins you get 3 silver and one copper; for 5 silver coins you get 3 gold and one copper. Nicholas only had silver coins. After several visits to the exchange office, his silver coins became smaller, no gold coins appeared, but 50 copper coins appeared. By how much did Nicholas's number of silver coins decrease? Solution. Let Nikolai first perform x operations of the second type, and then y operations of the first type. Then we have: Then there were 3y -5x = 90 – 100 = -10 silver coins, i.e. 10 less. Answer: 10

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

Water in the pit As a result of the flood, the pit was filled with water to a level of 2 meters. The construction pump continuously pumps out water, lowering its level by 20 cm per hour. Subsoil water, on the contrary, increases the water level in the pit by 5 cm per hour. How many hours of pump operation will it take for the water level in the pit to drop to 80 cm? Solution. As a result of pump operation and flooding with soil water, the water level in the pit decreases by 20-5 = 15 centimeters per hour. For the level to drop by 200-80=120 centimeters it takes 120:15=8 hours. Answer: 8.

Tank with a slot A full bucket of water with a volume of 8 liters is poured into a tank with a volume of 38 liters every hour, starting from 12 o’clock. But there is a small gap in the bottom of the tank, and 3 liters flow out of it in an hour. At what point in time (in hours) will the tank be completely filled? Solution. At the end of each hour, the volume of water in the tank increases by 8 − 3 = 5 liters. After 6 hours, that is, at 18 o’clock, there will be 30 liters of water in the tank. At 19:00, 8 liters of water will be added to the tank and the volume of water in the tank will become 38 liters. Answer: 19.

Well The oil company is drilling a well for oil production, which, according to geological exploration data, lies at a depth of 3 km. During the working day, drillers go 300 meters deep, but overnight the well “silts up” again, that is, it is filled with soil to a depth of 30 meters. How many working days will it take oilmen to drill a well to the depth of oil? Solution. Taking into account the siltation of the well, 300-30 = 270 meters pass during the day. This means that in 10 full days 2700 meters will be covered and on the 11th working day another 300 meters will be covered. Answer: 11.

Globe On the surface of the globe, 17 parallels and 24 meridians are drawn with a felt-tip pen. How many parts did the drawn lines divide the surface of the globe into? Solution. One parallel divides the surface of the globe into 2 parts. Two by three parts. Three by four parts, etc. 17 parallels divide the surface into 18 parts. Let's draw one meridian and get one whole (not cut) surface. Let's draw the second meridian and we already have two parts, the third meridian will divide the surface into three parts, etc. 24 meridians divided our surface into 24 parts. We get 18*24=432. All lines will divide the surface of the globe into 432 parts. Answer: 432.

The grasshopper jumps The grasshopper jumps along the coordinate line in any direction for a unit segment per jump. How many different points are there on the coordinate line at which the grasshopper can end up after making exactly 8 jumps, starting from the origin? Solution: After a little thought, we can notice that the grasshopper can only end up at points with even coordinates, since the number of jumps it makes is even. For example, if he makes five jumps in one direction, then in the opposite direction he will make three jumps and end up at points 2 or −2. The maximum grasshopper can be at points whose modulus does not exceed eight. Thus, the grasshopper can end up at points: −8, −6, −4, −2, 0, 2, 4, 6 and 8; only 9 points. Answer: 9.

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

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

Where does Petya live? Problem 1. The house where Petya lives has one entrance. There are six apartments on each floor. Petya lives in apartment No. 50. What floor does Petya live on? Solution: Divide 50 by 6, we get the quotient of 8 and the remainder is 2. This means that Petya lives on the 9th floor. Answer: 9. Problem 2. All entrances of the house have the same number of floors, and all floors have the same number of apartments. In this case, the number of floors in the house is greater than the number of apartments on the floor, the number of apartments on the floor is greater than the number of entrances, and the number of entrances is more than one. How many floors are there in the building if there are 455 apartments in total? Solution: The solution to this problem follows from factoring the number 455 into prime factors. 455 = 13*7*5. This means the house has 13 floors, 7 apartments on each floor in the entrance, 5 entrances. Answer: 13.

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

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

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

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

The doctor prescribed The doctor prescribed the patient to take the medicine according to the following regimen: on the first day he should take 3 drops, and on each subsequent day - 3 drops more than on the previous day. Having taken 30 drops, he drinks 30 drops of the medicine for another 3 days, and then reduces the intake by 3 drops daily. How many bottles of medicine should a patient buy for the entire course of treatment, if each bottle contains 20 ml of medicine (which is 250 drops)? Solution At the first stage of taking drops, the number of drops taken per day is an increasing arithmetic progression with the first term equal to 3, the difference equal to 3 and the last term equal to 30. Therefore: Then 3 + 3(n -1) = 30; 3+ 3 n -3=30; 3 n =30; n =10, i.e. 10 days have passed according to the scheme of increasing to 30 drops. We know the formula for the sum of ariths. progression: Let's calculate S10:

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

Household appliances store In a household appliances store, the volume of sales of refrigerators is seasonal. In January, 10 refrigerators were sold, and in the next three months, 10 refrigerators were sold. Since May, sales have increased by 15 units compared to the previous month. Since September, sales volume began to decrease by 15 refrigerators each month relative to the previous month. How many refrigerators did the store sell in a year? Solution. Let's sequentially calculate how many refrigerators were sold for each month and sum up the results: 10 4+(10+15)+(25+15)+(40+15)+(55+15)+(70-15)+ (55- 15)+(40-15)+ (25-15)= = 40+25+40+55+70+55+40+25+10=120+110+130=360 Answer: 360.

Boxes Boxes of two types, having the same width and height, are stacked in a warehouse in one row 43 m long, adjacent to each other in width. One type of box is 2m long, and the other is 5m long. What is the smallest number of boxes required to fill the entire row without creating empty spaces? Solution Because If you need to find the smallest number of boxes, then => you need to take the largest number of large boxes. So 5 · 7 = 35; 43 – 35 = 8 and 8:2 = 4; 4+7=11 So there are only 11 boxes. Answer: 11.

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

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

A group of tourists A group of tourists crossed a mountain pass. They covered the first kilometer of the climb in 50 minutes, and each subsequent kilometer took 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 top, the tourists began their descent, which was more gradual. The first kilometer after the summit was covered in an hour, and each next kilometer 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? Solution. The group spent 290 minutes going up the mountain, 10 minutes resting, and 210 minutes going down the mountain. In total, tourists spent 510 minutes on the entire route. Let's convert 510 minutes into hours and find that in 8.5 hours the tourists covered the entire route. Answer: 8.5

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