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

Task 20 Basic USE level

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

17) 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 4200 rubles, and for each next meter - 1300 rubles more than for the previous one. How much money will the owner have to pay the workers if they dig a well 11 meters deep ?(Answer: 117700)

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

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

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

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

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

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

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

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

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

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

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. Into how many parts did the drawn lines divide the surface of the globe? A meridian is an arc of a circle connecting the North and South Poles. A parallel is a circle lying in a plane parallel to the plane of the equator. (18 24 =432)

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

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

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

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

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

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

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

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

Problem #5922.

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

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

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

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

Substitute the data of the problem in this formula.

Answer: 89100.

Problem #5943.

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

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

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

Problem #5960.

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

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

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

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

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

There are only 6 of these points.

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

Answer: 6

Problem #5990

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

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

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

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

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

Answer: 8 days.

Task number 6010.

All entrances of the house have the same number of floors, and each floor has the same number of apartments. At the same time, the number of floors in the house more number apartments per floor, the number of apartments per floor is greater than the number of entrances, and the number of entrances is more than one. How many floors are there in a building if there are 105 apartments in total?

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

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

(1)

The last inequality reflects the condition "the number of floors in the building is greater than the number of apartments per floor, the number of apartments per floor is greater than the number of entrances, and the number of entrances is more than one."

That is, ( ) is the largest number.

We decompose 105 into prime factors:

Taking into account condition (1), .

Answer: 7.

Problem #6036.

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

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

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

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

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

Therefore, the number of mushrooms equals 19.

Answer: 19.

Problem number 6047.

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

Let on each floor of apartments.

Then the number of apartments in the first six entrances is

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

From here

The number of the last apartment in the sixth entrance -

The seventh entrance starts from the 325th apartment.

Therefore, apartment 333 is on the second floor.

Answer: 2

Problem number 6060.

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

Imagine a watermelon that we cut into pieces.

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

Now we will cut each slice.

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

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

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

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

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

Answer: 432.

Problem #6069

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

If you make 1 cut, you get 2 pieces.

If you make 2 cuts, you get 3 pieces.

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

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

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

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

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

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

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

Answer: 21.

Problem #9626.

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

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

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

Let's try this:

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

Consider this option:

This option satisfies the condition of the problem.

Answer: 10.

Problem #10041.

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

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

In addition, according to the condition

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

This equation must be solved in integers.

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

Let , then .

In this case, all conditions are met.

Problem #10056.

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

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

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

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

By the property of proportion, we get

From here.

Problem #10071.

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

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

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

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

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

Simplify the right and left sides, we get:

So, .

Answer: 18.

Problem #10086.

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

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

Let the number of rows in the table be .

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

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

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

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

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

The only natural that satisfies this system is

Answer: 17.

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

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

Let the unknown natural number be .

We get: ;

Let's consider various options.

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

Let's check: ; (1)

165 is divisible by 5.

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

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

Let's check: ; (2)

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

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

Let's check: ; (3)

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

Answer: 33

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

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

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

Answer: 85

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

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

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

Answer: 144

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

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

From here we get the system:

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

Answer: 7

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

Solution

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

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

We get: - the number of outgoing wires.

Number of wires stretched.

Answer: 40

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

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

Find the number of signatures:

Find the number of signed contracts:

Answer: 21

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

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

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

We get a system of inequalities:

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

Answer: 20

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

Solution

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

Let's find how many games there were.

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

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

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

Answer: 13

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

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

Got that

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

Answer: 3

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

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

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

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

We get the equality:

Single State exam mathematics basic level consists of 20 tasks. Task 20 tests the skills of solving logical problems. The student should be able to apply his knowledge to solve problems in practice, including arithmetic and geometric progression. Here you can learn how to solve task 20 of the Unified State Examination in mathematics at a basic level, as well as study examples and solutions based on detailed tasks.

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Two transverse stripes are marked on the tape on different sides from the middle

On the tape, on different sides from the middle, two transverse stripes are marked: blue and red. If you cut the tape along the blue strip, then one part will be longer than the other by A cm. If you cut along the red one, then one part will be longer than the other by B cm. Find the distance from the red to the blue strip.

The task about the tape is part of the USE in mathematics of the basic level for grade 11 at number 20.

Biologists have discovered a variety of amoeba

Biologists have discovered a variety of amoeba, each of which divides into two exactly in a minute. The biologist puts an amoeba in a test tube, and exactly after N hours the test tube is completely filled with amoeba. How many minutes will it take for the whole test tube to be filled with amoebas if we put not one, but K amoebas in it?

When demonstrating summer clothes, the outfits of each fashion model

When demonstrating summer clothes each fashion model's outfits differ in at least one of three elements: blouse, skirt and shoes. In total, the fashion designer prepared for the demonstration A types of blouses, B types of skirts and C types of shoes. How many different outfits will be shown in this demo?

The task about outfits is part of the USE in mathematics of the basic level for grade 11 at number 20.

A group of tourists overcame a mountain pass

A group of tourists overcame a mountain pass. They covered the first kilometer of the ascent in K minutes, and each next kilometer covered L minutes longer than the previous one. The last kilometer before the summit was covered in M minutes. After resting N minutes at the top, the tourists began their descent, which was more gentle. The first kilometer after the top was covered in P minutes, and each next one is R minutes faster than the previous one. How many hours did the group spend on the entire route if the last kilometer of the descent was covered in S minutes.