Task 19 profile level. Unified State Exam in Mathematics (profile)

home 30 different ones are written on the board natural numbers

, each of which is either even or its decimal notation ends in the number 7. The sum of the written numbers is 810.

A) Can there be exactly 24 even numbers on the board? Number sequence

is given by the general term formula: a_(n) = 1/(n^2+n) A) Find smallest value< 1/2017.

n , for which a_(n)

B) Find the smallest value of n at which the sum of the first n terms of this sequence will be greater than 0.99. B) Are there any members in this sequence that form?

arithmetic progression A) Let the product of eight different natural numbers be equal to A, and the product of the same numbers increased by 1 be equal to B. Find highest value

B/A.

B) Let the product of eight natural numbers (not necessarily different) be equal to A, and the product of the same numbers increased by 1 be equal to B. Can the value of the expression be equal to 210?

C) Let the product of eight natural numbers (not necessarily different) be equal to A, and the product of the same numbers increased by 1 is equal to B. Can the value of the expression B/A be equal to 63? With a natural number they produce next operation

: between each two of its adjacent digits the sum of these digits is written (for example, from the number 1923 the number 110911253 is obtained).

A) Give an example of a number from which 4106137125 is obtained

B) Can any number produce the number 27593118? In which greatest number

, a multiple of 9, can be obtained from a three-digit number whose decimal notation does not have nines? There are 32 students in the group. Each of them writes either one or two test papers

< 14.
, for each of which you can get from 0 to 20 points inclusive. Moreover, each of the two test papers separately gives an average of 14 points. Next, each student named his highest score (if he wrote one paper, he named it for it), from these scores the arithmetic mean was found and it is equal to S.
B) Could it be that 28 people write two tests and S=11?

Q) What is the maximum number of students who could write two tests if S=11?

There are 100 different natural numbers written on the board, the sum of which is 5130

A) Is it possible that the number 240 is written on the board?

Q) What is the smallest number of multiples of 16 that can be on the board?

There are 30 different natural numbers written on the board, each of which is either even or its decimal notation ends in the number 7. The sum of the written numbers is 810.

A) Can there be exactly 24 even numbers on the board?

B) Can exactly two numbers on the board end in 7?

Q) What is the smallest number of numbers ending in 7 that can be on the board?

Each of the 32 students either wrote one of two tests, or wrote both tests. For each work you could get an integer number of points from 0 to 20 inclusive. For each of the two tests separately GPA was 14. Then each student named the highest of his scores (if the student wrote one paper, then he named the score for it). The arithmetic mean of the named points turned out to be equal to S.

A) Give an example when S< 14

B) Could the value of S be equal to 17?

C) What is the smallest value that S could take if both test papers were written by 12 students?

19) There are 30 numbers written on the board. Each of them is either even or a decimal number ending in 3. Their sum is 793.

A) can there be exactly 23 even numbers on the board;
b) can only one of the numbers end in 3;
c) what is the smallest number of these numbers that can end in 3?

Several different natural numbers are written on the board, the product of any two of which is greater than 40 and less than 100.

A) Can there be 5 numbers on the board?

B) Can there be 6 numbers on the board?

Q) What is the largest value the sum of the numbers on the board can take if there are four of them?

Given numbers: 1, 2, 3, ..., 99, 100. Is it possible to divide these numbers into three groups so that

A) in each group the sum of the numbers was divided by 3.
b) in each group the sum of the numbers was divided by 10.
c) the sum of the numbers in one group was divided by 102, the sum of the numbers in another group was divided by 203, and the sum of the numbers in the third group was divided by 304?

a) Find a natural number n such that the sum 1+2+3+...+n is equal to a three-digit number, all of whose digits are the same.

B) The sum of the four numbers that make up an arithmetic progression is 1, and the sum of the cubes of these numbers is 0.1. Find these numbers.

A) Can the numbers 2, 3, 4, 5, 6, 7, 8, 9, 10 be divided into two groups with the same product of numbers in these groups?

B) Can the numbers 4, 5, 6, 7, 8, 9, 10, 12, 14 be divided into two groups with the same product of numbers in these groups?

Q) What is the smallest number of numbers that must be eliminated from the set 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 so that the remaining numbers can be divided into two groups with the same product of numbers in these groups? Give an example of such a division into groups.

Given a checkered square measuring 6x6.

A) Can this square be cut into ten pairwise different checkered polygons?
B) Can this square be cut into eleven pairwise different checkered polygons?
B) What is the largest number of pairwise different checkered rectangles that this square can be cut into?

Each cell of a 3 x 3 table contains numbers from 1 to 9 (Fig.). In one move it is possible to reach two adjacent numbers (cells
have a common side) add the same integer.

A) Is it possible to obtain a table in this way, in all cells of which there will be the same numbers?

B) Is it possible to obtain a table in this way consisting of one one (in the center) and eight zeros?

C) After several moves, the table contains eight zeros and some number N other than zero. Find all possible N.

A) Each point on the plane is colored in one of two colors. Are there necessarily two points of the same color on the plane that are exactly 1 m apart from each other?

B) Each point on the line is colored in one of 10 colors. Are there necessarily two points of the same color on a straight line, separated from each other by an integer number of meters?

B) Can any number produce the number 27593118? greatest number The vertices of a cube can be colored in Blue colour so that among blue peaks it was impossible to choose three that form equilateral triangle?

It is known about a five-digit natural number N that it is divisible by 12, and the sum of its digits is divisible by 12.

A) Can all five digits in N be different?
B) Find the smallest possible number N;
B) Find the largest possible number N;
D) What is the largest number of identical digits that can be contained in the number N? How many such numbers N are there (containing the largest number of identical digits in their notation)?

There are five sticks with lengths 2, 3, 4, 5, 6.

A) Is it possible to form an isosceles triangle using all the sticks?

B) Is it possible to form a right triangle using all the sticks?

In which smallest area can you make a triangle using all the sticks? (You can't break the sticks)

Three different natural numbers are the lengths of the sides of some obtuse triangle.

A) Can the ratio of the larger of these numbers to the smaller of them be equal to 3/2?

B) Can the ratio of the larger of these numbers to the smaller of them be equal to 5/4?

C) What is the smallest value that the ratio of the largest of these numbers to the smaller of them can take, if it is known that the average number is 18?

The finite sequence a1,a2,...,a_(n) consists of n greater than or equal to 3 not necessarily distinct natural numbers, and for all natural k less than or equal to n-2 the equality a_(k+2) = 2a_(k +1)-a_(k)-1.

A) Give an example of such a sequence for n = 5, in which a_(5) = 4.

B) Can a natural number appear three times in this sequence?

C) For what largest n can such a sequence consist only of three-digit numbers?

The integers x, y, and z, in that order, form a geometric progression.

A) Can the numbers x+3, y^2 and z+5 form an arithmetic progression in that order?

B) Can the numbers 5x, y and 3z form an arithmetic progression in that order?

B) Find all x, y and z such that the numbers 5x+3, y^2 and 3z+5 form an arithmetic progression in that order.

There are two natural numbers written on the board: 672 and 560. In one move, you can replace any of these numbers with the modulus of their difference or halve it (if the number is even).

A) Can there be two identical numbers on the board after a few moves?

B) Can the number 2 appear on the board in a few moves?

C) Find the smallest natural number that can appear on the board as a result of such moves.

Chess can be won, lost or drawn. The chess player writes down the result of each game he plays and after each game he calculates three indicators: “wins” - the percentage of victories, rounded to the nearest whole, “draws” - the percentage of draws, rounded to the nearest whole, and “defeats”, equal to the difference of 100 and the sum of the “wins” indicators " and "draws". (For example, 13.2 is rounded to 13, 14.5 is rounded to 15, 16.8 is rounded to 17).
a) Can the win rate be 17 at some point if less than 50 games have been played?
b) Can the “defeat” rate increase after a won game?
c) One of the games was lost. For what is the smallest number of games played, the “defeat” indicator can be equal to 1?

Let q be the least common multiple and d be the greatest common divisor of natural numbers x and y satisfying the equality 3x=8y–29.

There are two platoons in the company, the first platoon has fewer soldiers than the second, but more than 50, and together there are fewer soldiers than 120. The commander knows that a company can be lined up with several people in a row so that each row has the same number soldiers, more than 7, and in no row will there be soldiers from two different platoons.

A) How many soldiers are in the first platoon and how many in the second? Give at least one example.

B) Is it possible to build a company using the indicated method, 11 soldiers in one row?

Q) How many soldiers can there be in a company?

Let q be the least common multiple and d be the greatest common divisor of natural numbers x and y satisfying the equality 3x=8y-29.

A) Can q/d be equal to 170?

B) Can q/d be equal to 2?

B) Find the smallest value of q/d

Determine if two sequences have terms in common

A) 3; 16; 29; 42;... and 2; 19; 36; 53;...

B) 5; 16; 27; 38;... and 8; 19; thirty; 41;...

B) Determine the largest number of common terms that two arithmetic progressions 1 can have; ...; 1000 and 9; ...; 999, if it is known that for each of them the difference is an integer other than 1.

A) Can the number 2016 be represented as the sum of seven consecutive natural numbers?

A) Can the number 2016 be represented as the sum of six consecutive natural numbers?

B) Express the number 2016 as the sum of the greatest number of consecutive even natural numbers.

We call a set of numbers good if it can be divided into two subsets with the same sum of numbers.

A) Is the set (200;201;202;...;299) good?

B) Is the set (2;4;8;...;2^(100)) good?

C) How many good four-element subsets does the set (1;2;4;5;7;9;11) have?

The survey revealed that approximately 58% of respondents prefer an artificial Christmas tree to a natural one (the number 58 was obtained by rounding to the nearest whole number). From the same survey it followed that approximately 42% of respondents never noted New Year not at home.

A) Could exactly 40 people take part in the survey?
b) Could exactly 48 people take part in the survey?
c) What is the smallest number of people who could participate in this survey?

Vanya is playing a game. At the beginning of the game, two different natural numbers from 1 to 9999 are written on the board. In one turn of the game, Vanya must solve quadratic equation x^2-px+q=0, where p and q are two numbers, taken in the order chosen by Vanya, written on the board at the beginning of this move, and, if this equation has two different natural roots, replace the two numbers on the board with these roots . If this equation does not have two different natural roots, Vanya cannot make a move and the game ends.

A) Are there two numbers such that Vanya can make at least two moves when starting to play?
b) Are there two numbers with which Vanya can make ten moves when starting to play?
c) What is the maximum number of moves Vanya can make under these conditions?

30 natural numbers (not necessarily different) were written on the board, each of which is greater than 14, but does not exceed 54. The arithmetic mean of the written numbers was 18. Instead of each of the numbers, a number was written on the board that was half the original one. Numbers that subsequently turned out to be less than 8 were erased from the board.

We will call a four-digit number very happy if all the digits in its decimal notation are different, and the sum of the first two of these digits is equal to the sum of the last two of them. For example, 3140 is a very lucky number.
a) Are there ten consecutive four-digit numbers, among which two are very lucky?
b) Can the difference between two very lucky four-digit numbers equal 2015?
c) Find the smallest natural number for which there is no multiple of a very lucky four-digit number.

Students from a certain school wrote a test. A student could receive a non-negative integer number of points for this test. A student is considered to have passed the test if he scores at least 50 points. To improve the results, each test participant was given 5 points, so the number of people passing the test increased.

A) Could the average scores of participants who failed the test have dropped after this?

B) Could the average score of the participants who did not take the test decrease after this, and at the same time the average score of the participants who passed the test also decreased?

C) Let the initial average score of participants who passed the test be 60 points, those who did not pass the test be 40 points, and the average score of all participants be 50 points. After adding the points, the average score of the participants who passed the test became 63 points, and those who did not pass the test - 43. What is the smallest number of participants with which this situation is possible?

It is known about three different natural numbers that they are the lengths of the sides of some obtuse triangle.

A) Could the ratio of the larger of these numbers to the smaller of them be equal to 13/7?

B) Could the ratio of the larger of these numbers to the smaller of them be equal to 8/7?

C) What is the smallest value that the ratio of the largest of these numbers to the smaller of them can take, if it is known that the average of these numbers is 25?

Boys and girls take part in the chess tournament. For a victory in a chess game, 1 point is awarded, for a draw - 0.5 points, for a loss - 0 points. According to the rules of the tournament, each participant plays each other twice.

A) What is the maximum number of points that girls could score in total if five boys and three girls take part in the tournament?

B) What is the sum of the points scored by all participants if there are nine participants in total?

Q) How many girls could take part in the tournament if it is known that there are 9 times fewer of them than boys, and that the boys scored exactly four times as many points as the girls?

Given is an arithmetic progression (with a difference other than zero) made up of natural numbers whose decimal notation does not contain the number 9.

A) Can such a progression have 10 terms?
b) Prove that the number of its members is less than 100.
c) Prove that the number of terms of any such progression is not more than 72.
d) Give an example of such a progression with 72 terms.

A red pencil costs 18 rubles, a blue one costs 14 rubles. You need to buy pencils, having only 499 rubles and observing additional condition: The number of blue pencils should not differ from the number of red pencils by more than six.

A) Is it possible to buy 30 pencils?

B) Is it possible to buy 33 pencils?

Q) What is the largest number of pencils you can buy?

It is known that a, b, c, and d are pairwise distinct two-digit numbers.
a) Can the equality (a+c)/(b+d)=7/19 be satisfied?
b) Can the fraction (a+c)/(b+d) be 11 times less than the sum (a/c)+(b/d)
c) What is the smallest value the fraction (a+c)/(b+d) can take if a>3b and c>6d

It is known that a, b, c and d are pairwise distinct two-digit numbers.

A) Can the equality (3a+2c)/(b+d) = 12/19 be satisfied?

B) Can the fraction (3a+2c)/(b+d) be 11 times less than the sum 3a/b + 2c/d

C) What is the smallest value that the fraction (3a+2c)/(b+d) can take if a>3b and c>2d?

The natural numbers a, b, c and d satisfy the condition a>b>c>d.

A) Find the numbers a, b, c and d if a+b+c+d=15 and a2−b2+c2−d2=19.

B) Can there be a+b+c+d=23 and a2−b2+c2−d2=23?

C) Let a+b+c+d=1200 and a2−b2+c2−d2=1200. Find the number of possible values ​​of the number a.

Students from one school were writing a test. The result of each student is a non-negative integer number of points. A student is considered to have passed the test if he scores at least 85 points. Due to the fact that the tasks turned out to be too difficult, it was decided to add 7 points to all test participants, due to which the number of those passing the test increased.
a) Could it be that after this the average score of the participants who did not pass the test decreased?
b) Could it be that after this, the average score of the participants who passed the test decreased, and the average score of the participants who did not pass the test also decreased?
c) It is known that initially the average score of the test participants was 85, the average score of the participants who did not pass the test was 70. After adding points, the average score of the participants who passed the test became 100, and those who did not pass the test - 72. With what is the smallest number of participants Is this situation possible?

We call three numbers a good triple if they can be the lengths of the sides of a triangle.
We call three numbers an excellent triple if they can be the lengths of the sides of a right triangle.
a) Given 8 different natural numbers. Could it be? that among them there is not a single good three?
b) Given 4 different natural numbers. Could it turn out that among them you can find three excellent triplets?
c) Given 12 different numbers (not necessarily natural ones). What is the greatest number of excellent triplets that could be among them?

Several identical barrels contain a certain number of liters of water (not necessarily the same). You can transfer any amount of water from one barrel to another at one time.
a) Let there be four barrels containing 29, 32, 40, 91 liters. Is it possible to equalize the amount of water in barrels in no more than four transfers?
b) The path has seven barrels. Is it always possible to equalize the amount of water in all barrels in no more than five transfers?
c) For what is the least number of transfusions you can know to equalize the amount of water in 26 barrels?

There are 30 natural numbers (not necessarily different) written on the board, each of which is greater than 4, but does not exceed 44. The arithmetic mean of the written numbers was 11. Instead of each of the numbers, a number was written on the board that was half the original number. Numbers that then turned out to be less than 3 were erased from the board.
a) Could it turn out that the arithmetic mean of the numbers remaining on the board is greater than 16?
b) Could the arithmetic mean of the numbers remaining on the board be greater than 14 but less than 15?
c) Find the largest possible value of the arithmetic mean of the numbers remaining on the board.

In one of the tasks at an accounting competition, it is required to issue bonuses to employees of a certain department for a total amount of 800,000 rubles (the amount of the bonus for each employee is an integer multiple of 1000). The accountant is given a distribution of bonuses, and he must give them out without change or exchange, having 25 bills of 1000 rubles and 110 bills of 5000 rubles.
a) Will it be possible to complete the task if there are 40 employees in the department and everyone should receive the same amount?
b) Will it be possible to complete the task if the leading specialist needs to be given 80,000 rubles, and the rest is divided equally among 80 employees?
c) What is the largest number of employees in the department that will allow the task to be completed for any distribution of bonus amounts?

The number 2045 and several more (at least two) natural numbers not exceeding 5000 are written on the board. All numbers written on the board are different. The sum of any two of the written numbers is divided by any of the others.
a) Can exactly 1024 numbers be written on the board?
b) Can exactly five numbers be written on the board?
c) What is the smallest number of numbers that can be written on the board?

Several not necessarily different two-digit natural numbers without zeros in decimal notation were written on the board. The sum of these numbers turned out to be equal to 2970. In each number, the first and second digits were swapped (for example, the number 16 was replaced by 61)
a) Give an example of original numbers for which the sum of the resulting numbers is exactly 3 times less than the sum of the original numbers.
b) Could the sum of the resulting numbers be exactly 5 times less than the sum of the original numbers?
c) Find the smallest possible value of the sum of the resulting numbers.

An increasing finite arithmetic progression consists of various non-negative integers. The mathematician calculated the difference between the square of the sum of all terms of the progression and the sum of their squares. Then the mathematician added the next term to this progression and again calculated the same difference.
A) Give an example of such a progression if the second time the difference was 48 greater than the first time.
B) The second time the difference was 1440 greater than the first time. Could the progression initially consist of 12 members?
C) The second time the difference was 1440 greater than the first time. What is the largest number of members that could be in the progression at first?

The numbers from 9 to 18 are written once in a circle in some order. For each of ten pairs of adjacent numbers, their greatest common divisor is found.
a) Could it happen that all greatest common divisors are equal to 1? a) The set -8, -5, -4, -3, -1, 1, 4 is written on the board. What numbers were intended?
b) For some different conceived numbers in the set written on the board, the number 0 appears exactly 2 times.
What is the smallest number of numbers that could be conceived?
c) For some planned numbers, a set is written out on the board. Is it always possible to unambiguously determine the intended numbers from this set?

Several (not necessarily different) natural numbers are conceived. These numbers and all their possible sums (2, 3, etc.) are written on the board in non-decreasing order. If some number n written on the board is repeated several times, then one such number n is left on the board, and the remaining numbers equal to n are erased. For example, if the numbers are 1, 3, 3, 4, then the set 1, 3, 4, 5, 6, 7, 8, 10, 11 will be written on the board.
a) Give an example of planned numbers for which the set 1, 2, 3, 4, 5, 6, 7 will be written on the board.
b) Is there an example of such conceived numbers for which the set 1, 3, 4, 6, 7, 8, 10, 11, 12, 13, 15, 16, 17, 19, 20, 22 would be written on the board?
c) Give all examples of conceived numbers for which the set 7, 9, 11, 14, 16, 18, 20, 21, 23, 25, 27, 30, 32, 34, 41 will be written on the board.

There are stone blocks: 50 pieces of 800 kg each, 60 pieces of 1,000 kg each and 60 pieces of 1,500 kg each (the blocks cannot be split).
a) Is it possible to transport all these blocks simultaneously on 60 trucks, each with a carrying capacity of 5 tons, assuming that the selected blocks will fit into the truck?
b) Is it possible to transport all these blocks simultaneously on 38 trucks, each with a carrying capacity of 5 tons, assuming that the selected blocks will fit into the truck?
c) What is the smallest number of trucks, each with a carrying capacity of 5 tons, will be needed to remove all these blocks at the same time, assuming that the selected blocks will fit in the truck?

Given n different natural numbers that make up an arithmetic progression (n is greater than or equal to 3).

A) Can the sum of all these numbers be equal to 18?

B) What is the largest value of n if the sum of all given numbers is less than 800?

B) Find everything possible values n if the sum of all given numbers is 111?

Several (not necessarily different) natural numbers are conceived. These numbers and all their possible sums (2, 3, etc.) are written on the board in non-decreasing order. If some number n written on the board is repeated several times, then one such number n is left on the board, and the remaining numbers equal to n are erased. For example, if the numbers are 1, 3, 3, 4, then the set 1, 3, 4, 5, 6, 7, 8, 10, 11 will be written on the board.

A) Give an example of planned numbers for which the set 2, 4, 6, 8, 10 will be written on the board.

The cards are turned over and shuffled. On their blank sides they write again one of the numbers:

11, 12, 13, -14, -15, 17, -18, 19.
After this, the numbers on each card are added, and the resulting eight sums are multiplied.

A) Can the result be 0?

B) Could the result be 117?

Q) What is the smallest non-negative integer that can result?

Several integers are conceived. A set of these numbers and all their possible sums (2, 3, etc.) are written on the board in non-decreasing order. For example, if the numbers are 2, 3, 5, then the set 2, 3, 5, 5, 7, 8, 10 will be written on the board.

A) A set of -11, -7, -5, -4, -1, 2, 6 is written on the board. What numbers were intended?
b) For some different conceived numbers in the set written on the board, the number 0 appears exactly 4 times. What is the smallest number of numbers that could be conceived? a) How many numbers are written on the board?
b) Which numbers are written more: positive or negative?
c) What is the largest number of positive numbers that can be among them?

Unified State Examination in mathematics profile level

The work consists of 19 tasks.
Part 1:
8 short answer tasks of basic difficulty level.
Part 2:
4 short answer questions
7 tasks with detailed answers of a high level of difficulty.

Running time - 3 hours 55 minutes.

Examples of Unified State Examination tasks

Solving Unified State Examination tasks in mathematics.

For independent decision:

1 kilowatt-hour of electricity costs 1 ruble 80 kopecks.
The electricity meter showed 12,625 kilowatt-hours on November 1, and 12,802 kilowatt-hours on December 1.
How much should I pay for electricity for November?
Give your answer in rubles.

At the exchange office, 1 hryvnia costs 3 rubles 70 kopecks.
Vacationers exchanged rubles for hryvnia and bought 3 kg of tomatoes at a price of 4 hryvnia per 1 kg.
How many rubles did this purchase cost them? Round your answer to a whole number.

Masha sent SMS messages with New Year's greetings to my 16 friends.
The cost of one SMS message is 1 ruble 30 kopecks. Before sending the message, Masha had 30 rubles in her account.
How many rubles will Masha have left after sending all the messages?

The school has three-person camping tents.
What is the smallest number of tents you need to take on a camping trip involving 20 people?

The Novosibirsk-Krasnoyarsk train departs at 15:20 and arrives at 4:20 the next day (Moscow time).
How many hours does the train travel?

Solve the equation:

1/cos 2 x + 3tgx - 5 = 0

Please indicate the roots
belonging to the segment(-p; p/2).

Solution:

1) Let's write the equation like this:

(tg 2 x +1) + 3tgx - 5 = 0

Tg 2 x + 3tgx - 4 = 0

tgx = 1 or tgx = -4.

Hence:

X = n/4 + nk or x = -arctg4 + nk.

Segment (-p; p/2)

The roots belong to -3p/4, -arctg4, p/4.

Answer: -3p/4, -arctg4, p/4.

Do you know what?

If you multiply your age by 7, then multiply by 1443, the result will be your age written three times in a row.

We think of negative numbers as something natural, but this was not always the case. Negative numbers were first legalized in China in the 3rd century, but were used only for exceptional cases, as they were considered, in general, meaningless. A little later, negative numbers began to be used in India to denote debts, but in the west they did not take root - the famous Diophantus of Alexandria argued that the equation 4x+20=0 was absurd.

The American mathematician George Danzig, while a graduate student at the university, was once late for class and mistook the equations written on the blackboard for homework. It seemed more difficult to him than usual, but after a few days he was able to complete it. It turned out that he solved two “unsolvable” problems in statistics that many scientists had struggled with.

In Russian mathematical literature, zero is not a natural number, but in Western literature, on the contrary, it belongs to the set of natural numbers.

Used by us decimal system Numbers arose due to the fact that a person has 10 fingers on his hands. People did not immediately develop the ability to count abstractly, and it turned out to be most convenient to use their fingers for counting. The Mayan civilization and, independently of them, the Chukchi historically used the twenty-digit number system, using fingers not only on the hands, but also on the toes. The duodecimal and sexagesimal systems common in ancient Sumer and Babylon were also based on the use of hands: the phalanges of the other fingers of the palm, the number of which is 12, were counted with the thumb.

One lady friend asked Einstein to call her, but warned that her phone number was very difficult to remember: - 24-361. Do you remember? Repeat! Surprised, Einstein replied: “Of course I remember!” Two dozen and 19 squared.

Stephen Hawking is one of the leading theoretical physicists and popularizer of science. In his story about himself, Hawking mentioned that he became a professor of mathematics without receiving any mathematical education since high school. When Hawking began teaching mathematics at Oxford, he read the textbook two weeks ahead of his own students.

The maximum number that can be written in Roman numerals without violating Shvartsman's rules (rules for writing Roman numerals) is 3999 (MMMCMXCIX) - you cannot write more than three digits in a row.

There are many parables about how one person invites another to pay him for some service as follows: on the first square chessboard he will put one grain of rice, on the second - two, and so on: on each next cell twice as much as on the previous one. As a result, the one who pays in this way will certainly go bankrupt. This is not surprising: it is estimated that total weight rice will amount to more than 460 billion tons.

In many sources, often with the purpose of encouraging poorly performing students, there is a statement that Einstein failed mathematics at school or, moreover, generally studied very poorly in all subjects. In fact, everything was not like that: Albert began to show talent in mathematics at an early age and knew it far beyond the school curriculum.

Unified State Exam 2019 in mathematics task 19 with solution

Demo Unified State Exam option 2019 in mathematics

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Unified State Exam 2019 in mathematics task 19

Unified State Exam 2019 in mathematics profile level task 19 with solution

Unified State Exam in Mathematics
The number P is equal to the product of 11 different natural numbers greater than 1.

What is the smallest number of natural divisors (including one and the number itself) that the number P can have.

Any natural number N can be represented as a product:

N = (p1 x k1) (p2 x k2) ... etc.,

Where p1, p2, etc. - prime numbers,

And k1, k2, etc. - non-negative integers.

15 = (3 1) (5 1)

For example:

72 = 8 x 9 = (2 x 3) (3 2)

So, the total number of natural divisors of the number N is equal to

(k1 + 1) (k2 + 1) ...
N1 = (p1 x k) (p2 x k) ...
N2 = (p1 x k) (p2 x k) ...
...,
which means that
P = (p1 x (k + k + ... + k)) (p2 x (k + k + ... + k)) ...,

And the total number of natural divisors of P is equal to

(k + k + ... + k + 1) (k + k + ... + k + 1) ...

This expression takes on a minimum value if all numbers N1...N11 are successive natural powers of the same prime number, starting from 1: N1 = p, N2 = p 2 , ... N11 = p 1 1.

That is, for example,
N1 = 2 1 = 2,
N2 = 2 2 = 4,
N3 = 2 3 = 8,
...
N11 = 2 1 1 = 2048.

Then the number of natural divisors of P is equal to
1 + (1 + 2 + 3 + ... + 11) = 67.

Unified State Exam 2019 in mathematics profile level task 19 with solution

Solution:

Every natural number can be either even (2 k) or odd (2 k+1).

1. If the number is odd:
n = 2 k+1 = (k)+(k+1). Numbers k and k+1 are always relatively prime

(if there is some number d that is a divisor of x and y, then the number |x-y| must also be divisible by d. (k+1)-(k) = 1, that is, 1 must be divisible by d, that is, d=1, and this is a proof of mutual simplicity)

That is, we have proven that all odd numbers can be represented as the sum of two relatively prime numbers.
An exception according to the condition will be the numbers 1 and 3, since 1 cannot be represented at all as a sum of naturals, and 3 = 2+1 and nothing else, and one as a term does not fit according to the condition.

2. If the number is even:
n=2k
Here we have to consider two cases:

2.1. k - even, i.e. representable as k = 2 m.
Then n = 4 m = (2 m+1)+(2 m-1).
The numbers (2 m+1) and (2 m-1) can only have a common divisor (see above) that divides the number (2 m+1)-(2 m-1) = 2. 2 is divisible by 1 and 2.
But if the divisor is 2, then it turns out that the odd number 2 m+1 must be divisible by 2. This cannot happen, so only 1 remains.

So we proved that all numbers of the form 4 m (that is, multiples of 4) can also be represented as the sum of two relatively prime ones.
The exception here is the number 4 (m=1), which, although it can be represented as 1+3, unit as a term is still not suitable for us.

2.1. k - odd, i.e. representable as k = 2 m-1.
Then n = 2 (2 m-1) = 4 m-2 = (2 m-3)+(2 m+1)
The numbers (2 m-3) and (2 m+1) can have a common divisor that divides the number 4. That is, either 1, or 2, or 4. But neither 2 nor 4 are suitable, since (2 m+ 1) - the number is odd and cannot be divided by either 2 or 4.

So we proved that all numbers of the form 4 m-2 (that is, all multiples of 2, but not multiples of 4) can also be represented as the sum of two relatively prime ones.
The exceptions here are the numbers 2 (m=1) and 6 (m=2), for which one of the terms in the decomposition into a pair of relatively primes is equal to one.

The 19th task in the profile level of the Unified State Examination in mathematics is aimed at identifying students’ ability to operate with numbers, namely their properties. This task is the most difficult and requires a non-standard approach and good knowledge of the properties of numbers. Let's move on to consider a typical task.

Analysis of typical options for tasks No. 19 of the Unified State Examination in mathematics at the profile level

First version of the task (demo version 2018)

There are more than 40 but less than 48 integers written on the board. The arithmetic mean of these numbers is –3, the arithmetic mean of all positive ones is 4, and the arithmetic mean of all negative ones is –8.

a) How many numbers are written on the board?

b) Which numbers are written more: positive or negative?

c) What is the largest number of positive numbers that can be among them?

Solution algorithm:

1. We introduce the variables k, l, m.
2. Find the sum of a set of numbers.
3. We answer point a).
4. We determine which numbers are greater (point b)).
5. Determine how many positive numbers there are.

Solution:

1. Let k be positive numbers among the numbers written on the board. Negative numbers l and zero m.

2. The sum of the written numbers is equal to their number in a given entry on the board, multiplied by the arithmetic mean. Determine the amount:

4k−8 l+ 0⋅m = − 3(k + l+m)

3. Note that on the left in the equality just given, each of the terms is divisible by 4, therefore the sum of the number of each type of numbers k + l+ m is also divisible by 4. By condition total number written numbers satisfies the inequality:

40 < k + l+m< 48

Then k + l+ m = 44, because 44 is the only natural number between 40 and 48 that is divisible by 4.

This means that there are only 44 numbers written on the board.

4. Determine which type of numbers there are more: positive or negative. To do this, we present the equality 4k −8l = − 3(k + l+m) to a more simplified form: 5 l= 7k + 3m.

5. m≥ 0. This implies: 5 l≥ 7k, l> k. It turns out that there are more negative numbers written than positive ones. We substitute k + l+ m number 44 in equality

4k −8l = − 3(k + l+ m).

4k − 8 l= −132, k = 2 l − 33

k + l≤ 44, then it turns out: 3 l− 33 ≤ 44; 3l ≤ 77;l≤ 25; k = 2 l− 33 ≤17. From here we come to the conclusion that there are no more than 17 positive numbers.

If there are only 17 positive numbers, then the number 4 is written on the board 17 times, the number −8 is written 25 times, and the number 0 is written 2 times. Such a set meets all the requirements of the problem.

Answer: a) 44; b) negative; c) 17.

Second option 1 (from Yashchenko, No. 1)

There are 35 different natural numbers written on the board, each of which is either even or its decimal notation ends in the number 3. The sum of the written numbers is 1062.

a) Can there be exactly 27 even numbers on the board?

b) Can exactly two numbers on the board end in 3?

c) What is the smallest number of numbers ending in 3 that can be on the board?

Solution algorithm:

1. Let's give an example of a set of numbers that satisfies the condition (This confirms the possibility of a set of numbers).
2. We check the probability of the second condition.
3. We look for the answer to the third question by introducing the variable n.
4. We write down the answers.

Solution:

1. This approximate list of numbers on the board meets the given conditions:

3,13,23,33,43,53,63,73,2,4,6,…,50,52,56

This answers question a in the affirmative.

2. Let there be exactly two numbers written on the board, the last digit of which is 3. Then there are 33 even numbers written there. Their sum:

This contradicts the fact that the sum of the written numbers is 1062, that is, there is no affirmative answer to question b.

3. We assume that there are n numbers written on the board that end in 3, and (35 – n) of those written out are even. Then the sum of numbers that end in 3 is equal to

and the sum of even ones:

2+4+…+2(35 – n)=(35 – n)(36 – n)= n 2 -71 n+1260.

Then from the condition:

We solve the resulting inequality:

It turns out that . From here, knowing that n is a natural number, we get .

3. The smallest number of numbers ending in 3 can only be 5. And 30 even numbers are added, then the sum of all numbers is odd. This means there are more numbers that end in 3. than five, since the sum by condition is equal to an even number. Let's try to take 6 numbers, with the last digit being 3.

Let's give an example when 6 numbers end in three, and 29 even numbers. Their sum is 1062. The result is the following list:

3, 13, 23, 33, 43, 53, 2, 4, ..., 54, 56, 82.

Answer: a) yes; b) no; at 6.

Third option (from Yashchenko, No. 4)

Masha and Natasha took photographs for several days in a row. On the first day, Masha took m photographs, and Natasha - n photographs. On each subsequent day, each of the girls took one more photo than on the previous day. It is known that Natasha took a total of 1173 more photographs than Masha, and that they took photographs for more than one day.

a) Could they take photographs for 17 days?

b) Could they take photographs for 18 days?

c) What is the greatest total number of photographs that Natasha could take over all the days of photography, if it is known that on the last day Masha took less than 45 photographs?

Solution algorithm:

1. Let's answer question a).
2. Let's find the answer to question b).
3. Let's find the total number of photographs taken by Natasha.
4. Let's write down the answer.

Solution:

1. If Masha took m photographs on the 1st day, then in 17 days she took photographs pictures.

Unified State Exam 2017. Mathematics. Profile level. Task 19. Solving problems and equations in integers. Sadovnichy Yu.V.

M.: 2017. - 128 p.

This book is devoted to problems in which the properties of integers are used. Using the example of problems similar to problems from the Unified State Exam variants, as well as tasks offered at various mathematical Olympiads, an attempt was made to systematize them by type and outline the main methods of solution. The author hopes that this book will be useful for high school students self-study to the Unified State Examination, as well as mathematics teachers, club leaders and all those who want to independently learn to solve interesting mathematical problems.

Format: pdf

Size: 1.4 MB

Watch, download:drive.google

TABLE OF CONTENTS
INTRODUCTION 4
CHAPTER 1. FIRST ORDER DIOPHANTINE EQUATIONS WITH TWO UNKNOWN 6
Problems for independent solution 11
CHAPTER 2. SECOND ORDER DIOPHANTINE EQUATIONS WITH TWO UNKNOWN 12
Problems for independent solution 20
CHAPTER 3. OTHER EQUATIONS IN INTEGERS 22
Problems for independent solution 25
CHAPTER 4. TEXT PROBLEMS USING INTEGER EQUATIONS 28
Problems for independent solution 33
CHAPTER 5. VARIABLE ESTIMATES. ORGANIZATION OF SEARCH 36
Problems for independent solution 45
CHAPTER 6. INEQUALITIES IN WHOLE NUMBERS. GRAPHIC ILLUSTRATIONS 51
Problems for independent solution 60
CHAPTER 7. DIVISION PROBLEMS 62
Problems for independent solution 68
CHAPTER 8. TEXT PROBLEMS USING THE DIVISION OF INTEGERS 70
Problems for independent solution 75
CHAPTER 9. EXTREME PROBLEMS IN INTEGERS 79
Problems for independent solution 87
CHAPTER 10. INTEGER PROGRESSIONS 91
Problems for independent solution 97
CHAPTER 11. INTEGERS AND SQUARE TRINOMIAL 99
Problems for independent solution 105
CHAPTER 12. TASKS SIMILAR TO TASKS 19 FROM USE 107
Problems for independent solution 113
CHAPTER 13. PROBLEMS OF MATHEMATICAL OLYMPIADS 115
Problems for independent solution 120
ANSWERS TO PROBLEMS FOR INDEPENDENT STUDY
SOLUTIONS 124

In recent years, interest in problems that use the properties of integers has increased significantly. This is determined, first of all, by the changed format of the United state exam mathematics. In the Unified State Exam variants recent years The high level problem (problem 19) is traditionally associated with integers. In addition, such problems are found in almost every version of various Olympiads held for high school students and providing benefits for admission to universities.
Integer problems have always been considered one of the most challenging problems offered to high school students. This is due to the lack of a single method or even several methods for solving them. At the same time, the solution to most of these problems, with the possible exception of problems examined in special courses in physics and mathematics schools, does not contain theoretical material that goes beyond the scope of the high school mathematics course curriculum. Moreover, the theory, in a sense, is generally reduced to a minimum here. For example, to solve problems involving integers, it is not at all necessary to know all the trigonometry formulas. But what is absolutely necessary is the ability to think logically, to embrace the entire problem, as chess players say, to “calculate several moves ahead.”

In task 19 of the basic level, problems are proposed on the topic “Divisibility of natural numbers”. To solve such a problem, you need to know well the signs of divisibility of natural numbers.

Signs of divisibility.

Signs of divisibility by 2, 3, 4, 6, 8, 9, 11, 5, 25, 10, 100, 1000.

1. Test of divisibility by 2 . A number is divisible by 2 if its last digit is zero or divisible by 2. Numbers divisible by two are called even, those not divisible by two are called odd.

2. Test of divisibility by 4 . A number is divisible by 4 if its last two digits are zeros or form a number that is divisible by 4.

3. Test of divisibility by 8 . A number is divisible by 8 if its last three digits are zeros or form a number that is divisible by 8.

4. Signs of divisibility into 3 And 9 . A number is divisible by 3 if its sum of digits is divisible by 3. A number is divisible by 9 if its sum of digits is divisible by 9.

5. Test of divisibility by 6 . A number is divisible by 6 if it is divisible by 2 and 3.

6. Test of divisibility by 5 . A number is divisible by 5 if its last digit is zero or 5.

7. Test of divisibility by 25 . A number is divisible by 25 if its last two digits are zeros or form a number that is divisible by 25.

8. Test of divisibility by 10 . A number is divisible by 10 if its last digit is zero.

9. Test of divisibility by 100 . A number is divisible by 100 if its last two digits are zeros.

10. Test of divisibility by 1000 . A number is divisible by 1000 if its last three digits are zeros.

11. Test of divisibility by 11 . Only those numbers are divisible by 11 if the sum of the digits in odd places is either equal to the sum of the digits in even places or differs from it by a number divisible by 11. (For example, 12364 is divisible by 11, because 1+3+4=2+6.)

Za-da-nie 19 (1). For example, a three-digit number, the sum of the digits is equal to 20, and the sum of the squares of the digits is divided by 3, but not de-lit -sya on 9.

Solution.

Divide the number 20 into weaker, different ways:

1) 20 = 9 + 9 + 2

2) 20 = 9 + 8 + 3

3) 20 = 9 + 7 + 4

4) 20 = 9 + 6 + 5

5) 20 = 8 + 8 + 4

6) 20 = 8 + 7 + 5.

Find the sum of squares in each expansion and check whether it is divisible by 3 and not divisible by 9?

We note that if in the decomposition 2 numbers are divisible by 3, then the sum of squares is not divisible by 3.

9 2 +9 2 +2 2 is not divisible by 3

When dividing with the help of (1)−(4), the sums of squares of numbers are not divisible by 3.

When dividing the method (5), the sum of the squares is divided by 3 and 9.

The disposition in the sixth way satisfies the conditions for doing so. Thus, the condition is satisfied by any number, such as the digits 5, 7 and 8, for example , numbers 578 or 587 or 785, etc.