Task 16:

Is it possible to exchange 25 rubles with ten banknotes in denominations of 1, 3 and 5 rubles? Solution:

Answer: No

Petya bought a common notebook with a volume of 96 sheets and numbered all its pages in order with numbers from 1 to 192. Vasya tore out 25 sheets from this notebook and added up all 50 numbers that are written on them. Could he have made 1990? Solution:

On each sheet, the sum of the page numbers is odd, and the sum of the 25 odd numbers is odd.

The product of 22 integers is equal to 1. Prove that their sum is not equal to zero. Solution:

Among these numbers - even number"minus units", and in order for the sum to be equal to zero, there must be exactly 11 of them.

Is it possible to make a magic square from the first 36 prime numbers? Solution:

Among these numbers, one (2) is even and the rest are odd. Therefore, in the line where there is a deuce, the sum of the numbers is odd, and in the others it is even.

Numbers from 1 to 10 are written in a row. Is it possible to place the signs “+” and “-” between them so that the value of the resulting expression is equal to zero?

Note: Please note that negative numbers are also odd and even. Solution:

Indeed, the sum of numbers from 1 to 10 is 55, and by changing the signs in it, we change the entire expression to an even number.

The grasshopper jumps in a straight line, and the first time he jumped 1 cm in some direction, the second time he jumped 2 cm, and so on. Prove that after 1985 jumps he cannot be where he started. Solution:

Note: The sum 1 + 2 + … + 1985 is odd.

The numbers 1, 2, 3, ..., 1984, 1985 are written on the board. It is allowed to erase any two numbers from the board and write down the modulus of their difference instead. In the end, only one number will remain on the board. Can it be zero? Solution:

Check that the indicated operations do not change the parity of the sum of all the numbers written on the board.

Is it possible to cover a chessboard with 1 × 2 dominoes in such a way that only cells a1 and h8 remain free? Solution:

Each domino covers one black and one white square, and when the a1 and h8 squares are thrown out, there are 2 less black squares than white ones.

To the 17-digit number was added the number written in the same digits, but in reverse order. Prove that at least one digit of the resulting sum is even. Solution:

Analyze two cases: the sum of the first and last digits of the number is less than 10, and the sum of the first and last digits of the number is not less than 10. If we assume that all the digits of the sum are odd, then in the first case there should not be a single carry in the digits (which, obviously , leads to a contradiction), and in the second case, the presence of a carry when moving from right to left or from left to right alternates with the absence of a carry, and as a result we get that the digit of the sum in the ninth digit is necessarily even.

There are 100 people in the people's squad, and every evening three of them go on duty. Can it turn out after some time that everyone was on duty with everyone exactly once? Solution:

Since on each watch in which this person participates, he is on duty with two others, then all the rest can be divided into pairs. However, 99 is an odd number.

There are 45 points marked on the straight line that lie outside the segment AB. Prove that the sum of the distances from these points to point A is not equal to the sum of the distances from these points to point B. Solution:

For any point X lying outside AB, we have AX - BX = ± AB. If we assume that the sums of distances are equal, then we get that the expression ± AB ± AB ± … ± AB, in which 45 terms are involved, is equal to zero. But this is impossible.

There are 9 numbers arranged in a circle - 4 ones and 5 zeros. Every second, the following operation is performed on the numbers: zero is put between adjacent numbers if they are different, and one if they are equal; after that, the old numbers are erased. Can all numbers become the same after some time? Solution:

It is clear that a combination of nine ones before nine zeros cannot be obtained. If there were nine zeros, then on the previous move, zeros and ones should have alternated, which is impossible, since there are only an odd number of them.

25 boys and 25 girls sit at a round table. Prove that one of the people sitting at the table has both neighbors boys. Solution:

Let us carry out our proof by contradiction. We number all those sitting at the table in order, starting from some place. If on k-th place a boy is sitting, it is clear that the (k - 2)-th and (k + 2)-th places are occupied by girls. But since there are equal numbers of boys and girls, then for any girl sitting in the nth place, it is true that the (n - 2)th and (n + 2)th places are occupied by boys. If we now consider only those 25 people who are sitting in "even" places, then we get that among them boys and girls alternate if they go around the table in some direction. But 25 is an odd number.

The snail crawls along the plane at a constant speed, turning at right angles every 15 minutes. Prove that it can return to the starting point only after an integer number of hours. Solution:

It is clear that the number a of the sections in which the snail crawled up or down is equal to the number of sections in which it crawled to the right or to the left. It only remains to note that a is even.

Three grasshoppers play leapfrog on a straight line. Each time one of them jumps over the other (but not over two at once!). Can they return to their original positions after the 1991 jump? Solution:

Denote the grasshoppers A, B and C. Let's call the grasshoppers' arrangements ABC, BCA and CAB (from left to right) correct, and ACB, BAC and CBA incorrect. It is easy to see that with any jump the type of arrangement changes.

There are 101 coins, of which 50 are counterfeit, differing in weight by 1 gram from the real ones. Petya took one coin and for one weighing on the scales with an arrow showing the difference in weights on the cups, he wants to determine whether it is fake. Can he do it? Solution:

You need to put this coin aside, and then divide the remaining 100 coins into two piles of 50 coins, and compare the weights of these piles. If they differ by an even number of grams, then the coin we are interested in is real. If the difference between the weights is odd, then the coin is counterfeit.

Is it possible to write out the numbers from 1 to 9 in a row one time so that there is an odd number of digits between one and two, two and three, ..., eight and nine? Solution:

Otherwise, all the numbers in the row would be in places of the same parity.

This work Petya bought a common notebook with a volume of 96 sheets and numbered all its pages in order with numbers from 1 to 192. Vasya pulled out (Control) in the subject (AHD and the financial analysis), was custom-made by the specialists of our company and passed its successful defense. Work - Petya bought a common notebook with a volume of 96 sheets and numbered all its pages in order with numbers from 1 to 192. Vasya pulled out on the subject of AHD and financial analysis reflects its topic and the logical component of its disclosure, the essence of the issue under study is revealed, the main provisions and leading ideas are highlighted this topic.
Work - Petya bought a common notebook with a volume of 96 sheets and numbered all its pages in order with numbers from 1 to 192. Vasya tore it out, contains: tables, drawings, the latest literary sources, the year of submission and defense of the work - 2017. In the work, Petya bought a common notebook volume of 96 sheets and numbered all its pages in order by numbers from 1 to 192. Vasya pulled out (AHD and financial analysis) the relevance of the research topic is revealed, the degree of development of the problem is reflected, based on a deep assessment and analysis of scientific and methodological literature, in the work on the subject of AHD and financial analysis, the object of analysis and its issues are comprehensively considered, both from the theoretical and practical sides, the goal and specific tasks of the topic under consideration are formulated, there is a logic of presentation of the material and its sequence.

Sections: Maths

Dear participant of the Olympiad!

School Mathematics Olympiad is held in one round.
There are 5 tasks of different difficulty levels.
There are no special requirements for the design of the work. The form of presentation of the solution of problems, as well as the methods of solution, can be any. If you have any individual thoughts about a particular task, but you cannot bring the solution to the end, do not hesitate to state all your thoughts. Even partially solved problems will be evaluated by the corresponding number of points.
Start solving tasks that seem easier to you, and then move on to the rest. This way you save time.

We wish you success!

school stage All-Russian Olympiad schoolchildren in mathematics

Grade 5

Exercise 1. In the expression 1*2*3*4*5, replace the "*" with action signs and place the brackets like this. To get an expression whose value is 100.

Task 2. It is required to decipher the record of arithmetic equality, in which the numbers are replaced by letters, and different numbers are replaced by different letters, the same ones are the same.

FIVE - THREE \u003d TWO It is known that instead of the letter BUT you need to put in the number 2.

Task 3. How to divide 80 kg of nails into two parts - 15 kg and 65 kg using pan scales without weights?

Task 4. Cut the figure shown in the figure into two equal parts so that each part has one star. You can only cut along grid lines.

Task 5. A cup and saucer together cost 25 rubles, while 4 cups and 3 saucers cost 88 rubles. Find the price of the cup and the price of the saucer.

6th grade.

Exercise 1. Compare fractions without bringing them to a common denominator.

Task 2. It is required to decipher the record of arithmetic equality, in which the numbers are replaced by letters, and different numbers are replaced by different letters, the same ones are the same. It is assumed that the original equality is true and written according to the usual rules of arithmetic.

WORK
+ WILL
LUCK

Task 3. Three friends came to the summer camp to rest: Misha, Volodya and Petya. It is known that each of them has one of the following surnames: Ivanov, Semenov, Gerasimov. Misha is not Gerasimov. Volodya's father is an engineer. Volodya is in 6th grade. Gerasimov is in 5th grade. Ivanov's father is a teacher. What is the last name of each of the three friends?

Task 4. Divide the figure along the grid lines into four identical parts so that each part has one point.

Task 5. The jumping dragonfly slept for half the time of every day of the red summer, danced for a third of the time of every day, and sang for the sixth part. The rest of the time she decided to devote to preparing for the winter. How many hours a day did the Dragonfly prepare for winter?

7th grade.

Exercise 1. Solve the rebus if you know that the largest digit in the number STRONG is 5:

DECIDE
IF
STRONG

Task 2. Solve the equation│7 - x│ = 9.3

Task 3. After seven washes, the length, width and thickness of the soap had halved. How many of the same washes will last the remaining soap?

Task 4 . Divide the rectangle of 4 × 9 cells along the sides of the cells into two equal parts so that you can then make a square out of them.

Task 5. A wooden cube was painted with white paint on all sides, and then sawn into 64 identical cubes. How many cubes turned out to be colored on three sides? From two sides?
On the one side? How many cubes are not colored?

8th grade.

Exercise 1. What two digits end the number 13!

Task 2. Reduce the fraction:

Task 3. The school drama circle, preparing for the production of an excerpt from the fairy tale by A.S. Pushkin about Tsar Saltan, decided to distribute the roles between the participants.
- I will be Chernomor, - Yura said.
- No, I will be Chernomor, - said Kolya.
- All right, - Yura conceded to him, - I can play Gvidon.
- Well, I can become Saltan, - Kolya also showed compliance.
- I agree to be only Guidon! Misha said.
The wishes of the boys were satisfied. How were the roles distributed?

Task 4. IN isosceles triangle ABC with base AB = 8m median AD is drawn. The perimeter of triangle ACD is greater than the perimeter of triangle ABD by 2m. Find AS.

Task 5. Nikolai bought a common notebook of 96 sheets and numbered the pages from 1 to 192. His nephew Arthur tore out 35 sheets from this notebook and added up all 70 numbers that were written on them. Could he get 2010.

Grade 9

Exercise 1. Find the last digit of 1989 1989 .

Task 2. The sum of the roots of some quadratic equation is 1 and the sum of their squares is 2. What is the sum of their cubes?

Task 3. Using three medians m a , m b and m c ∆ ABC find the length of the side AC = b.

Task 4. Reduce the fraction .

Task 5. In how many ways can you choose a vowel and a consonant in the word "kamzol"?

Grade 10.

Exercise 1. Currently there are coins of 1, 2, 5, 10 rubles. Indicate all the amounts of money that can be paid with both an even and an odd number of coins.

Task 2. Prove that 5 + 5 2 + 5 3 + … + 5 2010 is divisible by 6.

Task 3. In a quadrilateral ABCD diagonals intersect at a point M. It is known that AM = 1,
VM = 2, CM = 4. At what values DM quadrilateral ABCD is a trapezoid?

Task 4. Solve System of Equations

Task 5. Thirty schoolchildren - tenth graders and eleventh graders - shook hands. At the same time, it turned out that each tenth grader shook hands with eight eleventh graders, and each eleventh grader shook hands with seven tenth graders. How many tenth graders and how many eleventh graders?