Unified State Examination Russian language task 19 training exercises. Unified State Exam in Mathematics (profile)
The most difficult task on punctuation in the Unified State Examination in the Russian language requires you to be very careful. We've sorted out the possible options for you. syntactic constructions, showed how to reason. Mastering a skill is a matter of practice.
Task formulation:
Place punctuation marks: indicate all the numbers in their place
There must be commas in the sentence.
In this task you will encounter complex sentences consisting of three or more simple ones, connected by coordinating and subordinating connections. We talked about coordinating connections and coordinating conjunctions in task 15, about subordinating connection between sentences - in task 18.
Reason in the same way as in task 18:
We read the sentence, making semantic pauses;
We divide difficult sentence into simple ones (each simple sentence has a grammatical basis and expresses a thought);
Let's look at how the sentences are connected (the place of the subordinating conjunction is at the beginning of the subordinate clause).
Let's dwell on the difficulties that may arise.
1. Pay attention to this diagram (union...), , (union...).
The sentence begins with a subordinating conjunction, then it will not be at the junction, at the beginning of the next sentence (the main one). Most often in such constructions there are conjunctions if, when, so that, as soon as, since and etc.
If look at the clouds for a long time, you can see What they look like white animal figures. As soon as the rain stopped, a light fog hung over the village, as if the roofs of the houses began to smoke slightly.
2. With another sequence of subordination, two conjunctions may be nearby, but at the same time relate to different offers. Let's consider the option if there are subordinating conjunctions at the junction: , (what if…), …).
It seemed to me, What, If We won't train every day, we won't have a chance to win.(Main sentence: it seemed to me. First subordinate clause: that we will have no chance of winning. Second clause: if we don't train daily.) Commas are used at sentence boundaries. If you “straighten” the sentence, you get a more understandable construction: It seemed to me that we would have no chance of winning if we did not train every day.
Signs are placed differently in the case when the union If a continuation appears in the form of the words TO, SO, BUT. See how the scheme changes:
, (What(if...), then...).
Therefore, if you see a junction of conjunctions, read the sentence further and check if there is a “tail” THAT(less often SO, BUT). THAT as if replacing a comma at the junction between conjunctions.
The old man sat so quietly what if would not be a slight cough, That one could not even guess about his presence. Anton Prokofievich, by the way, had some trousers of such a strange quality, what when he put them on That The dogs always bit him on the calves.
3. At the junction of conjunctions there may be a coordinating and subordinating conjunction: AND WHEN; AND IF; AND ALTHOUGH etc. If AND connects sentences, then the signs are placed according to the rules discussed in paragraph 2. On the rifts the raft was thrown towards the shores, and to it didn’t break on the sharp rocks, we leaned on the oars.(Commas are used at all sentence boundaries: on the rifts the raft was thrown towards the shores; and we leaned on the oars; so that it does not break on sharp stones.) The patient needs rest and if we don't want to disturb him, That must leave the room.(There is no comma at the junction of conjunctions, because there is a “tail” THAT: the patient needs rest; and must leave the room; if we don't want to disturb him... then.)
And if the union AND connects homogeneous members of a sentence, then a comma is not placed before it . Mumu did not go to the manor house and, when Gerasim carried firewood into the rooms, she remained at the porch.(Main sentence: Mumu did not go to the manor house and stayed at the porch; subordinate clause: when Gerasim carried firewood into the rooms.)
4. Subordinate clauses can be homogeneous and connected by a conjunction AND. In such cases, a comma is not placed between them (just as a comma is not placed between homogeneous members of a sentence connected by the conjunction I). I didn't have time to tell you What already done And What I'm still going to do it. Sentence pattern: , (that...) and (that...)
Let's complete the task:
The regiment stretched out like a long snake (1) and (2) when the rays of the sun hit the bayonets and rifle barrels (3) you could see (4) how the weapons glittered.
We divide sentences into simple ones, focusing on intonation, the semantic independence of each sentence, and conjunctions: [ long snake the regiment spread out], And [it was obvious] – conjunction And connected two sentences;
And , (When the rays of the sun fell on bayonets and rifle barrels) – comma between AND - WHEN is placed because after the sentence No THAT ; (When the rays of the sun fell on bayonets and rifle barrels),[...it was obvious], (How weapons gleamed). Answer: commas 1, 2, 3, 4
Training exercises for completing task No. 19 of the Unified State Exam in the Russian language
Block 1.
Place punctuation marks: indicate all the numbers that should be replaced by commas in the sentence.
Everyone is so used to them (the clock) (1) that (2) if they disappeared (3) somehow miraculously from the wall (4) it would be sad, as if one’s own voice had died and nothing could fill the empty space. (Bulgakov)
2. After the third bell sounded (1) the curtain trembled and slowly moved up (2) and (3) as soon as the audience saw their favorite (4) the walls of the theater literally trembled with applause and enthusiastic screams.
3. The first (1) thing we saw near the house (2) was a slender obelisk of black marble (3) and (4) when I read the inscription on the other side of the base (5) it became clear (6) that the obelisk was erected on the centenary of birth of Lermontov.
4. A huge cloud was approaching (1) behind which was a veil of rain (2) and (3) when the whole sky was covered with a dense curtain (4) large drops began to pound on the ground.
I am simply not ready to (1) say goodbye to my passion for painting (2) and (3) if I am destined to one day become a real artist (4) I will certainly become one.
I move forward with faith (1) that I will achieve the desired goal (2) and that (3) if God wants (4) I will be justified in the eyes of those (5) whom I love.
7. As soon as the sun rose (1) it became clear (2) that (3) if you go further (4) you could get stuck in a swamp (5) and the lieutenant gave the order to stop.
At first I thought (1) that I wouldn’t understand anything in the chess textbook (2) but (3) when I started reading (4) I saw (5) that it was written very simply and clearly.
9. Hadji Murat was sitting next to him in the room (1) and (2) although he did not understand the conversation (3) he felt (4) that they were arguing about him.
10. He wanted to assure himself (1) that there was no danger (2) and that the horsemen on the road simply seemed to the boy out of fear (3) and (4) although he managed to deceive the child’s mind for short minutes (5) but deep down he I clearly felt the approach of an inevitable tragedy.
On the outskirts of the city there was a wonderful park with shady alleys and gazebos for relaxation (1) and (2) although getting to it was not very convenient (3) the townspeople loved this place (4) and often spent holidays here.
The regiment stretched out like a long snake (1) and (2) when the rays of the sun hit the bayonets and rifle barrels (3) you could see (4) how the weapons glittered.
13. I didn’t know (1) how long I wandered through the forests (2) and (3) when I returned to the forester’s house (4) it turned out (5) that they had been waiting for me there for a long time.
The swans flew up screaming, made several farewell circles over the lake (1) where they spent the summer (2) and (3) when the white-winged flock disappeared into the foggy distance (4) the old huntsman and I (5) looked at the sky in silence for a long time.
15. Leonid Andreev took thousands of photographs of his relatives and friends at that time (1) and (2) when we came to visit him (3) he forced us (4) to look at all these thousands of photographs (5) because he wanted to surprise everyone with your passion.
16. A few days later (1) when the resentment began to fade (2) and (3) Andrei’s act no longer seemed so bad (4) as Vovka thought at first (5) the friends decided to meet and talk.
I remembered (1) that it was necessary to change the guard in the garden (2) and (3) as soon as Semyonov was free (4) I put him at his post.
We rinsed our clothes (1) and (2) while they were drying (3) on the hot sand (4) we swam.
Volodka knew (1) that he couldn’t lie (2) and (3) that from the expression on his face Yulka would immediately guess (4) what happened on Domnikovka.
It was necessary to rest (1) but Ivan felt (2) that (3) if he sat down (4) he would probably never get up again.
The German stood in the shadows (1) and (2) when (3) Sashka, passing forward, touched his shoulder (4) he felt (5) the German tremble.
It was a blue evening, but (1) when (2) the fire flared up (3) twilight thickened around the fire (4) and it began to seem (5) that it was already real night.
We argued with my brother about the books we read (1) and (2) if mother (3) sometimes tried to get a word in (4) we politely fell silent.
Vasya went with a lantern to the locomotive (1) because (2) the car was having difficulty (3) and he wanted to stay near it (4) as if by doing so he could share its fate.
There was nothing special in the rubber mask and the corrugated tube, but (1) as soon as (2) the major took out the box (3) it became clear (4) that the secret was in it.
A warm breeze slightly rustled the leaves of the trees (1) and (2) if (3) it weren’t for the clanking of shovels and the alarming horns of cars on the highway (4) then it wouldn’t have looked like war.
Block 2.
Place punctuation marks: indicate all the numbers (numbers) that should be replaced by commas in the sentence.
1. The sun had already risen (1) when the travelers looked around at the top of the hill (2) and (3) although there was not a single cloud (4) the sky was a strange whitish color (5) and closer to the horizon it became leaden gray.
2. At first no one could understand (1) how the boat went against the current without a sail and a motor (2) but (3) when the people went down to the river (4) everyone saw a team of dogs pulling the boat.
3. Belikov wore dark glasses, a sweatshirt, stuffed his ears with cotton wool (1) and (2) when he sat on a cab (3) ordered the top to be raised (4) so that no one could invade his cramped little world.
4. Some new ideas (1) and (2) came to my mind if you come (3) then I will be happy to tell you about (4) what worries me.
5. Romashov walked slowly along the highway (1) and (2) while he looked at the magical fire of the sunset (3) it seemed to him (4) as if there was some kind of mysterious life behind the bright dawn.
6. The territorial structure of the population and economy of foreign Europe developed in the 19th century (1) when perhaps the main factor of location (2) was natural resources (3) and (4) when the coal and metallurgical regions of Great Britain, France, Germany, Belgium arose , Poland, Czech Republic, and other countries.
7. I didn’t know (1) what Gregory was thinking about now (2) but I wanted (3) for (4) him to experience the same feelings (5) as me.
8. In any role, a talented actor feels free and natural (1) and (2) when he expresses the character of his hero on stage (3) and experiences his fate (4), he usually reaches a complete feeling (5) that he is the same hero.
9. The mother’s face, after finding out all the circumstances of the children’s willful behavior, became stern, even somehow haggard (1) and was followed by a stern and skillful reprimand (2) which (3) despite the fact that the children fully admitted their guilt (4) they still I had to listen.
10. In such weather (1) when nature seemed meek and thoughtful (2) Ivan Ivanovich and Burkin (3) were imbued with love for this field (4) and both thought about (5) how great (6) and how beautiful this a country.
11. A small incident happened to Matvey (1) which he remembered all his life (2) and (3) although he could not consider himself guilty (4) his conscience was uneasy.
12. After the performance of the young soloist, the audience felt (1) that (2) even if the performer did not manage to fully realize the director’s plan on stage (3) they were still present at the birth of a great talent (4) and the entire hall of thousands literally exploded applause.
13. Soul A.P. Chekhova always suffered from boredom and idleness in life (1) and (2) when enormous fame came to the writer (3) when it came devoted love to him everything (4) that was smart and honest in Russian society (5) he did not withdraw into the unattainability of cold greatness.
14. Korolev explained to them (1) that they would serve in the airfield service battalion (2) and (3) that (4) if their battalion did not exist (5) the planes would not be able to fly and fight.
15. For hundreds of years there (1) where the great pine tree stood (2) everything was unchanged (3) but (4) when it fell (5) a lot changed.
Block 3.
Task 19
1 option
At sunset it began to rain (1) which immediately dispelled the stuffiness that had accumulated in the air (2) and (3) while it made a full and monotonous noise in the garden around the house (4) the sweet freshness of wet greenery came through the open windows in the hall.
When Ivan Aristarkhovich appeared at the door of the dressing room (1) he habitually leaned over (2) and (3) all the actors got the impression (4) that their artistic director was very tall(5) although in fact the doorway was simply quite low.
It is well known (1) that (2) if an athlete does not train regularly (3) then (4) no matter how hard he tries (5) good results he can't achieve it.
The prince was not expected at the estate (1) since no one knew (2) whether he would come (3) and (4) so his appearance came as a surprise to everyone.
On the stone terrace of one of the most beautiful buildings in the city (1) there were two (2) and (3) while the shadows steadily lengthened (4) they watched (5) as the dazzling sun lit up in the windows of the upper floors.
It seemed to me (1) that no one could disturb (2) the peace that surrounded me (3) and the more unexpected was the sudden appearance of Alexei and his friends.
The birds could not be heard (1) because they do not sing during the hot hours (2) and there was silence in the frozen forest (3).
When Ivan returned home in the evening (1) all the impressions of the day washed over him (2) and (3) since he was overcome by the most contradictory feelings (4) he began to look for the reasons for his emotional excitement.
Ganin went ashore (1) and (2) when he saw the blue Turk on a huge pile of oranges at the pier (3) he felt piercingly and clearly (4) how far away the warm bulk of his homeland was from him.
Task 19
Option 2
Indicate all the numbers that should be replaced by commas in the sentence
1. This long row seemed especially difficult to Levin (1) but then (2) when the row was reached to the end (3) and Titus began to follow the tracks with slow steps (4) Levin followed his swath in the same way.
2. A few hours later (1) Ivan became exhausted (2) and (3) when he realized (4) that he couldn’t cope with the papers (5) he cried quietly and bitterly.
3. When the artist lived in Crimea (1) he devoted all his time to contemplating pictures of nature (2) and (3) if the weather was favorable for walks (4) he spent hours studying on the seashore the pattern of waves endlessly running one after another.
The snow covered the tracks of the travelers (1) and it became clear (2) that (3) if the snowfall did not stop by night (4) then it would be difficult to find the way back.
I thought about the people (1) whose lives (2) were connected with this story (3) and I wanted to know (4) what happened to them.
Elena was so daydreaming (1) that (2) when she heard the doorbell (3) she did not immediately understand (4) what was happening.
Everyone loved me (1) and (2) although I was incredibly naughty (3) I was forgiven for everything (4) no matter what I did.
They say (1) that kindness cures loneliness (2) and (3) when I settled in the village (4) I had the opportunity to verify this.
When it was necessary to rush to the gymnasium (1) Nikolenka tried his best to keep up with his older brother (2) and (3) since he always moved quickly (4) the first-grader often had to catch up with him by skipping.
Task 19
Option 3
Indicate all the numbers that should be replaced by commas in the sentence
Lucy was gently persistent (1) and (2) although it was difficult to remember everything (3) gradually the old woman told (4) how it was.
Those greeting him constantly looked at their watches (1) and (2) when a train appeared in the distance (3) the crowd moved towards him (4) although this could not speed up the meeting with loved ones.
According to the calendar, we arrived in Boldino at the same time as the poet (1) and (2) if we take into account the difference between the new and old style (3) then ten days earlier (4) when the color green still reigned everywhere in nature.
There is an opinion (1) that the weather affects a person’s well-being (2) and (3) I have been convinced of this more than once.
A belated lightning flashed directly overhead (1) and (2) while it was shining (3) I saw (4) some kind of flickering white dot on the shore.
The rest of the day dragged on unbearably long for Zakhar (1) and (2) when the sun set (3) and the gray shadows began to cover the ground thicker (4) he felt relief.
After all the guests had left (1) the hostess wanted to be alone (2) and (3) when Anton asked permission to spend the evening with the neighbors (4) she did not stop her son.
Pyotr Ivanovich always tried to avoid conversations at the table (1) and (2) when he was invited to have a meal (3) he simply sat down (4) and ate in silence.
I don’t remember (1) how I got to the place (2) but (3) when I woke up (4) my friends were already standing next to me.
Task 19
Option 4
Indicate all the numbers that should be replaced by commas in the sentence
In any role, a talented actor feels free and natural (1) and (2) when he expresses the character of his hero on stage (3) he usually reaches the full feeling (4) that he is that same hero.
The sister tried to tell Kitty (1) what the doctor was talking about (2) but (3) although he spoke for a very long time and very smoothly (4) she was unable to convey the meaning of what he said.
It’s always difficult to start doing a job you don’t like (1) and (2) in order to delay the unpleasant moment at least a little (3) we often look for any excuses (4) that can somehow justify our lack of will.
After the third bell sounded (1) the curtain trembled and slowly moved up (2) and (3) as soon as the audience saw their favorite (4) the walls of the theater literally trembled with applause and enthusiastic screams.
All the guests left (1) the hostess wanted to be alone (2) and (3) when Anton asked permission to spend the evening with the neighbors (4) she did not stop her son.
Dawn is far away (1) and the transparent silence of the night floats over the sleeping forest (2) and (3) when you get used to it (4) every rustle and whisper begins to be clearly heard.
The entrance door suddenly swung open (1) and an unkempt-looking, strong young man jumped out into the street (2) who (3) if Alexey had not managed to step aside at the last moment (4) would probably have run straight into him.
It was already turning blue over the Volga summer night(1) and (2) when we found ourselves on the shore (3) we saw (4) the lights on the masts of passing ships flickering in the distance.
Tatyana Afanasyevna gave her brother a sign (1) that the patient wanted to sleep (2) and (3) when everyone slowly left the room (4) she sat down at the spinning wheel again.
Task 19
Option 5
Indicate all the numbers that should be replaced by commas in the sentence
His hand trembled (1) and (2) when Nikolai handed the horse to the horse breeder (3) he felt (4) the blood rushing to his heart.
Snow covered the tanks (1) and (2) when the tankers climbed out of the tower to breathe (3) it instantly covered their hot faces (4) as if trying to cool them down.
And the old woman kept talking and talking about her happiness (1) and (2) although her words were familiar (3) their grandson’s heart suddenly ached sweetly (4) as if everything he heard was happening to him.
Startsev avoided conversations (1) and (2) when he was invited to eat (3) he sat down (4) and ate in silence.
Elena did not have time to leave the stage with the other actors (1) and (2) when the curtain opened (3) the noisy wave of the hall (4) covered her.
The smell of fog is stronger (1) and (2) when we step into the meadow (3) the smell of mown, still damp grass is overwhelming (4) although signs of its first withering are already visible.
Lisa walked into the deserted square (1) and (2) when her legs began to fall heavily from the cobblestones (3) she remembered (4) how she returned to this square on a sunny day after her first meeting with Tsvetukhin.
Katya listened to the story about the latest achievements in the field of nuclear physics very carefully (1) and (2) if Konstantinov had not realized (3) that the scope of his scientific interests could not truly excite such a young person (4) he would have continued his reasoning.
Now I will have to leave for a while (1) but (2) when I return to Moscow again (3) I will be sincerely glad to see you (4) if you deign to agree to a meeting.
Task 19
Option 6
Indicate all the numbers that should be replaced by commas in the sentence
1. Alexey was alone in the trench (1) and (2) when the carts disappeared (3) and (4) the field was cleared of dust (5) he decided to look around.
Katya was preparing very seriously for the first exam in her life (1) and (2) when she found herself in the classroom in front of the sitting teachers (3) she felt joyful (4) because there was an opportunity to show off her accumulated knowledge.
IN parental home everything was as before (1) and (2) if Volodya seemed to have his home space as if it had narrowed (3) it was only because (4) that during the years of absence he had matured and grown a lot.
At night, timber was brought to the river (1) and (2) when a white fog shrouded the banks (3) all eight companies laid planks (4) on the wreckage of the bridges.
Such fatigue set in (1) that (2) even if there had not been an order (3) to rest (4) people would not have been able to take a single step further.
The hostess realized (1) that (2) if now the guests find themselves in the hall again (3) they will no longer see the distant alley in the rays of the setting sun (4) and she suggested taking a walk in the garden.
The mosquitoes (1) and (2) sang an endless song as dusk deepened (3) and all other sounds fell silent (4) the sound of a distant waterfall began to reach me.
After the instructor’s comments (1) the guys walked faster (2) and (3) when it began to get dark (4) there were only three kilometers left to the place where they would spend the night.
He continued his journey (1) but (2) when only twelve miles remained (3) suddenly the tire whistled and sank (4) because a sharp pebble again fell under the wheel.
Answers
| 1 option | Option 2 | Option 3 | Option 4 | Option 5 | Option 6 |
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This task consists of a sentence and punctuation options. You must select all the correct punctuation options.
Algorithm for completing the task:
- Highlight the semantic parts in the sentence and determine their syntactic role.
- Determine how the parts of the sentence are connected, separate them with appropriate punctuation marks.
- Analyze how each part is complicated, check the punctuation marks for them.
- Compare the result with the punctuation options.
- Write down the correct sequence of numbers.
Garik had a very important matter (1) but (2) if we take into account his frivolous appearance(3) it seemed (4) that he was by no means preparing for a serious event.
Let's go through the commas:
1) A comma separates the sentence “Garik had a very important matter” and the sentence “it seemed”, connected by a coordinating connection..
2) There is no comma, since the conjunction “If” has a correlative word “Then”.
3) A comma highlights the subordinate clause “if we accept ... appearance.”
4) A comma highlights the subordinate clause “that he had prepared... for... the event.”
Answer: 1,3,4.
Test options for task 19 from the Unified State Examination in Russian:
Try to solve them yourself and compare with the answers at the end of the page
Example 1:
Place punctuation marks: indicate all the numbers that should be replaced by commas in the sentence.
Let such heroes grow up in Rus' at all times (1) so that (2) when the time comes (3) no one will ever be able to overcome Russia (4) and will not even be able to think about it.
Example 2:
Place punctuation marks: indicate all the numbers that should be replaced by commas in the sentence.
Olga walked into the deserted square (1) and (2) when her heels began to fall heavily from the round cobblestones of the pavement (3) she remembered (4) how she had once returned home along this road.
Example 3:
Place punctuation marks: indicate all the numbers that should be replaced by commas in the sentence.
Tatyana Afanasyevna gave her brother a sign (1) that the patient wanted to sleep (2) and (3) when everyone slowly left the room (4) she sat down at the spinning wheel again.
Example 4:
Place punctuation marks: indicate all the numbers that should be replaced by commas in the sentence.
I calmed down a little (1) and (2) when my mother left for work (3) I took up my usual chores (4) although the mood was not at all joyful.
Example 5:
Place punctuation marks: indicate all the numbers that should be replaced by commas in the sentence.
All the guests left (1) the hostess wanted to be alone (2) and (3) when Anton asked permission to spend the evening with the neighbors (4) she did not stop her son.
Example 6:
Place punctuation marks: indicate all the numbers that should be replaced by commas in the sentence.
Now I will have to leave for a while (1) but (2) when I return to Moscow again (3) I will be sincerely glad to see you (4) if you deign to agree to a meeting.
Example 7:
Place punctuation marks: indicate all the numbers that should be replaced by commas in the sentence.
So much has been written about Maxim Gorky (1) that (2) if he had not been an inexhaustible person (3) it would have been impossible to add a single line to what (4) has already been written about him.
Example 8:
Example 9:
Place punctuation marks: indicate all the numbers that should be replaced by commas in the sentence.
I knew (1) that it had rained at night (2) and (3) that (4) if I now touch the lilac branches (5) dew will fall from the bushes.
Example 10:
Place punctuation marks: indicate all the numbers that should be replaced by commas in the sentence.
Some new ideas came to my mind (1) and (2) if you come (3) I will be happy to tell you about (4) what worries me now.
Example 11:
Place punctuation marks: indicate all the numbers that should be replaced by commas in the sentence.
If Irina got comfortable in Ferapontovo and managed to fall in love with it (1), then Victor came here for the first time (2) and (3) although from the stories he knew a lot (4) he was amazed at everything (5) he saw.
Answers:
1) 1,2,3
2) 1,2,3,4
3) 1,2,3,4
4) 2,3,4
5) 1,2,4
6) 1,3,4
7) 1,3,4
8) 1,4
9) 1,4,5
10) 1,2,3,4
11) 1,3,4,5
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?
The numerical sequence is given by the general term formula: a_(n) = 1/(n^2+n)
A) Find smallest value n , for which a_(n)< 1/2017.
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 terms in this sequence that form an 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?
The following operation is performed with a natural number: 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?
C) What is the largest multiple of 9 that can be obtained from a three-digit number, in decimal notation which has no nines?
There are 32 students in the group. Each of them writes either one or two test papers, 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.
< 14.
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?
B) Is it possible that there is no number 16 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 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?
In which greatest number The vertices of a cube can be colored in Blue colour so that among the blue vertices it is impossible to choose three that form an 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 identical numbers can be contained in the notation of 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?
Q) What is the smallest area that can be folded into 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?
End sequence a1,a2,...,a_(n) consists of n greater than or equal to 3 not necessarily different natural numbers, and for all natural k less than or equal to n-2 the equality a_(k+2) = 2a_(k+1) holds )-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 a company, in the first platoon there are fewer soldiers than in 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 in each row there will be 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) Represent 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 an 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?
Q) Find all possible values of 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 positive numbers maybe among them?
