Every year, many different Olympiads are held for schoolchildren of any schools in the Russian Federation, allowing students to show their knowledge and skills in subjects included in the list of programs of general educational institutions of the country. Participation in such events is considered a very prestigious and responsible task, in which schoolchildren demonstrate the knowledge accumulated over the years of study and defend their honor own school. If you win, you have the opportunity to earn some privilege for further admission to Russian universities and receive a small monetary reward.
Historical summary
For the first time, Russian educational authorities provided the opportunity for competition between young students back in 1886. In times of prosperity Soviet Union such a movement received additional impetus for further development. In the 60s of the last century, school Olympiads began to be held in almost every discipline related to the general education program of compulsory education. Initially, such competitions were more of an all-Russian scale, which in the future became all-Union.
To find out exactly what subjects such a competition will consist of in the future, all school Olympiads for 2017-2018 should be announced.
Present time
Next academic year, the best schoolchildren will be able to test their knowledge in competitions in several categories of disciplines.
1. Natural sciences: geography, physics, biology, chemistry, ecology and astronomy.
2. Humanities: history, social studies, economics and law.
3. Exact sciences: mathematics, computer science.
4. Philology: English, French, Chinese, Italian and Russian, as well as Russian literature.
5. Other disciplines: physical education, life safety, technology and world artistic culture.
In each of the listed disciplines, there are two blocks of tasks: a part aimed at finding practical skills and a part testing the theoretical basis of each participant.
The main stages of the Russian Olympiads
The All-Russian Olympiad consists of the organization and further conduct of 4 stages intellectual competition held at different levels. Representatives of regional educational institutions and schools determine final schedule each Olympiad and its location. Of course, the exact list of each competition for next year has not yet been compiled, but current applicants for participation should be guided by the following dates.
1. School stage. The competition between rivals from the same educational application starts almost at the beginning academic year– September-October 2017. The Olympiads concern students of the same parallel, starting from the fifth grade. Members of the city-level methodological commission are responsible for developing tasks.
2. Municipal stage. The next stage at which competitions are held between the winners of the previous level of grades 7-11 from the same city. The duration of the Olympiad is December 2017-January 2018. The organizers of such an event are representatives of the educational sphere at the regional level, while officials are responsible for the place, time and procedure of the competition itself.
3. Regional stage. The next level of the All-Russian Olympiad, held in January-February. It is attended by schoolchildren who took leading places in similar competitions at the city level, as well as the winners of the regional selection of the past year.
4. All-Russian stage. The highest level of the subject Olympiad is organized by representatives of the Ministry of Education of the Russian Federation in March-April 2018. The winners of the regional Olympiad and the winners of the past year will be able to take part in it. The exception is schoolchildren who took 1st place, but are behind participants from other cities. The winners of the marked stage receive the right to participate in a similar competition international level, scheduled for next summer.
List of school Olympiads with their main features
Any school Olympiad consists of 3 main stages, each of which is characterized by distinctive properties. For example, the winners have a number of privileges over their opponents from the other two groups - the opportunity to enroll in the university on the basis of which the Olympiad itself was held. In this case, entrance exams for enrollment in the first year are canceled automatically. The winners or prize-winners of the 3rd stage in this sense do not have any concessions.
Today it is already known that the list of school Olympiads of the 1st level consists of the following areas and disciplines.
1. Lomonosov Olympiad, consisting of a huge number of different items.
2. “Nanotechnologies - a breakthrough into the future” - an all-Russian Olympiad for every interested student.
3. All-Siberian Chemistry Olympiad.
4. “Young talents” – geography.
5. Open Olympiad in programming.
6. Astronomy Olympiad for schoolchildren from St. Petersburg.
7. Open Olympiad “culture and art”.
8. All-Russian Economic Olympiad for schoolchildren named after N. D. Kondratiev in economics.
9. Moscow Olympiad in physics, mathematics, computer science.
The list of Level II Olympiads consists of the following areas.
1. Herzen Olympiad in foreign languages.
2. South Russian Olympiad for schoolchildren “Architecture and Art” with the following items: painting, drawing, composition and drawing.
3. Interregional Olympiad of MPGU in Law.
4. All-Siberian Open Olympiad in Informatics, Mathematics, Biology.
5. Interregional Olympiad " Highest standard"in computer science, literature, history of world civilization and oriental studies.
6. Interregional Olympiad “Future Researchers – Future of Science” in Biology.
7. City Olympics open type in physics.
8. Interdisciplinary Olympiad named after V.I. Vernadsky in social studies and history.
9. Engineering Olympiad in physics.
10. Eurasian linguistic Olympiad in a foreign language at the interregional level.
The 2017-2018 Level III Olympics are represented by the following list of competitions.
1. “Mission accomplished. Your calling is a financier!” from economics.
2. Herzen Olympiad in geography, biology and pedagogy.
3. “In the beginning was the Word...” in history and literature.
4. All-Russian tournament of young physicists.
5. All-Russian Sechenov Olympiad in chemistry and biology.
6. All-Russian chemical tournament.
7. “Learn to build the future” from urban planning and architectural graphics.
8. All-Russian Tolstoy Olympiad in history, literature and social studies.
9. All-Russian Olympiad of representatives of musical institutions of the Russian Federation on string instruments, music pedagogy, folk orchestra instruments, choral conducting and performance.
10. All-Russian competition of scientific works “Junior” in engineering and natural sciences.
The noted list of the most relevant Olympiads in Russia has been in effect for the past few years. True, having familiarized yourself with all the competitions, a completely logical question arises: what is the difference between the tasks of all levels? First of all, we are talking about the level of preparation of schoolchildren.
To become not only an ordinary representative of the Olympiad, but even to take a prize place, you should have enough high level preparation. On some Internet portals you can find Olympiad tasks from previous years to check your own level using ready-made answers, find out the approximate start time of the competition and some organizational issues.
All-Russian Olympiads for schoolchildren are held under the auspices of the Russian Ministry of Education and Science after official confirmation of the calendar of their dates. Such events cover almost all disciplines and subjects included in the compulsory curriculum of secondary schools.
By participating in such competitions, students are given the opportunity to gain experience in answering questions in intellectual competitions, as well as expand and demonstrate their knowledge. Schoolchildren begin to calmly respond to various forms of knowledge testing, and are responsible for representing and defending the level of their school or region, which develops a sense of duty and discipline. In addition, a good result can bring a well-deserved cash bonus or advantages during admission to the country's leading universities.
Olympiads for schoolchildren in the 2017-2018 academic year are held in 4 stages, divided by territorial aspect. These stages in all cities and regions are carried out within the general calendar periods established by the regional leadership of educational municipal departments.
Schoolchildren taking part in the competition gradually go through four levels of competition:
- Level 1 (school). In September-October 2017, competitions will be held within each individual school. All parallels of students are tested independently from each other, starting from the 5th grade and ending with graduates. Assignments for this level are prepared by methodological commissions at the city level, and they also provide assignments for district and rural secondary schools.
- Level 2 (regional). In December 2017 - January 2018, the next level will be held, in which the winners of the city and district - students in grades 7-11 - will take part. Tests and tasks at this stage are developed by the organizers of the regional (third) stage, and all questions regarding preparation and locations for conducting are assigned to local authorities.
- Level 3 (regional). Duration: from January to February 2018. Participants are the winners of the Olympiads of the current and completed year of study.
- Level 4 (All-Russian). Organized by the Ministry of Education and runs from March to April 2018. The winners of regional stages and the winners of last year participate in it. However, not all winners of the current year can take part in the All-Russian Olympiads. The exception is children who took 1st place in the region, but are significantly behind the other winners in points.
Winners All-Russian level If desired, they can take part in international competitions taking place during the summer holidays.
List of disciplines
In the 2017-2018 academic season, Russian schoolchildren can test their strength in the following areas:
- exact sciences – analytical and physical and mathematical direction;
- natural sciences - biology, ecology, geography, chemistry, etc.;
- philological sector – various foreign languages, native language and literature;
- humanitarian direction - economics, law, historical sciences, etc.;
- other subjects - art and, BJD.
This year, the Ministry of Education officially announced the holding of 97 Olympiads, which will be held in all regions of Russia from 2017 to 2018 (9 more than last year).
Benefits for winners and runners-up
Each Olympiad has its own level: I, II or III. Level I is the most difficult, but it gives its graduates and prize-winners the most advantages when entering many prestigious universities in the country.
Benefits for winners and runners-up come in two categories:
- admission without exams to the chosen university;
- awarding the highest Unified State Exam score in the discipline in which the student received a prize.
The most famous level I state competitions include the following Olympiads:
- St. Petersburg Astronomical Institute;
- "Lomonosov";
- St. Petersburg State Institute;
- "Young Talents";
- Moscow school;
- "Highest standard";
- "Information Technology";
- “Culture and art”, etc.
Level II Olympics 2017-2018:
- Hertsenovskaya;
- Moscow;
- "Eurasian linguistic";
- "Teacher of the school of the future";
- Lomonosov Tournament;
- "TechnoCup" etc.
TO competition III level 2017-2018 include the following:
- "Star";
- "Young Talents";
- Competition of scientific works "Junior";
- "Hope of Energy";
- "Step into the future";
- “Ocean of Knowledge”, etc.
According to the Order “On Amendments to the Procedure for Admission to Universities”, winners or prize-winners final stage have the right to admission without entrance examinations to any university in a field corresponding to the profile of the Olympiad. At the same time, the correlation between the direction of training and the profile of the Olympiad is determined by the university itself and without fail publishes this information on its official website.
The right to use the benefit is retained by the winner for 4 years, after which it is canceled and admission occurs on a general basis.
Preparation for the Olympics
The standard structure of Olympiad tasks is divided into 2 types:
- testing theoretical knowledge;
- the ability to translate theory into practice or demonstrate practical skills.
A decent level of preparation can be achieved using the official website of the Russian state Olympiads, which contains tasks from past rounds. They can be used both to test your knowledge and to identify problem areas in preparation. There, on the website you can check the dates of the rounds and get acquainted with the official results.
Video: assignments for the All-Russian Olympiad for schoolchildren appeared online
It has become a good tradition to hold the All-Russian school Olympiad. Its main task is to identify gifted children, motivate schoolchildren to study subjects in depth, develop creative abilities and innovative thinking in children.
The Olympic movement is becoming increasingly popular among schoolchildren. And there are reasons for this:
- winners of the All-Russian round are admitted to universities without competition if the core subject is an Olympiad subject (the winners’ diplomas are valid for 4 years);
- participants and winners receive additional chances upon admission to educational institutions (if the subject is not in the profile of the university, the winner receives an additional 100 points upon admission);
- significant monetary reward for prizes (60 thousand, 30 thousand rubles;
- and, of course, fame throughout the country.
Before becoming a winner you must go through all stages All-Russian Olympiad:
- The primary school stage at which they determine worthy representatives to the next level, to be carried out in September-October 2017. Organization and conduct school stage carried out by specialists from the methodological office.
- The municipal stage is held between schools in a city or district. It takes place at the end of December 2017. – early January 2018
- The third round is more difficult. Talented students from all over the region take part in it. The regional stage takes place in January-February 2018.
- The final stage determines the winners of the All-Russian Olympiad. In March-April, the best children in the country compete: the winners of the regional stage and the winners of last year’s Olympiad.
Organizers final round are representatives of the Ministry of Education and Science of Russia, they also sum up the results.
You can show your knowledge in any subject: mathematics, physics, geography, even physical education and technology. You can compete in erudition in several subjects at once. There are 24 disciplines in total.
Olympic subjects are divided into areas:
№ | Direction | Items |
1 | Exact disciplines | mathematics, computer science |
2 | Natural sciences | geography, biology, physics, chemistry, ecology, astronomy |
3 | Philological disciplines | literature, Russian language, foreign languages |
4 | Humanities | economics, social studies, history, law |
5 | Others | art, technology, physical culture, basics of life safety |
The peculiarity of the final stage of the Olympiad consists of two types of tasks: theoretical and practical. For example, to get good results in geography, students must complete 6 theoretical problems, 8 practical tasks, and also answer 30 test questions.
The first stage of the Olympiad begins in September, which means that those wishing to take part in the intellectual marathon should prepare in advance. But first of all, you must have good base school level, which constantly needs to be replenished with additional knowledge that goes beyond the school curriculum.
The official website of the Olympiad www.rosolymp.ru posts tasks from previous years. These materials can be used in preparation for the intellectual marathon. And of course, you can’t do without the help of teachers: additional classes after school, classes with tutors.
The winners of the final stage will take part in international olympiads. They form the Russian national team, which will prepare at training camps in 8 subjects.
To provide methodological assistance, orientation webinars are held on the site, the Central Organizing Committee of the Olympiad and subject-methodological commissions have been formed.
Tasks and keys for the school stage of the All-Russian Olympiad for schoolchildren in mathematics
Download:
Preview:
School stage
4th grade
1. Area of rectangle 91
Preview:
Objectives of the All-Russian Olympiad for Schoolchildren in Mathematics
School stage
5th grade
The maximum score for each task is 7 points
3. Cut the figure into three identical (matching when overlapping) figures:
4. Replace letter A
Preview:
Objectives of the All-Russian Olympiad for Schoolchildren in Mathematics
School stage
6th grade
The maximum score for each task is 7 points
Preview:
Objectives of the All-Russian Olympiad for Schoolchildren in Mathematics
School stage
7th grade
The maximum score for each task is 7 points
1. - various numbers.
4. Replace the letters Y, E, A and R with numbers so that you get the correct equality:
YYYY ─ EEE ─ AA + R = 2017 .
5. Something lives on the island number of people, including her
Preview:
Objectives of the All-Russian Olympiad for Schoolchildren in Mathematics
School stage
8th grade
The maximum score for each task is 7 points
AVM, CLD and ADK respectively. Find∠ MKL.
6. Prove that if a, b, c and - whole numbers, then fractionswill be an integer.
Preview:
Objectives of the All-Russian Olympiad for Schoolchildren in Mathematics
School stage
9th grade
The maximum score for each task is 7 points
2. Numbers a and b are such that the equations And also has a solution.
6. At what natural x expression
Preview:
Objectives of the All-Russian Olympiad for Schoolchildren in Mathematics
School stage
10th grade
The maximum score for each task is 7 points
4 – 5 – 7 – 11 – 19 = 22
3. In Eq.
5. In triangle ABC drew a bisector BL. It turned out that . Prove that the triangle ABL – isosceles.
6. By definition,
Preview:
Objectives of the All-Russian Olympiad for Schoolchildren in Mathematics
School stage
11th grade
The maximum score for each task is 7 points
1. The sum of two numbers is 1. Can their product be greater than 0.3?
2. Segments AM and BH ABC.
It is known that AH = 1 and . Find side length B.C.
3. and inequality true for all values X ?
Preview:
4th grade
1. Area of rectangle 91. The length of one of its sides is 13 cm. What is the sum of all sides of the rectangle?
Answer. 40
Solution. We find the length of the unknown side of the rectangle from the area and the known side: 91:13 cm = 7 cm.
The sum of all sides of the rectangle is 13 + 7 + 13 + 7 = 40 cm.
2. Cut the figure into three identical (matching when overlapping) figures:
Solution.
3. Recreate the example for addition, where the digits of the terms are replaced by asterisks: *** + *** = 1997.
Answer. 999 + 998 = 1997.
4 . Four girls were eating candy. Anya ate more than Yulia, Ira – more than Sveta, but less than Yulia. Arrange the girls' names in ascending order of the candies eaten.
Answer. Sveta, Ira, Julia, Anya.
Preview:
Keys for the school mathematics Olympiad
5th grade
1. Without changing the order of the numbers 1 2 3 4 5, place arithmetic signs and parentheses between them so that the result is one. You cannot “glue” adjacent numbers into one number.
Solution. For example, ((1 + 2) : 3 + 4) : 5 = 1. Other solutions are possible.
2. Geese and piglets were walking in the barnyard. The boy counted the number of heads, there were 30, and then he counted the number of legs, there were 84. How many geese and how many piglets were there in the schoolyard?
Answer. 12 piglets and 18 geese.
Solution.
1 step. Imagine that all the piglets raised two legs up.
Step 2. There are 30 ∙ 2 = 60 legs left standing on the ground.
Step 3. Raised up 84 - 60 = 24 legs.
Step 4 Raised 24: 2 = 12 piglets.
Step 5 30 - 12 = 18 geese.
3. Cut the figure into three identical (matching when overlapping) figures:
Solution.
4. Replace letter A by a non-zero number to obtain a true equality. It is enough to give one example.
Answer. A = 3.
Solution. It is easy to show that A = 3 is suitable, let us prove that there are no other solutions. Let's reduce the equality by A . We'll get it.
If A ,
if A > 3, then .
5. Girls and boys went into a store on their way to school. Each student bought 5 thin notebooks. In addition, each girl bought 5 pens and 2 pencils, and each boy bought 3 pencils and 4 pens. How many notebooks were purchased if the children bought 196 pens and pencils in total?
Answer. 140 notebooks.
Solution. Each of the students bought 7 pens and pencils. A total of 196 pens and pencils were purchased.
196: 7 = 28 students.
Each student bought 5 notebooks, which means they bought a total
28
⋅ 5=140 notebooks.
Preview:
Keys for the school mathematics Olympiad
6th grade
1. There are 30 points on a straight line, the distance between any two adjacent ones is 2 cm. What is the distance between the two extreme points?
Answer. 58 cm.
Solution. Between the extreme points there are 29 pieces of 2 cm each.
2 cm * 29 = 58 cm.
2. Will the sum of the numbers 1 + 2 + 3 + ......+ 2005 + 2006 + 2007 be divisible by 2007? Justify your answer.
Answer. Will.
Solution. Let's imagine this amount in the form of the following terms:
(1 + 2006) + (2 + 2005) + …..+ (1003 + 1004) + 2007.
Since each term is divisible by 2007, the entire sum will be divisible by 2007.
3. Cut the figure into 6 equal checkered figures.
Solution. This is the only way to cut a figurine
4. Nastya arranges the numbers 1, 3, 5, 7, 9 in the cells of a 3 by 3 square. She wants the sum of the numbers along all horizontals, verticals and diagonals to be divisible by 5. Give an example of such an arrangement, provided that Nastya is going to use each number no more than two times.
Solution. Below is one of the arrangements. There are other solutions.
5. Usually dad comes to pick up Pavlik after school by car. One day, classes ended earlier than usual and Pavlik walked home. 20 minutes later he met his dad, got into the car and arrived home 10 minutes early. How many minutes earlier did classes end that day?
Answer. 25 minutes earlier.
Solution. The car arrived home earlier because it didn’t have to drive from the meeting place to the school and back, which means the car covers twice this distance in 10 minutes, and one way in 5 minutes. So, the car met Pavlik 5 minutes before the usual end of classes. By this time, Pavlik had already been walking for 20 minutes. Thus, classes ended 25 minutes early.
Preview:
Keys for the school mathematics Olympiad
7th grade
1. Find the solution to a number puzzle a,bb + bb,ab = 60, where a and b - various numbers.
Answer. 4.55 + 55.45 = 60
2. After Natasha ate half of the peaches from the jar, the level of the compote dropped by one third. By what part (from the obtained level) will the compote level decrease if you eat half of the remaining peaches?
Answer. One quarter.
Solution. From the condition it is clear that half of the peaches occupy a third of the jar. This means that after Natasha ate half of the peaches, there were equal amounts of peaches and compote left in the jar (one third each). This means that half of the number of remaining peaches is a quarter of the total volume of contents
banks. If you eat this half of the remaining peaches, the compote level will drop by a quarter.
3. Cut the rectangle shown in the figure along the grid lines into five rectangles of varying sizes.
Solution. For example, like this
4. Replace the letters Y, E, A and R with numbers so that you get the correct equation: YYYY ─ EEE ─ AA + R = 2017.
Answer. With Y=2, E=1, A=9, R=5 we get 2222 ─ 111 ─ 99 + 5 = 2017.
5. Something lives on the island number of people, including e m each of them is either a knight who always tells the truth, or a liar who always lies e t. Once all the knights said: “I am only friends with one liar,” and all the liars: “I am not friends with knights.” Who is more on the island, knights or knaves?
Answer. There are more knights
Solution. Every liar is friends with at least one knight. But since each knight is friends with exactly one liar, two liars cannot have a common knight friend. Then each liar can be matched with his knight friend, which means that there are at least as many knights as there are liars. Since the total number of inhabitants on the island e number, then equality is impossible. This means there are more knights.
Preview:
Keys for the school mathematics Olympiad
8th grade
1. There are 4 people in the family. If Masha's scholarship is doubled, the total income of the entire family will increase by 5%, if instead mom's salary is doubled - by 15%, if dad's salary is doubled - by 25%. By what percentage will the income of the entire family increase if grandfather’s pension is doubled?
Answer. By 55%.
Solution . When Masha's scholarship doubles, the total family income increases exactly by the amount of this scholarship, so it is 5% of income. Likewise, mom and dad's salaries are 15% and 25%. This means that grandfather’s pension is 100 – 5 – 15 - 25 = 55%, and if e double, then family income will increase by 55%.
2. On sides AB, CD and AD of square ABCD equilateral triangles are constructed on the outside AVM, CLD and ADK respectively. Find∠ MKL.
Answer. 90°.
Solution. Consider a triangle MAK: Angle MAK equals 360° - 90° - 60° - 60° = 150°. MA = AK according to the condition, it means a triangle MAK isosceles,∠ AMK = ∠ AKM = (180° - 150°) : 2 = 15°.
Similarly we find that the angle DKL equal to 15°. Then the required angle MKL is equal to the sum of ∠ MKA + ∠ AKD + ∠ DKL = 15° + 60° + 15° = 90°.
3. Nif-Nif, Naf-Naf and Nuf-Nuf were sharing three pieces of truffle weighing 4 g, 7 g and 10 g. The wolf decided to help them. He can cut off any two pieces at the same time and eat 1 g of truffle each. Will the wolf be able to leave equal pieces of truffle for the piglets? If so, how?
Answer. Yes.
Solution. The wolf can first cut 1 g three times from pieces of 4 g and 10 g. You will get one piece of 1 g and two pieces of 7 g. Now it remains to cut six times and eat 1 g each from pieces of 7 g, then the piglets you will get 1 g of truffle.
4. How many four-digit numbers are there that are divisible by 19 and end in 19?
Answer. 5.
Solution. Let - such a number. Thenis also a multiple of 19. But
Since 100 and 19 are relatively prime, a two-digit number is divisible by 19. And there are only five of them: 19, 38, 57, 76 and 95.
It is easy to verify that all the numbers 1919, 3819, 5719, 7619 and 9519 are suitable for us.
5. A team of Petya, Vasya and a single-seater scooter is participating in the race. The distance is divided into sections same length, their number is 42, at the beginning of each there is a checkpoint. Petya runs the section in 9 minutes, Vasya – in 11 minutes, and on a scooter, either of them covers the section in 3 minutes. They start at the same time, and at the finish line the time of the one who came last is taken into account. The guys agreed that one would ride the first part of the journey on a scooter, run the rest, and the other would do the opposite (the scooter can be left at any checkpoint). How many sections must Petya cover on a scooter for the team to show the best time?
Answer. 18
Solution. If the time of one becomes less than the time of another of the guys, then the time of the other and, consequently, the time of the team will increase. This means that the guys’ time must coincide. Having indicated the number of sections Petya passes through x and solving the equation, we get x = 18.
6. Prove that if a, b, c and - whole numbers, then fractionswill be an integer.
Solution.
Let's consider , by convention this is an integer.
Then will also be an integer as the difference N and double the integer.
Preview:
Keys for the school mathematics Olympiad
9th grade
1. Sasha and Yura have now been together for 35 years. Sasha is now twice as old as Yura was then, when Sasha was as old as Yura is now. How old is Sasha now and how old is Yura?
Answer. Sasha is 20 years old, Yura is 15 years old.
Solution. Let Sasha now x years, then Yura , and when Sasha wasyears, then Yura, according to the condition,. But the time passed equally for both Sasha and Yura, so we get the equation
from which .
2. Numbers a and b are such that the equations And have solutions. Prove that the equationalso has a solution.
Solution. If the first equations have solutions, then their discriminants are non-negative, whence And . Multiplying these inequalities, we get or , from which it follows that the discriminant of the last equation is also non-negative and the equation has a solution.
3. The fisherman caught a large number of fish weighing 3.5 kg. and 4.5 kg. His backpack holds no more than 20 kg. Which Weight Limit can he take fish with him? Justify your answer.
Answer. 19.5 kg.
Solution. The backpack can hold 0, 1, 2, 3 or 4 fish weighing 4.5 kg.
(no more, because). For each of these options, the remaining backpack capacity is not divisible by 3.5, and in the best case it will be possible to pack kg. fish.
4. The shooter fired ten times at a standard target and scored 90 points.
How many hits were there on the seven, eight and nine, if there were four tens, and there were no other hits or misses?
Answer. Seven – 1 hit, eight – 2 hits, nine – 3 hits.
Solution. Since the shooter hit only seven, eight and nine in the remaining six shots, then in three shots (since the shooter hit seven, eight and nine at least once each) he will scorepoints Then for the remaining 3 shots you need to score 26 points. What is possible with the only combination 8 + 9 + 9 = 26. So, the shooter hit the seven once, the eight - 2 times, and the nine - 3 times.
5 . The midpoints of adjacent sides in a convex quadrilateral are connected by segments. Prove that the area of the resulting quadrilateral is half the area of the original one.
Solution. Let's denote the quadrilateral by ABCD , and the midpoints of the sides AB, BC, CD, DA for P, Q, S, T respectively. Note that in the triangle ABC segment PQ is the midline, which means it cuts off the triangle from it PBQ four times less area than area ABC. Likewise, . But triangles ABC and CDA in total they make up the entire quadrilateral ABCD means Similarly we get thatThen the total area of these four triangles is half the area of the quadrilateral ABCD and the area of the remaining quadrilateral PQST is also equal to half the area ABCD.
6. At what natural x expression is the square of a natural number?
Answer. At x = 5.
Solution. Let . Note that – also the square of some integer, less than t. We get that . Numbers and – natural and the first is greater than the second. Means, A . Solving this system, we get, which gives.
Preview:
Keys for the school mathematics Olympiad
10th grade
1. Arrange the modulus signs so that you get the correct equality
4 – 5 – 7 – 11 – 19 = 22
Solution. For example,
2. When Winnie the Pooh came to visit the Rabbit, he ate 3 plates of honey, 4 plates of condensed milk and 2 plates of jam, and after that he could not go outside because he had become very fat from such food. But it is known that if he ate 2 plates of honey, 3 plates of condensed milk and 4 plates of jam or 4 plates of honey, 2 plates of condensed milk and 3 plates of jam, he could easily leave the hole of the hospitable Rabbit. What makes you fatter: jam or condensed milk?
Answer. From condensed milk.
Solution. Let us denote by M the nutritional value of honey, by C the nutritional value of condensed milk, and by B the nutritional value of jam.
By condition, 3M + 4C + 2B > 2M + 3C + 4B, whence M + C > 2B. (*)
According to the condition, 3M + 4C + 2B > 4M + 2C + 3B, whence 2C > M + B (**).
Adding inequality (**) with inequality (*), we obtain M + 3C > M + 3B, whence C > B.
3. In Eq. one of the numbers is replaced with dots. Find this number if it is known that one of the roots is 2.
Answer. 2.
Solution. Since 2 is the root of the equation, we have:
where do we get that, which means the number 2 was written instead of an ellipsis.
4. Marya Ivanovna came out from the city into the village, and Katerina Mikhailovna came out to meet her from the village into the city at the same time. Find the distance between the village and the city if it is known that the distance between pedestrians was 2 km twice: first, when Marya Ivanovna walked half the way to the village, and then when Katerina Mikhailovna walked a third of the way to the city.
Answer. 6 km.
Solution. Let us denote the distance between the village and the city as S km, the speeds of Marya Ivanovna and Katerina Mikhailovna as x and y , and calculate the time spent by pedestrians in the first and second cases. In the first case we get
In the second. Hence, excluding x and y, we have
, from where S = 6 km.
5. In triangle ABC drew a bisector BL. It turned out that . Prove that the triangle ABL – isosceles.
Solution. By the bisector property we have BC:AB = CL:AL. Multiplying this equality by, we get , from where BC:CL = AC:BC . The last equality implies the similarity of triangles ABC and BLC at angle C and adjacent sides. From the equality of the corresponding angles in similar triangles we obtain, from where to
triangle ABL vertex angles A and B are equal, i.e. it is isosceles: AL = BL.
6. By definition, . Which factor should be deleted from the product?so that the remaining product becomes the square of some natural number?
Answer. 10!
Solution. Note that
x = 0.5 and is 0.25.2. Segments AM and BH - the median and altitude of the triangle, respectively ABC.
It is known that AH = 1 and . Find side length B.C.
Answer. 2 cm.
Solution. Let's draw a segment MN, it will be the median of the right triangle B.H.C. , drawn to the hypotenuse B.C. and is equal to half of it. Then– isosceles, therefore, therefore, AH = HM = MC = 1 and BC = 2MC = 2 cm.
3. At what values of the numerical parameter and inequality true for all values X ?
Answer . .
Solution . When we have , which is incorrect.
At 1 reduce the inequality by, keeping the sign:
This inequality is true for everyone x only at .
At reduce inequality by, changing the sign to the opposite:. But the square of a number is never negative.
4. There is one kilogram of 20% saline solution. The laboratory assistant placed the flask with this solution in an apparatus in which water is evaporated from the solution and at the same time a 30% solution of the same salt is added to it at a constant rate of 300 g/hour. The evaporation rate is also constant and amounts to 200 g/h. The process stops as soon as there is a 40% solution in the flask. What will be the mass of the resulting solution?
Answer. 1.4 kilograms.
Solution. Let t be the time during which the device worked. Then, at the end of the work, the result in the flask was 1 + (0.3 – 0.2)t = 1 + 0.1t kg. solution. In this case, the mass of salt in this solution is equal to 1 · 0.2 + 0.3 · 0.3 · t = 0.2 + 0.09t. Since the resulting solution contains 40% salt, we get
0.2 + 0.09t = 0.4(1 + 0.1t), that is, 0.2 + 0.09t = 0.4 + 0.04t, hence t = 4 hours. Therefore, the mass of the resulting solution is 1 + 0.1 · 4 = 1.4 kg.
5. In how many ways can you choose 13 different numbers from all natural numbers from 1 to 25 so that the sum of any two chosen numbers does not equal 25 or 26?
Answer. The only one.
Solution. Let's write all our numbers in the following order: 25,1,24,2,23,3,...,14,12,13. It is clear that any two of them equal a sum of 25 or 26 if and only if they are adjacent in this sequence. Thus, among the thirteen numbers we have chosen there should be no neighboring ones, from which we immediately obtain that these must be all members of this sequence with odd numbers - there is only one choice.
6. Let k be a natural number. It is known that among the 29 consecutive numbers 30k+1, 30k+2, ..., 30k+29 there are 7 primes. Prove that the first and last of them are simple.
Solution. Let's cross out numbers that are multiples of 2, 3 or 5 from this series. There will be 8 numbers left: 30k+1, 30k+7, 30k+11, 30k+13, 30k+17, 30k+19, 30k+23, 30k+29. Let us assume that among them there is a composite number. Let us prove that this number is a multiple of 7. The first seven of these numbers give different remainders when divided by 7, since the numbers 1, 7, 11, 13, 17, 19, 23 give different remainders when divided by 7. This means that one of these numbers are a multiple of 7. Note that the number 30k+1 is not a multiple of 7, otherwise 30k+29 will also be a multiple of 7, and the composite number must be exactly one. This means that the numbers 30k+1 and 30k+29 are prime.
- Contest
- Olympics
- Competition-game
- Subject week
- Family competition
- Children with disabilities
- Control test
- Summer camp
- Tests online
Distance Olympics of the Snail Center
Goals and objectives of distance Olympiads of the Snail Center:
- checking students' knowledge level
- developing the skill of self-appropriation of knowledge
- formation and development of skills for independent search and analysis of information
- formation and development of skills in using Internet services in education
- increasing motivation to study the subject
Olympics
They give the participant the opportunity to test and deepen their knowledge of a specific school discipline or even one section of it. All tasks of distance olympiads are divided by age groups and comply with school programs and Federal State Educational Standards requirements.
Competition-game
They give the participant the opportunity to test and deepen their knowledge of a specific school discipline or even one section of it. All tasks of distance Olympiads are divided by age groups and correspond to school programs and the requirements of the Federal State Educational Standard.
Subject week
They give the participant the opportunity to test and deepen their knowledge of a specific school discipline or even one section of it. All tasks of distance Olympiads are divided by age groups and correspond to school programs and the requirements of the Federal State Educational Standard.
Family competition
They give the participant the opportunity to test and deepen their knowledge of a specific school discipline or even one section of it. All tasks of distance Olympiads are divided by age groups and correspond to school programs and the requirements of the Federal State Educational Standard.
Specialist. competitions
They give the participant the opportunity to test and deepen their knowledge of a specific school discipline or even one section of it. All tasks of distance Olympiads are divided by age groups and correspond to school programs and the requirements of the Federal State Educational Standard.