The museum has 16 halls located. I (school) stage of the All-Russian Mathematical Olympiad with solutions and criteria

Stepnovsky district

2016/17 academic year
Mathematics
5th grade

1. Find among the numbers of the form 3a + 1 the first three numbers that are multiples of 5.

2. A kid can eat 600 g of jam in 6 minutes, and Carlson can eat 2 times faster. How long will it take them to eat this jam together?

3. The sailboat sets sail on Monday at noon. The voyage will last 100 hours. Give the day and hour of his return to the port.

4. Restore the recording:
*2*3
**
***87
*****
2*004*

5. The stepmother, leaving for the ball, gave Cinderella a bag in which poppy seeds and millet were mixed, and ordered her to sort them out. When Cinderella left for the ball, she left three bags: in one there was millet, in the other there was poppy seed, and in the third there was a mixture that had not yet been sorted. In order not to mix up the bags, Cinderella glued signs to each of them: “Poppyseed”, “Millet”, “Mixture”. The stepmother returned from the ball first and deliberately swapped the signs so that each bag had the wrong entry. The Fairy's student managed to warn Cinderella that now not a single inscription on the bags corresponds to reality. Then Cinderella took out only one single grain from one bag and, looking at it, immediately guessed where it was. How did she do it?
Stepnovsky district
School stage of the All-Russian Olympiad for schoolchildren
2016/17 academic year
Mathematics
6th grade
1.

2. Some island has an unusually regular climate: it always rains on Mondays and Wednesdays, fog on Saturdays, but sunny on other days. In the morning of what day of the week should a group of tourists begin their vacation if they want to stay there for 44 days and capture as much as possible? sunny days?
A) on Monday; B) on Wednesday; C) on Thursday; D) on Friday; E) on Tuesday.
3. For a two-digit number "n", the tens digit is twice as large as the units digit. Then the number "n" is required:
even; B) odd; C) less than 20; D) divisible by 3; E) is divisible by 6.

4. How much water must be added to 600 g of liquid containing 40% salt to obtain a 12% solution of this salt?

5. Place 8 kids and 9 geese in 5 barns so that in each barn there are both kids and geese, and the number of their legs is 10.

Stepnovsky district
School stage of the All-Russian Olympiad for schoolchildren
2016/17 academic year
Mathematics
7th grade

Tanya and Vanya were eating watermelon. Tanya ate half a third of a quarter of a watermelon, and Vanya ate a quarter of a half of a third of a watermelon. Who ate more watermelon?

It's half past nine. Why equal to the angle between the hour and minute hands?

Cowboy Bill walked into a gun shop and asked the seller for a Colt 3 before
·lara and six boxes of cartridges, the price of which he did not know. The seller demanded $11.80 ($1 = 100 cents) and Bill pulled out a revolver. Then the seller recalculated the purchase price and corrected the error. How did Bill realize that the salesman was trying to shortchange him?

Cut a square with a side of 4 cm into 5 rectangles with a perimeter of 8 cm. (Any square is also a rectangle).

The museum has 16 halls, arranged as shown in the picture. Half of them display paintings, and half exhibit sculptures. From any room you can get into any adjacent room (which has a common wall). During any tour of the museum, the halls alternate: a hall with paintings – a hall with sculptures – a hall with paintings, etc. The inspection begins in Hall A, where the paintings hang, and ends in Hall B.

Mark with crosses all the rooms in which paintings hang.
b) The tourist wants to see as many halls as possible (go from hall A to hall B), but at the same time visit each hall no more than once. Which greatest number can he see the halls? Draw some of his route longest length and prove that he could not have seen more halls.
Stepnovsky district
School stage of the All-Russian Olympiad for schoolchildren
2016/17 academic year
Mathematics
8th grade

What number ends in the sum 13EMBED Equation.31415?

Three mathematicians were traveling in different carriages of the same train. When the train approached the station, mathematicians counted 7, 12 and 15 benches on the platform. As the train pulled away, each of them counted several more benches, with one of them counting three times as many as the other. How much did the third one count?

Find 3 numbers that have the following properties: they are integers, positive, and the sum of their reciprocals is 1.

The company makes a lemon drink by diluting lemon juice with water. At first, the company produced a drink containing 15% lemon juice. Over time CEO gave instructions to reduce the lemon juice content to 10%. By what percentage will the quantity of lemon drink produced increase with the same volume of lemon supplies?

One of the angles of a triangle is 120° greater than the other. Prove that the bisector of a triangle drawn from the vertex of the third angle is twice as long as the altitude drawn from the same vertex.
5-6 4Figure 15-6 4

History of the Russian state in XIV-XVI centuries. Continuation. In the previous part we got acquainted with.

1. Formation of a unified Russian state in the XIV-XVI centuries
2.Russian culture of the 14th - early 16th centuries
3.Russian town and village of the 15th–17th centuries
4.Craft and trade of the 16th–17th centuries

History of the Russian state in the XIV-XVI centuries. Hall 13
Formation of a unified Russian state in the XIV-XVI centuries

(The fight of Rus' against foreign invaders. The unification of Russian lands).

This hall is called “Moscow” - it is dedicated to the rise of the Moscow principality in the 14th–15th centuries. and the formation of a unified Russian state.

The composition of the vault painting includes ornamental motifs from the famous Monomakh cap.

The hat, first mentioned in the spiritual letter (testament) of Ivan Kalita (1341), was the most important regalia of Russian princes and tsars, a symbol of Russian autocracy.
On the walls of the hall are reproduced friezes and columns that decorated the churches of the Moscow principality: the Assumption Cathedral of the Kremlin and the Cathedral of the Savior on Gorodok in Zvenigorod.

Above the exit from the hall is a painting by artists V.N. Sigorsky and N.P. Smolyak “The Moscow Kremlin at the beginning of the 16th century,” painted in 1947 for the 800th anniversary of Moscow. It depicts the Kremlin as it might have been at the beginning of the 16th century. The painting was painted based on the results of studying archaeological and written sources dating back to the rise of Moscow.

The exposition of the hall is divided into two large sections, located opposite each other and thereby symbolizing the historical confrontation between Rus' and the Golden Horde, the gradual rise and strengthening of one and the decline and collapse of the other.

One of the showcases is dedicated to the Battle of Kulikovo. The key exhibit is the chain mail found on the Kulikovo field. It weighs 10.3 kg and consists of 3000 rings.

It is almost impossible to find any weapons or ammunition on the medieval battlefield. Metal was expensive, all iron objects were collected and immediately taken out to repair or beat “swords into plowshares.” The discovery of this chain mail is a great success for archaeologists.
The Battle of Kulikovo became the most important historical event not only of the 14th century, but of the entire Russian Middle Ages. Before her, Russian people considered the Tatars invincible, and the yoke as God's punishment for sins. The Battle of Kulikovo did not protect the Russian principalities from raids, but it contributed to raising the spirit of all of Rus': it is possible and necessary to fight the enemy, he can be defeated.

On the other side of the central showcase are the remains of Ivan Kalita's oak Kremlin. These are fragments of the Octagonal Tower. The rise of Moscow began with the construction of the “city of oaks”.

The panel with a panorama of the Kremlin and a view of Red Square still looks very impressive. This is roughly what the bank of the Moscow River and the square looked like in the first half of the 16th century.

Showcases 1-2 display the famous Simferopol treasure.

318 items made of gold, silver and precious stones with a total weight of 2584 grams were discovered during excavation work near Simferopol in the 60s of the twentieth century. The treasure includes a silver paiza of Khan Keldibek (XIV century)
Paiza is special sign Tatar official-Baskak, vested with power. The Khan gave out paiza to his associates.
On the wall is an icon of Metropolitan Alexy, an outstanding church figure of the 14th century.
The so-called “Greater Zion” is also interesting. This is an ark made in the shape Orthodox church during the reign of Ivan III. There is a similar ark in .

Hall 14
Russian culture of the XIV - early XVI centuries. History of the Russian state in the XIV-XVI centuries

The main themes of the story: culture, Moscow - the successor of Kyiv, icon painting.

The key exhibit of this hall is a carved wooden gate from a temple built over the burial place of heroes of the Battle of Kulikovo.

ROYAL GATES. Russia XVI century Wood, gesso, tempera, carving, gilding, silvering. Reconstruction of the 19th century. From the Church of the Nativity of the Virgin Mary in the village of Monastyrshchino, Tula region.

The wooden church fell into disrepair in the 19th century, it was dismantled, a new stone church was erected, and the royal doors were moved to Moscow. The gates are a highly artistic work of Russian carvers. From a distance it looks like they are made of metal.
The exhibition includes an ancient chronicle. This is a handwritten book that includes the most famous Russian chronicles. (I will add a photo later).

In the central display case there is a veil - a wonderful example of ancient Russian facial embroidery. Ancient sewing is very sensitive to light, so various shrouds from the collection of the State Historical Museum are displayed here in turn. You should pay attention to the label.

EUCHARIST WITH THE LIVES OF THE MOTHER OF GOD, JOACHIM AND ANNA. Altarpiece, “Suzdal air”, Moscow, 1410-1416. Taffeta, spinning, silver and gold threads, “split and attach” sewing.

The main theme of the story is about the technique of embroidery - split stitch (when the needle splits, splits the thread of the previous stitch) and pin stitch - when the stitches are golden or silver thread secured with silk thread.
In the next showcase there are handwritten books, dedicated to the Battle of Kulikovo - “Zadonshchina” and “The Tale of the Massacre of Mamaev.”

The white stone details of the columns come from the Kremlin, from the old sovereign's palace, presumably built under Ivan III. In the general photo of the hall they can be seen in the corners next to the side windows.

Several carved white stone details are located near the wall to the left and right of the arch, where the “Old Testament Trinity” icon is placed.

Hall 15
Russian city and Russian village in the 15th–17th centuries. Feudal relations in the Russian state. History of the Russian state in the XIV-XVI centuries

Upon entering the hall, the large lattice on the right wall immediately catches your eye. It comes from the Novodvinsk fortress - this is an example protective structure. The Novodvinsk fortress was built to protect Arkhangelsk in early XVIII century in the image of the “cities” of the 17th century.

This is the last fortress in Russia, built in the image of the Kremlin, surrounded by walls with towers. Later they didn’t build like this anymore, because the development of artillery made the fortress walls completely useless.
The bell at the grille was cast by German craftsmen, it comes from.
This room features mica windows. Muscovy was rich in mica; it is not for nothing that the Latin name for mica sounds like “muscovite”. However, even in our country, mica was an expensive material. Such windows could only be used by the nobility and, perhaps, by hundreds of merchants.

All houses were strictly divided into female and male halves. The display case to the right of the grille displays the personal belongings of the thrifty owner of the house.

In the center of the display case is the shirt worn by the head of the family. His wife had to sew such a shirt. A decent woman could not afford for her husband to wear a shirt sewn by another woman.

In the same display case there is a remote ladle for treating guests.

A copy of “Domostroy” is also on display here. This is a book about how to run a house in a good, kind home, what you need to do to maintain a house, a wife, and raise children. The head of the family himself read it aloud in the evenings to his family.

To the left of the grille there is a display case with women's clothes.

The most curious item is the dress of a rich woman. It was found during the demolition of the Kitaygorod wall. They discovered a cache, and it only contained this dress. Apparently this item was stolen and hidden. Then something could happen to the thief and he did not take the stolen goods, so only this one dress was in the cache.

The uniqueness of this dress is that women's clothing Almost none of the pre-Petrine era has survived. The dress has a slit for a belt. On the one hand, the rules ordered a woman to gird herself and wear a belt. On the other hand, wear loose clothing so that all the curves of your figure are not visible. Therefore, the lower clothing was belted, and the upper one was spacious.

In the same showcase, next to the dress (see top photo) there is a headdress married woman. Nobody knows its name. But this is not a hair maker. Hair fibers are presented in.
There is also a cradle here, and above the cradle there is a horn. It was used for drinking infant. This horn is decorated; it simultaneously served for feeding and was the baby's first toy.

Another display case shows a small egg-little. This is a clay vessel in which pennies were stored and collected. One penny could buy 10 eggs or a pound of cucumbers in August.

Hall 16.
Craft and trade of the 16th–17th centuries. History of the Russian state in the XIV-XVI centuries

This room shows the reconstruction of part of the house. In the 17th century they appeared in Rus' European customs. Therefore, a chair was placed in the reconstruction, although in Rus' they sat on benches. IN XVI-XVII centuries a chair is always “a little bit of a throne”. Furniture on which only one person could sit showed the special status of that person. There is also a painting here biblical story- the court of King Solomon. However, the picture in medieval Rus'– something out of the ordinary. We didn't have a tradition of decorating the house with paintings. The cabinet was made by a Russian master; it resembles a chest.

Outlandish dishes - funny cups. Such dishes were not used, they were kept in the house for prestige. An interesting cup with a mill on top. Such a cup, if wine was poured into it, could not be placed - only drunk. When a person drank this cup, it was poured again. Very soon the guest got drunk, became cheerful and could play in the same cup: it has a spout, it is possible to blow into it and the wheel of the mill rotates.

Watches in Rus' were a luxury item. Here is a watch in the form of a gold box. Their hands are stationary, but the dial, on the contrary, rotates.
Salt shaker. Salt has always been a luxury item, so expensive utensils were used for this expensive product. One could leave the feast “without a meal.” Salt was mined in a rather labor-intensive way; salt deposits in Russia were discovered quite late.

This room displays a reconstruction of the saltworks. In the old days, this was the most important enterprise after, perhaps, the foundry cannon yard.

In some places salt waters came quite close to the surface of the earth. People learned to extract salt by digging wells and pumping out natural brine. It was fed through wooden chutes and settled in wooden settling tanks. When the brine concentration increased, the salt was evaporated or, in Old Russian, “boiled” over a fire. In the bottom photo you can see the boiling trays at the bottom left. This is where the names of salt production places come from - salt pans.

In Rus', food was never salted entirely. Each person added salt to their dish separately. Salt shakers were betrayed to each other, since then the sign has been preserved that if you spill salt, it will lead to a quarrel. Proverbs about salt have also been preserved.
Gingerbread boards are also displayed in one of the showcases. different sizes.
The lamp at the top is made according to the model of temple chandeliers of the 17th century.

The rest of the display cases display things that tell about other crafts. Below are the works of blacksmiths: locks, axes, chests, saddlers: saddle. In the center of the display case there are precious dishes - feet, ladles, glasses made by Russian goldsmiths.

Among them are the production of fabrics,

shoemaking, shoe production,

pottery craft and “tseninnoe”, i.e. tile business.

1. Vasya can get the number 100 using ten triplets, brackets and arithmetic signs:

100 = (33:3 – 3:3) · (33:3 – 3:3)

Improve his result: use fewer threes and get the number 100.

(It is enough to give one example).

2. Cut the figure into 3 equal parts.

3. How to measure 8 liters of water, being near a river and having two buckets with a capacity of 10 liters and 6 liters?

(8 liters of water should be in one bucket).

(Write a solution to the problem, not just an answer).

5. The museum has 16 halls, located as shown in the picture. Half of them display paintings, and half sculptures. From any room you can get into any adjacent room (which has a common wall). During any tour of the museum, the halls alternate: a hall with paintings – a hall with sculptures – a hall with paintings, etc. The inspection begins in Hall A, where the paintings hang, and ends in Hall B.

a) Mark with crosses all the rooms in which paintings hang.

b) The tourist wants to see as many halls as possible (go from hall A to hall B), but at the same time visit each hall no more than once. What is the largest number of halls he can watch? Draw some route of his greatest length and prove that he could not have seen more halls.

“Evaluation plus example Evaluation plus example is a special mathematical reasoning that is used in some problems on...”

I. V. Yakovlev | Mathematics materials | MathUs.ru

Evaluation plus example

Estimate plus example is a special mathematical reasoning that is used in some problems involving finding the largest or smallest values. The essence of this

The reasoning is best understood through concrete examples.

Task 1. What greatest number three-cell corners can be cut from checkered

square 8 8?

Solution. There are 64 cells in a square. Therefore, cutting out 22 or more corners will not work: after all, then the total number of cells in them will be no less than 22 · 3 = 66. This means that the number of corners is no more than 21 (estimate).

You can cut out 21 corners, an example is shown in the figure.

Therefore, the largest possible number of corners is 21.

The logic of the reasoning is clear: we have shown that the number of corners does not exceed the number 21 (estimate) and sometimes is equal to it (example). This means that 21 is the maximum number of corners.

Problem 2. What is the smallest number of coins of 3 and 5 kopecks that can reach the amount of 37 kopecks?

Solution. If the number of coins does not exceed seven, then the amount will be no more than 7 5 = 35 kopecks.

Therefore, seven or less coins will not be enough for us.

Let's assume there are eight coins. All of them cannot be five kopecks (8 · 5 = 40). Seven five-kopeck coins and one three-kopeck coin give a total of 38 kopecks. If there are no more than six five-kopeck coins, then the amount does not exceed 6·5+2·3 = 36 kopecks. This means that it is also impossible to get 37 kopecks with eight coins.

So, there must be at least nine coins. Let's give an example of a suitable set of nine coins: five five-kopeck coins and four three-kopeck coins (5 5 + 4 3 = 37).

Therefore, the smallest possible number of coins is nine.

Please note: you do not have to explain to anyone how you came up with the example! When writing a solution, it is enough to simply give an example. There is no need to describe the reasons for which your example was built.

1. What is the largest number of three-square corners that can be cut from a 5 7 checkered rectangle?

2. (Conquer Sparrow Hills!, 2016, 5–6.1) 30 schoolchildren and their parents are going on an excursion to St. Petersburg, some of whom are driving cars. Each car can accommodate 5 people, including the driver. What is the smallest number of parents needed to be invited on a field trip?

3. (All-Russian, 2014, Stage I, 5.4) Snow White entered the room where there was round table there were 30 chairs. There were dwarves sitting on some of the chairs. It turned out that Snow White could not sit without anyone sitting next to her. What is the smallest number of dwarves that could be at the table? (Explain how the dwarves were supposed to sit and why, if there were fewer dwarfs, Snow White would have found a chair with no one sitting next to it.)

4. (Conquer the Sparrow Hills!, 2016, 5–6.5; 7–8.4; 9.2) Find the largest natural number that cannot be represented as the sum of two composite numbers.

5. (All-Russian, 2014, stage I, 6–7.5) The museum has 16 halls, located as shown in the figure. Half of them display paintings, and half sculptures. From any room you can get into any adjacent room (which has a common wall). During any tour of the museum, the halls alternate: a hall with paintings, a hall with sculptures, a hall with paintings, etc. The inspection begins in hall A, in which paintings hang, and ends in hall B.

a) Mark with crosses all the rooms in which paintings hang.

6. (Mathematical Holiday, 2008, 6.2) The hare bought seven drums of different sizes and seven pairs of sticks of different lengths for her seven bunnies. If the bunny sees that he has a larger drum and longer sticks than one of his brothers, he begins to drum loudly.

What is the largest number of bunnies that can start drumming?

7. What is the smallest number of cells on the 8 8 board that can be colored so that there is at least one colored cell: a) in any square 2 2; b) in any corner of three cells?

8. What is the largest number of different (in shape or area) rectangles that a rectangle of 5 6 cells can be cut into? You can only cut along the grid lines.

9. (Mathematical Holiday, 1991, 6.2) An electrician was called to repair a garland of four light bulbs connected in series, one of which had burned out. It takes 10 seconds to unscrew any light bulb from a garland, and 10 seconds to screw it in. The time spent on other activities is small. In what shortest time can an electrician know to find a burnt-out light bulb if he has one spare light bulb?

10. (Mathematical holiday, 2016, 6.3) Equilateral triangle with side 8 were divided into equilateral triangles with side 1 (see figure). What is the smallest number of triangles that must be shaded so that all the points of intersection of the lines (including those at the edges) are the vertices of at least one shaded triangle?

11. (Mathematical holiday, 1990, 5.3) 48 blacksmiths must shoe 60 horses. What is the least time they will spend on work if each blacksmith spends five minutes on one horseshoe? (A horse cannot stand on two legs.) 12. (Conquer the Sparrow Hills!, 2017, 5–6.4) Masha has 2 kg of Swallow candies, 3 kg of Truffle candies, 4 kg of Bird’s Milk candies and 5 kg of Citron candies. What is the largest number New Year's gifts it can be made if each gift must contain 3 various types sweets, 100 grams each?

13. (Kurchatov, 2017, 6.4) Alexey wrote several consecutive natural numbers. It turned out that only two of the written numbers have a sum of digits divisible by 8: the smallest and the largest. Which maximum amount numbers could be written on the board?

14. (Mathematical holiday, 2015, 6.5) The monkey becomes happy when he eats three different fruits. What is the largest number of monkeys that can be made happy with 20 pears, 30 bananas, 40 peaches and 50 tangerines? Justify your answer.

15. (Mathematical holiday, 2014, 6.5) Mom baked three pies with rice, three with cabbage and one with cherries and laid them out on a dish in a circle (see picture). Then I put the dish in the microwave to heat it up. All pies look the same. Masha knows how they lay, but does not know how the dish turned. She wants to eat a cherry pie, but considers the rest to be tasteless.

How can Masha achieve this for sure by biting as few tasteless pies as possible?

16. (Mathematical holiday, 2006, 6.5) The grandfather called his grandson to his village: Look what an extraordinary garden I planted! I have four pear trees growing there, and I also have apple trees, and they are planted so that exactly two pears grow at a distance of 10 meters from each apple tree. Well, what’s interesting here, the grandson answered. You only have two apple trees. But I didn’t guess right, the grandfather smiled. There are more apple trees in my garden than pears. Draw how apple and pear trees could grow in your grandfather’s garden. Try to place as many apple trees as possible in the picture without violating the conditions. If you think you have placed the maximum number of apple trees possible, try to explain why this is so.

– – –

18. (Mathematical holiday, 2012, 6.5) Replace the same letters in the equality PIE = PIECE + PIECE + PIECE +... + PIECE the same numbers, and different are different so that the equality is true and the number of pieces of the pie is the largest possible.

19. (Moscow Oral Olympiad, 2014, 6.5) On a checkered board measuring 4 4, Petya paints several squares. Vasya will win if he can cover all these cells with corners of three cells that do not intersect and do not extend beyond the border of the square. What is the smallest number of cells that Petya must paint so that Vasya does not win?

20. (Mathematical holiday, 2013, 6.6) Thirty-three heroes hired to guard Lukomorye for 240 coins. The cunning guy Chernomor can divide the heroes into detachments of arbitrary size (or write everyone into one detachment), and then distribute all the salaries between the detachments. Each detachment divides its coins equally, and gives the remainder to Chernomor.

What is the largest number of coins that Chernomor can get if:

a) Chernomor distributes salaries between units as it pleases;

b) does Chernomor distribute salaries equally between units?

21. (Moscow Oral Olympiad, 2013, 6.6) To play the hat, Nadya wants to cut a sheet of paper into 48 identical rectangles. What is the minimum number of cuts she will have to make if any pieces of paper can be rearranged, but cannot be folded, and Nadya is able to cut as many layers of paper as she wants at the same time? (Each cut is a straight line from edge to edge of the piece.) 22. (Moscow Oral Olympiad, 2015, 6.6) From the same number of squares with sides 1, 2 and 3, make a square of the smallest possible size.

23. (Moscow Oral Olympiad, 2008, 6.6) Find the largest number of colors in which the edges of the cube can be colored (each edge with one color) so that for each pair of colors there are two adjacent edges colored in these colors. Edges that have a common vertex are considered adjacent.

24. (Moscow Oral Olympiad, 2006, 6.6) Speaking in the arena with 10 lions and 15 tigers, the trainer Pasha lost control over them, and the animals began to devour each other. A lion will be satisfied if he eats three tigers, and a tiger if he eats two lions. Determine the largest number of predators that could be fed and how this could happen.

25. (Moscow Oral Olympiad, 2013, 6.7) The Rabbit poured three kilograms of honey into five pots standing in a row (not necessarily into each one and not necessarily equally). Winnie the Pooh can take any two pots, standing nearby. What is the largest amount of honey that Winnie the Pooh is guaranteed to eat?

26. (Moscow Oral Olympiad, 2012, 6.7) A five-digit number is called indecomposable if it cannot be decomposed into the product of two three-digit numbers. What is the greatest number of indecomposable five-digit numbers that can be in a row?

27. (Moscow Oral Olympiad, 2002, 6.7) Each of the 50 products must first be painted and then packaged. Painting time 10 minutes, packaging time 20 minutes. After painting, the part must dry for 5 minutes. How many painters and how many packers do you need to hire to complete the job in the shortest possible time if you cannot hire more than 10 people?

28. (Moscow Oral Olympiad, 2017, 6–7.8) In each square of the board measuring 5 5 there is a cross or a zero, and no three crosses are in a row, neither horizontally, nor vertically, nor diagonally. What is the largest number of crosses that can be on the board?

29. (Mathematical holiday, 1993, 6.8) 100 fat men weighing from 1 to 100 kg are training in a sports club. What is the smallest number of teams they can be divided into so that no team has two fat people, one of whom weighs twice as much as the other?

30. (Moscow Oral Olympiad, 2016, 6.9) The store sells boxes of chocolates. Among them there are at least five boxes of different prices (no two of them cost the same). Whatever two boxes Vasya buys, Petya can always buy two boxes, spending the same amount of money. What is the smallest number of boxes of chocolates that should be on sale?

31. (Moscow Oral Olympiad, 2012, 6.9) The plan of the Shah's palace is a square measuring 66, divided into rooms measuring 1 1. In the middle of each wall between the rooms there is a door.

The Shah told his architect: Break down some of the walls so that all the rooms become 2 1 in size, no new doors appear, and the path between any two rooms passes through no more than N doors. What is the smallest value of N that the check must call so that the order can be executed?

32. (MMO, 1989, 7) In a dark room on a shelf, 4 pairs of socks of two different sizes and two different colors. What is the smallest number of socks that needs to be transferred from the shelf into a suitcase without leaving the room so that it contains two pairs of different sizes and colors?

33. (Lomonosov, 2012, 7–8.1) Digital Watch show the time in a standard format (for example, 20:27). Find the greatest possible value of the product of digits on such a clock.

34. (Lomonosov, 2014, 7.2) Find the smallest integer n 3 such that there is no convex n-gon each interior angle of which is an even number of degrees.

35. (Conquer the Sparrow Hills!, 2014, 7.2) Find the smallest possible value of the expression |2015m5 2014n4 | provided that m, n are natural numbers.

36. (Mathematical Holiday, 1997, 7.2) In Mexico, environmentalists achieved the adoption of a law according to which every car should not be driven at least one day a week (the owner tells the police the car number and the day of the week of the car). In some family, all adults want to travel every day (each on their own business!). How many cars (at least) should a family have if there are a) 5 adults in it?

b) 8 people?

37. (MMO, district tour, 2008, 7.3) New Year's garland, hanging along the school corridor, consists of red and blue light bulbs. Next to every red light there is always a blue one. What is the maximum number of red light bulbs that can be in this garland if there are 50 light bulbs in total?

38. (Kurchatov, 2014, 7.4) Of ten different numbers, two three-digit and one four-digit number were made. These three numbers were multiplied. What is the largest number of zeros that a product can end with?

39. (Lomonosov, 2013, 7.4) A flea jumps along a number line, and the length of each jump cannot be less than n. She starts her movement from the origin of coordinates and wants to visit all whole points belonging to the segment (and only them!) exactly once.

At what highest value n will she succeed?

40. (Lomonosov, 2012, 7.4) In the elections to the city council, 22,410 votes were cast for 7 parties. One of the parties received more votes than each of the others. What is the smallest number of votes she could receive?

41. (Moscow Oral Olympiad, 2005, 7.4) The frame of a cube with edges of length 1 is smeared with honey.

There is a beetle at the top of the cube. What is the minimum distance he must crawl to eat all the honey?

42. (Archimedes Tournament, 2012.5) The bag contains gold coins doubloons, ducats and piastres, identical to the touch. If you take 10 coins out of a bag, then there will definitely be at least one doubloon among them; if you take out 9 coins, then among them there will definitely be at least one ducat; if you take out 8 coins, then there will definitely be at least one piastre among them. What is the largest number of coins that could be in the bag?

43. (All-Russian, 2014, stage II, 7.5) In the total +1 + 3 + 9 + 27 + 81 + 243 + 729, you can cross out any terms and change some signs in front of the remaining numbers from + to. Masha wants to use this method to first obtain an expression whose value is 1, then, starting from the beginning, to obtain an expression whose value is 2, then (starting again from the beginning) to obtain 3, and so on. Up to what is the largest integer she can do this without missing any gaps?

44. (Mathematical holiday, 2003, 7.5) In honor of the holiday, 1% of the soldiers in the regiment received new uniforms. The soldiers are arranged in a rectangle so that the soldiers in the new uniform are in at least 30% of the column in at least 40% of the ranks. What is the smallest number of soldiers that could be in the regiment?

45. (Archimedes Tournament, 2014.6) Dunno rearranged the numbers in a certain number A and got the number B. Then he calculated the difference A B and got a number written using only units (other numbers were not used). What is the smallest number he could get?

46. (Mathematical holiday, 2005, 7.6) On the island of Bad luck with a population of 96 people, the government decided to carry out five reforms. Exactly half of all citizens are dissatisfied with each reform. A citizen goes to a rally if he is dissatisfied with more than half of all reforms.

What is the maximum number of people the government can expect at a rally? (Give an example and prove that this is no longer possible.) 47. (Mathematical holiday, 2008, 7.6) Vasya stood at the bus stop for a while. During this time, one bus and two trams passed. After some time, a Spy came to the same stop. While he was sitting there, 10 buses passed by. What is the minimum number of trams that could pass during this time? Both buses and trams run at regular intervals, with buses running every 1 hour.

48. (Mathematical holiday, 2012, 7.6) Having defeated Kashchei, Ivan demanded gold to ransom Vasilisa from the robbers. Kashchei brought him to the cave and said:

The chest contains gold bars. But you can’t just take them away: they are enchanted. Put one or more in your bag. Then I will transfer one or more from the bag to the chest, but always a different number. So we will take turns moving them: you into the bag, I into the chest, each time a new number. When a new shift becomes impossible, you can take away your bag of bullion.

What is the largest number of ingots that Ivan can carry away, no matter how Kashchei acts, if the chest initially contains a) 13; b) 14 gold bars? How can he do this?

49. (Mathematical holiday, 2010, 7.6) It’s easy to place a set of ships for playing Sea battle on the board 1010 (see picture). What is the smallest square board that this set can be placed on? (Remember that according to the rules, ships should not even touch at the corners.) 50. (Mathematical Holiday, 2013, 7.6) Alice the fox and Basilio the cat grew 20 counterfeit bills on a tree and now write seven-digit numbers into them. Each bill has 7 empty cells for numbers. Basilio names one number at a time, 1 or 2 (he doesn’t know the others), and Alice enters the named number in any empty cell of any bill and shows the result to Basilio.

When all the cells are filled, Basilio takes as many bills as possible from different numbers(out of several with the same number, he takes only one), and Alice takes the rest.

What is the largest number of banknotes that Basilio can get, no matter how Alice acts?

51. (Lomonosov, 2015, 7.6) Find the largest possible value

– – –

if it is known that x and y are relatively prime numbers.

52. (Lomonosov, 2012, 7.7) For what smallest number n can we mark n points on the plane so that there are three squares, all of whose vertices are marked points?

53. (Conquer the Sparrow Hills!, 2015, 7.7) The numbers 1, 2,..., 2016 were divided into pairs, and it turned out that the product of the numbers in each pair does not exceed some natural number N.

For what smallest N is this possible?

54. (Moscow Oral Olympiad, 2014, 7.9) 2014 points are marked on the circle. In one of them sits a grasshopper, which makes clockwise jumps of either 57 divisions or 10. It is known that he visited all the marked points, making the least number of jumps of length 10. Which?

All-Russian Olympiad for schoolchildren 2013-2014 in Moscow Typical tasks of the 1st (school) stage of the Olympiad in mathematics Information about the stages All-Russian Olympiad for mathematics, see the website http://vos.olimpiada.ru/ 5th grade 1. Vasya can...

All-Russian Olympiad for schoolchildren 2013-2014 in Moscow Typical tasks of the first (school) stage of the Olympiad in mathematics For information about the stages of the All-Russian Olympiad in mathematics, see the website http://vos.olimpiada.ru/ 6th grade 1. Vasya can get the number 100 using ten triples, parentheses and arithmetic symbols:)3:33:33()3:33:33(100  . Improve his result: use fewer triples and get the number 100. (Just give one example). 2 . Cut the figure into 3 equal parts. 3. How to measure 2 liters of water, being near the river and having two buckets with a capacity of 10 liters and 6 liters? (2 liters of water should be in one bucket). 4. Dad, Masha and Yasha go to school. While dad takes 3 steps, Masha takes 5 steps. While Masha takes 3 steps, Yasha takes 5 steps. Masha and Yasha counted that together they took 400 steps. How many steps did dad take? (Write the solution to the problem, not just the answer 5. The museum has 16 halls, arranged as shown in the picture.Half of them display paintings, and half exhibit sculptures. From any room you can get into any adjacent room (which has a common wall). During any tour of the museum, the halls alternate: a hall with paintings – a hall with sculptures – a hall with paintings, etc. The inspection begins in hall A, in which the paintings hang, and ends in hall B. a) Mark with crosses all the halls in which the paintings hang. b) The tourist wants to see as many halls as possible (go from hall A to hall B), but at the same time visit each hall no more than once. What is the largest number of halls he can watch? Draw some route of his greatest length and prove that he could not have seen more halls. B A All-Russian Olympiad for schoolchildren 2013-2014 in Moscow Typical tasks of the 1st (school) stage of the Olympiad in mathematics For information about the stages of the All-Russian Olympiad in mathematics, see the website http://vos.olimpiada.ru/ 6th grade 1. Vasya can get the number 100 , using ten triples, parentheses and arithmetic signs:)3:33:33()3:33:33(100  . Improve his result: use fewer triples and get the number 100. (Just give one example) 2. Cut the figure into 3 equal parts. 3. How to measure 2 liters of water, being near the river and having two buckets with a capacity of 10 liters and 6 liters? (2 liters of water should be in one bucket). 4. Dad, Masha and Yasha go to school. While dad takes 3 steps, Masha takes 5 steps. While Masha takes 3 steps, Yasha takes 5 steps. Masha and Yasha counted that together they took 400 steps. How many steps did dad take? (Write a solution to the problem, not just the answer). 5. The museum has 16 halls, located as shown in the picture. Half of them display paintings, and half sculptures. From any room you can get into any adjacent room (which has a common wall). During any tour of the museum, the halls alternate: a hall with paintings – a hall with sculptures – a hall with paintings, etc. The inspection begins in hall A, in which the paintings hang, and ends in hall B. a) Mark with crosses all the halls in which the paintings hang. b) The tourist wants to see as many halls as possible (go from hall A to hall B), but at the same time visit each hall no more than once. What is the largest number of halls he can watch? Draw some route of his greatest length and prove that he could not have seen more halls. B A

All-Russian Olympiad for schoolchildren 2013-2014 in Moscow Typical tasks of the first (school) stage of the Olympiad in mathematics For information about the stages of the All-Russian Olympiad in mathematics, see the website http://vos.olimpiada.ru/ 7th grade 1. Vasya can get the number 100 using ten sevens, parentheses and arithmetic signs:)7:77:77()7:77:77(100  . Improve his result: use fewer sevens and get the number 100. (Just give one example). 2 . The clock is half past nine. What is the angle between the hour and minute hands? (Justify your answer). 3. Let's call a number mirror if from left to right it is “read” the same way as from right to left. For example, the number 12321 is mirror. a) Write some five-digit mirror number that is divisible by 5. b) How many five-digit mirror numbers are there that are divisible by 5? 4. Sasha, Lyosha and Kolya started at the same time in the 100 m race. When Sasha finished, Lyosha was ten meters behind him, and when Lyosha finished, Kolya was ten meters behind him. How far apart were Sasha and Kolya when Sasha finished? (It is assumed that all the boys run with the regulars, but of course not at equal speeds .) 5. The museum has 16 halls, located as shown in the picture. Half of them display paintings, and half sculptures. From any room you can get into any adjacent room (which has a common wall). During any tour of the museum, the halls alternate: a hall with paintings – a hall with sculptures – a hall with paintings, etc. The inspection begins in hall A, in which the paintings hang, and ends in hall B. a) Mark with crosses all the halls in which the paintings hang. b) The tourist wants to explore as many halls as possible (go from hall A to hall B) so that he can visit each hall no more than once. What is the largest number of halls he can watch? Draw some route of his greatest length and prove that he could not have seen more halls. B A All-Russian Olympiad for schoolchildren 2013-2014 in Moscow Typical tasks of the first (school) stage of the Olympiad in mathematics For information about the stages of the All-Russian Olympiad in mathematics, see the website http://vos.olimpiada.ru/ 7th grade 1. Vasya can get the number 100 , using ten sevens, parentheses and arithmetic signs:)7:77:77()7:77:77(100  . Improve his result: use fewer sevens and get the number 100. (Just give one example) 2. The clock is half past nine. What is the angle between the hour and minute hands? (Justify your answer). 3. Let's call a number mirror if from left to right it is “read” the same way as from right to left. For example, the number 12321 is mirror. a ) Write some five-digit mirror number that is divisible by 5. b) How many five-digit mirror numbers are there that are divisible by 5? 4. Sasha, Lyosha and Kolya started at the same time in the 100 m race. When Sasha finished, Lyosha was ten meters behind him, and when Lyosha finished, Kolya was ten meters behind him. How far apart were Sasha and Kolya when Sasha finished? (It is assumed that all the boys are running at constant, but, of course, not equal speeds.) 5. The museum has 16 halls, located as shown in the figure. Half of them display paintings, and half sculptures. From any room you can get into any adjacent room (which has a common wall). During any tour of the museum, the halls alternate: a hall with paintings – a hall with sculptures – a hall with paintings, etc. The inspection begins in hall A, in which the paintings hang, and ends in hall B. a) Mark with crosses all the halls in which the paintings hang. b) The tourist wants to explore as many halls as possible (go from hall A to hall B) so that he can visit each hall no more than once. What is the largest number of halls he can watch? Draw some route of his greatest length and prove that he could not have seen more halls. B A

All-Russian Olympiad for schoolchildren 2013-2014 in Moscow Typical tasks of the first (school) stage of the Olympiad in mathematics For information about the stages of the All-Russian Olympiad in mathematics, see the website http://vos.olimpiada.ru/ 8th grade 1. Replace 2223 *)( )2( xx asterisk (*) per monomial so that after squaring and bringing similar terms, four terms are obtained. 2. What is the ratio of the area of the shaded part to the white part (the vertices of all squares except the largest are in the middles of the corresponding sides )? 3. Let's call a number mirror if from left to right it is “read” the same way as from right to left. For example, the number 12321 is mirror. a) Write some mirror five-digit number that is divisible by 5. b) How many five-digit mirror numbers are there? numbers that are divisible by 5? 4. Sasha, Lyosha and Kolya started at the same time in the 100 m race. When Sasha finished, Lyosha was ten meters behind him, and when Lyosha finished, Kolya was ten meters behind him. How far apart were Sasha and Kolya when Sasha finished? (It is assumed that all the boys run at constant, but, of course, not equal speeds.) 5. Petya cut a 2x1 paper parallelepiped along its edges and obtained a development. Then Dima cut off one square from this development, and nine squares remained, as in the picture. Where could the cut square be? Draw a full development and mark the cut square on it. (It is enough to give one correct version of the development). 6. Each of the 10 dwarves either always tells the truth or always lies. It is known that each of them loves exactly one type of ice cream: cream, chocolate or fruit. First, Snow White asked those who like ice cream to raise their hands, and everyone raised their hands, then those who like chocolate ice cream - and half of the dwarfs raised their hands, then those who like popsicles - and only one dwarf raised their hand. How many of the gnomes are truthful? All-Russian Olympiad for schoolchildren 2013-2014 in Moscow Typical tasks of the first (school) stage of the Olympiad in mathematics For information about the stages of the All-Russian Olympiad in mathematics, see the website http://vos.olimpiada.ru/ 8th grade 1. Replace 2223 *)( )2( xx asterisk (*) per monomial so that after squaring and bringing similar terms, four terms are obtained. 2. What is the ratio of the area of the shaded part to the white part (the vertices of all squares except the largest are in the middles of the corresponding sides )? 3. Let's call a number mirror if from left to right it is “read” the same way as from right to left. For example, the number 12321 is a mirror number. a) Write some five-digit mirror number that is divisible by 5. b) How many five-digit mirror numbers are there that are divisible by 5? 4. Sasha, Lyosha and Kolya started at the same time in the 100 m race. When Sasha finished, Lyosha was ten meters behind him, and when Lyosha finished, Kolya was ten meters behind him. How far apart were Sasha and Kolya when Sasha finished? (It is assumed that all the boys run at constant, but, of course, not equal speeds.) 5. Petya cut a 2x1 paper parallelepiped along its edges and obtained a development. Then Dima cut off one square from this development, and nine squares remained, as in the picture. Where could the cut square be? Draw a full development and mark the cut square on it. (It is enough to give one correct version of the development). 6. Each of the 10 dwarves either always tells the truth or always lies. It is known that each of them loves exactly one type of ice cream: cream, chocolate or fruit. First, Snow White asked those who like ice cream to raise their hands, and everyone raised their hands, then those who like chocolate ice cream - and half of the dwarfs raised their hands, then those who like popsicles - and only one dwarf raised their hand. How many of the gnomes are truthful?

All-Russian Olympiad for schoolchildren 2013-2014 in Moscow Typical tasks of the first (school) stage of the Olympiad in mathematics For information about the stages of the All-Russian Olympiad in mathematics, see the website http://vos.olimpiada.ru/ Grade 9 1. Replace 2324 *)( )3( xx asterisk (*) per monomial so that after squaring and bringing similar terms, four terms are obtained. 2. What is the ratio of the area of the shaded part to the white part (the vertices of all squares except the largest are in the middles of the corresponding sides )? 3. Let's call a number mirror if from left to right it is "read" the same way as from right to left. For example, the number 12321 is mirror. How many five-digit mirror numbers are there that are divisible by 5? 4. Vasya thought of two numbers. Their sum is equal their product and is equal to their quotient. What numbers did Vasya have in mind? 5. Petya cut a 2x1 paper parallelepiped along its edges and got a development. Then Dima cut off one square from this development, and nine squares remained, as in the picture. Where could the cut square be? Draw a full development and mark the cut square on it. (It is enough to give one correct version of the development). 6. Each of the 10 dwarves either always tells the truth or always lies. It is known that each of them loves exactly one type of ice cream: cream, chocolate or fruit. First, Snow White asked those who like ice cream to raise their hands, and everyone raised their hands, then those who like chocolate ice cream - and half of the dwarfs raised their hands, then those who like popsicles - and only one dwarf raised their hand. How many of the gnomes are truthful? All-Russian Olympiad for schoolchildren 2013-2014 in Moscow Typical tasks of the first (school) stage of the Olympiad in mathematics For information about the stages of the All-Russian Olympiad in mathematics, see the website http://vos.olimpiada.ru/ Grade 9 1. Replace 2324 *)( )3( xx asterisk (*) per monomial so that after squaring and bringing similar terms, four terms are obtained. 2. What is the ratio of the area of the shaded part to the white part (the vertices of all squares except the largest are in the middles of the corresponding sides )? 3. Let's call a number mirror if from left to right it is "read" the same way as from right to left. For example, the number 12321 is mirror. How many five-digit mirror numbers are there that are divisible by 5? 4. Vasya thought of two numbers. Their sum is equal their product and is equal to their quotient. What numbers did Vasya have in mind? 5. Petya cut a 2x1 paper parallelepiped along its edges and obtained a development. Then Dima cut off one square from this development, and nine squares remained, as in the picture. Where could the cut square be? Draw a full development and mark the cut square on it. (It is enough to give one correct version of the development). 6. Each of the 10 dwarves either always tells the truth or always lies. It is known that each of them loves exactly one type of ice cream: cream, chocolate or fruit. First, Snow White asked those who like ice cream to raise their hands, and everyone raised their hands, then those who like chocolate ice cream - and half of the dwarfs raised their hands, then those who like popsicles - and only one dwarf raised their hand. How many of the gnomes are truthful?

All-Russian Olympiad for schoolchildren 2013-2014 in Moscow Typical tasks of the I (school) stage of the Olympiad in mathematics, grade 5. Brief solutions. 1. Vasya can get the number 100 using ten twos, parentheses and arithmetic symbols:)2:22:22()2:22:22(100  . Improve his result: use fewer twos and get the number 100 . (It is enough to give one example.) Solution. For example: 1) 2:222:222100  , 2))2222()2222(100  . There are other solutions. 2. Cut the figure into 3 equal parts. Solution. See figure. 3. How to measure 8 liters of water, being near a river and having two buckets with a capacity of 10 liters and 6 liters? (8 liters of water should be in one bucket). Solution. Let's write down the filling sequence in the form of a table buckets: Bucket with a capacity of 10 l Bucket with a capacity of 6 l Comment First 0 l 0 l 1 step 10 l 0 l The first bucket was filled from the river 2 step 4 l 6 l Poured from the first bucket into the second before it was filled 3 step 4 l 0 l Poured from the second into the river 4 step 0 l 4 l Poured from the first bucket into the second 5 step 10 l 4 l The first bucket was filled from the river 6 step 8 l 6 l Poured from the first bucket into the second until it was full

4. Snow White entered a room where there were 30 chairs around a round table. There were dwarves sitting on some of the chairs. It turned out that Snow White could not sit without anyone sitting next to her. What is the smallest number of dwarves that could be at the table? (Explain how the gnomes were supposed to sit and why, if there were fewer gnomes, there would be a chair that no one sits next to). Answer. 10. Solution. If there were three empty chairs in a row at a table somewhere, Snow White could sit without anyone sitting next to her. This means that no matter what three consecutive chairs we take, at least one of them should have a gnome sitting on it. Since there are 30 chairs in total, there cannot be less than 10 gnomes. Let us show that it is possible to seat 10 gnomes in such a way that the conditions of the problem are met: we seat the gnomes in two chairs: on the first chair, on the fourth chair, on the seventh, etc. Then the condition of the problem will be fulfilled. 5. Dad, Masha and Yasha are going to school. While dad takes 3 steps, Masha takes 5 steps. While Masha takes 3 steps, Yasha takes 5 steps. Masha and Yasha counted that together they took 400 steps. How many steps did dad take? Answer. 90 steps. Solution. 1 way. Let's call a distance equal to 3 steps of Masha and 5 steps of Yasha the Giant's step. While the Giant takes one step, Masha and Yasha take 8 steps together. Since they took 400 steps together, the Giant would have taken 400:8 = 50 giant steps during this time. If the Giant took 50 steps, then Masha took 150 steps. Let us now count them as “fives”. 150 is 30 times 5 steps. This means that dad took 3 steps 30 times, that is, 90 steps. Method 2. While Masha takes 1553  steps, dad takes 933  steps, and Yasha takes 2555  steps. Together during this time Masha and Yasha will take 15+25=40 steps. And while they take 400 steps, dad will also take 10 times more steps, i.e. 90109  steps.

5th grade. Recommendations for verification. Each task is scored out of 7 points. Each score is an integer from 0 to 7. Below are some guidelines for checking. Naturally, compilers cannot foresee all cases. When evaluating a solution, one must proceed from whether the given solution is generally correct (although, perhaps, with shortcomings) - then the solution is scored at least 4 points. Or it is incorrect (although, perhaps, with significant progress) - in this case, the score should not be higher than 3 points. Task 1. Any correct example – 7 points. Two or more examples, some of which are true and some are false – 5 points. Task 2. Correct cutting – 7 points. No justification required. Cutting into unequal shapes equal area– 2 points. Problem 3. Correct algorithm – 7 points. Reasonable promotions, for example, measured 4 l - up to 3 points. Problem 4. Complete solution – 7 points. An example of seating arrangement is given and there is reasoning why there cannot be fewer gnomes with some gaps - 5-6 points. A correct example of seating arrangements is given, but it is not explained why there cannot be fewer gnomes - 3 points. The answer is given, it is explained why there cannot be fewer gnomes. But how the gnomes sit is not explained - 3 points. Only answer – 1 point. Problem 5. Complete solution – 7 points. The solution in the figure (by cells, etc.) without sufficient explanation – 4-5 points. Correct solution with an arithmetic error – 4 points. The only answer is 0 points.

All-Russian Olympiad for schoolchildren 2013-2014 in Moscow Typical tasks of the 1st (school) stage of the Olympiad in mathematics, grade 6. Brief solutions. 1. Vasya can get the number 100 using ten triples, parentheses and arithmetic symbols:)3:33:33()3:33:33(100  . Improve his result: use fewer triples and get the number 100 (It is enough to give one example.) Solution: For example: 1) 3:333:333100  , 2) 3:3333100  . There are other solutions. 2. Cut the figure into 3 equal parts. Solution. See the picture. 3. How to measure 2 liters of water, being near a river and having two buckets with a capacity of 10 liters and 6 liters? (2 liters of water should be in one bucket). Solution. Let's write down the sequence of filling buckets in the form of a table: Bucket with a capacity of 10 l Bucket with a capacity of 6 l Comment First 0 l 0 l 1 step 10 l 0 l The first bucket was filled from the river 2 step 4 l 6 l Poured from the first bucket into the second before it is filled 3 step 4 l 0 l Poured from the second bucket into the river 4 step 0 l 4 l Poured from the first bucket into the second step 5 10 l 4 l The first bucket was filled from the river 6 step 8 l 6 l Poured from the first bucket into the second until it was full 7 step 8 l 0 l Pour from the second bucket into the river 8 step 2 l 6 l Pour from the first bucket into the second until it is full

4. Dad, Masha and Yasha are going to school. While dad takes 3 steps, Masha takes 5 steps. While Masha takes 3 steps, Yasha takes 5 steps. Masha and Yasha counted that together they took 400 steps. How many steps did dad take? Answer. 90 steps. Solution. 1 way. Let's call a distance equal to 3 steps of Masha and 5 steps of Yasha the Giant's step. While the Giant takes one step, Masha and Yasha take 8 steps together. Since they took 400 steps together, the Giant would have taken 400:8 = 50 giant steps during this time. If the Giant took 50 steps, then Masha took 150 steps. Let us now count them as “fives”. 150 is 30 times 5 steps. This means that dad took 3 steps 30 times, that is, 90 steps. Method 2. While Masha takes 1553  steps, dad takes 933  steps, and Yasha takes 2555  steps. Together during this time Masha and Yasha will take 15+25=40 steps. And while they take 400 steps, dad will also take 10 times more steps, i.e. 90109  steps. The museum has 16 halls, arranged as shown in the picture. Half of them display paintings, and half exhibit sculptures. From any room you can get into any adjacent room (which has a common wall). During any tour of the museum, the halls alternate: a hall with paintings – a hall with sculptures – a hall with paintings, etc. The inspection begins in hall A, in which the paintings hang, and ends in hall B. a) Mark with crosses all the halls in which the paintings hang. Solution. See the picture. b) The tourist wants to see as many halls as possible (go from hall A to hall B), but at the same time visit each hall no more than once. What is the largest number of halls he can watch? Draw some route of his greatest length and prove that he could not have seen more halls. Answer. 15. Solution. One of the possible routes is shown in the figure. Let us prove that if a tourist wants to visit each hall no more than once, he will not be able to see more than 15 halls. Note that the route begins in the hall with paintings (A) and ends in the hall with paintings (B). This means that the number of halls with paintings that a tourist passed through per more number halls with sculptures. Since there are no more than 8 halls with paintings that a tourist could go through, there are no more than 7 halls with sculptures. So, the route cannot pass through more than 15 halls. B A X X X X X XX X B A B A

6th grade. Recommendations for verification. Each task is scored out of 7 points. Each score is an integer from 0 to 7. Below are some guidelines for checking. Naturally, compilers cannot foresee all cases. When evaluating a solution, one must proceed from whether the given solution is generally correct (although, perhaps, with shortcomings) - then the solution is scored at least 4 points. Or it is incorrect (although, perhaps, with significant progress) - in this case, the score should not be higher than 3 points. Task 1. Any correct example – 7 points. Two or more examples, some of which are true and some are false – 5 points. Task 2. Correct cutting – 7 points. No justification required. Cutting into unequal shapes of equal area – 2 points. Problem 3. Correct algorithm – 7 points. Reasonable promotions, for example, measured 8 l - up to 3 points. Problem 4. Complete solution – 7 points. The solution in the figure (by cells, etc.) without sufficient explanation – 4-5 points. Correct solution with an arithmetic error – 4 points. The only answer is 0 points. Problem 5. a) Correct solution – 1 point. b) An example of the correct route is given (of course, not necessarily the same as in the solution above) and it is proven that the route cannot be longer - 6 points. An example of a correct route is given, but it is not proven that the route cannot be longer – 2 points.

All-Russian Olympiad for schoolchildren 2013-2014 in Moscow Typical tasks of the 1st (school) stage of the Olympiad in mathematics, grade 7. Brief solutions. 1. Vasya can get the number 100 using ten sevens, brackets and arithmetic signs:)7:77:77()7:77:77(100  . Improve his result: use fewer sevens and get the number 100 (It is enough to give one example.) Solution: For example: 1) 7:777:777100  , 2) 7:77:77777100  . There are other solutions. 2. It’s half past nine. What is the angle between the hour and minute hands? Answer. 750. Solution. At the moment when the clock shows half past nine, the minute hand points to the number 6, and the hour hand to the middle of the arc between the numbers 8 and 9 (see figure). If we draw two rays from the center of the clock to adjacent numbers on the dial, then between them there will be an angle of 3600:12=300. The angle between the clock hands when they show half past eight is two and a half times greater. Therefore, it is equal to 750. 3. Let's call a number mirror if from left to right it is “read” the same way as from right to left. For example, the number 12321 is a mirror number. a) Write some five-digit mirror number that is divisible by 5. b) How many five-digit mirror numbers are there that are divisible by 5? a) Solution. Any mirror number ending in 5. For example, 51715. b) Answer. 100. Solution. A number that is divisible by 5 must end in 5 or 0. A mirror number cannot end in 0, since then it must begin with 0. So, the first and last digits are 5. The second and third digits can be anything - from the combination 00 to the combination 99 - there are 100 options in total. Since the fourth digit repeats the second, in total different numbers will be 100. 4. Sasha, Lyosha and Kolya started at the same time in the 100 m race. When Sasha finished, Lyosha was ten meters behind him, and when Lyosha finished, Kolya was ten meters behind him. How far apart were Sasha and Kolya when Sasha finished? (It is assumed that all the boys are running at constant, but, of course, not equal speeds.) Answer. 19 m. Solution. Kolya's speed is 0.9 of Lyosha's speed. At the moment when Sasha finished, Lyosha ran 90 m, and Kolya 81909.0  m. Therefore, the distance between Sasha and Kolya was 19 m. 12 9 8 6 O9 8

5. The museum has 16 halls, located as shown in the picture. Half of them display paintings, and half exhibit sculptures. From any room you can get into any adjacent room (which has a common wall). During any tour of the museum, the halls alternate: a hall with paintings – a hall with sculptures – a hall with paintings, etc. The inspection begins in hall A, in which the paintings hang, and ends in hall B. a) Mark with crosses all the halls in which the paintings hang. Solution. See the picture. b) The tourist wants to see as many halls as possible (go from hall A to hall B), but at the same time visit each hall no more than once. What is the largest number of halls he can watch? Draw some route of his greatest length and prove that he could not have seen more halls. Answer. 15. Solution. One of the possible routes is shown in the figure. Let us prove that if a tourist wants to visit each hall no more than once, he will not be able to see more than 15 halls. Note that the route begins in the hall with paintings (A) and ends in the hall with paintings (B). This means that the number of halls with paintings that the tourist passed through is one more than the number of halls with sculptures. Since there are no more than 8 halls with paintings that a tourist could go through, there are no more than 7 halls with sculptures. So, the route cannot pass through more than 15 halls. X X X X X XX X B A B A

7th grade. Recommendations for verification. Each task is scored out of 7 points. Each score is an integer from 0 to 7. Below are some guidelines for checking. Naturally, compilers cannot foresee all cases. When evaluating a solution, one must proceed from whether the given solution is generally correct (although, perhaps, with shortcomings) - then the solution is scored at least 4 points. Or it is incorrect (although, perhaps, with significant progress) - in this case, the score should not be higher than 3 points. Task 1. Any correct example – 7 points. Two or more examples, some of which are true and some are false – 5 points. Problem 2. Correct solution – 7 points. Correct answer with insufficient justification – 5 points. An incorrect answer that is a multiple of 150 due to an arithmetic error, but the overall solution is correct – 3 points. Only answer – 1 point. Task 3. a) Any correct example of a number – 2 points. b) Correct reasoned decision – 5 points. Partially correct reasoning with an incorrect answer - up to 1-2 points. The only correct answer is 0 points. Problem 4. Complete solution – 7 points. It was found how far Kolya ran, but the distance between Sasha and Kolya was not found - 5 points. There has been significant progress, for example, the speed ratios of runners have been found to be 2-3 points. The only answer is 0 points. Problem 5. a) Correct solution – 1 point. b) An example of the correct route is given (of course, not necessarily the same as in the solution above) and it is proven that the route cannot be longer - 6 points. An example of the correct route is given, but it is not proven that the route cannot be longer – 2 points.

All-Russian Olympiad for schoolchildren 2013-2014 in Moscow Typical tasks of the 1st (school) stage of the Olympiad in mathematics, grade 8. Brief solutions. 1. Replace the asterisk (*) in the expression 2223 *)()2( xx) with a monomial so that after squaring and bringing similar terms, you get four terms. Solution: Replace the asterisk (*) with 2x:   2223)2()2(xxx  23436 4444 xxxxx 44 246  xxx 2. What is the ratio of the area of the shaded part to the white part? (The vertices of all squares except the largest are in the middles of the corresponding sides.) Answer 5:3. Solution. Consider the “quarter” of this figure (the upper right “quarter” is taken in the figure). Divide the shaded area into equal triangles as shown in the figure. The shaded area consists of five equal triangles, and the white area is made up of three similar equal triangles. Area ratio: 5:3. 3. Let's call a number mirror if from left to right it is “read” the same way as from right to left. For example, the number 12321 is a mirror number. a) Write some five-digit mirror number that is divisible by 5. b) How many five-digit mirror numbers are there that are divisible by 5? a) Solution. Any mirror number ending in 5. For example, 51715. b) Answer. 100. Solution. A number that is divisible by 5 must end in 5 or 0. A mirror number cannot end in 0, since then it must begin with 0. So, the first and last digits are 5. The second and third digits can be anything - from the combination 00 to the combination 99 - there are 100 options in total. Since the fourth digit repeats the second, there will be 100 different numbers in total. 4. Sasha, Lyosha and Kolya start at the same time in the 100 m race. When Sasha finished, Lyosha was ten meters behind him, and when Lyosha finished, Kolya was behind him ten meters. How far apart were Sasha and Kolya when Sasha finished? (It is assumed that all the boys are running at constant, but, of course, not equal speeds.) Answer. 19 m. Solution. Kolya's speed is 0.9 of Lyosha's speed. At the moment when Sasha finished, Lyosha ran 90 m, and Kolya 81909.0  m. Therefore, the distance between Sasha and Kolya was 19 m.

5. Petya cut a 2x1 paper parallelepiped along its edges and obtained a development. Then Dima cut off one square from this development, and nine squares remained, as in the picture. Where could the cut square be? Draw a full development and mark the cut square on it. (It is enough to give one correct version of the development). Solution. There are 5 options for where the cut square could be: 6. Each of the 10 gnomes either always tells the truth or always lies. It is known that each of them loves exactly one type of ice cream: cream, chocolate or fruit. First, Snow White asked those who like ice cream to raise their hands, and everyone raised their hands, then those who like chocolate ice cream - and half of the dwarfs raised their hands, then those who like popsicles - and only one dwarf raised their hand. How many of the gnomes are truthful? Answer. 4. Solution. The gnomes who always tell the truth raised their hand once, and the gnomes who always lie raised their hand twice. A total of 16 hands were raised (10+5+1). If all the dwarves told the truth, 10 hands would be raised. If one truthful dwarf is replaced by one liar, then the number of raised hands will increase by 1. Since 6 “extra” hands were raised, then 6 dwarves lied, and 4 told the truth.

8th grade. Recommendations for verification. Each task is scored out of 7 points. Each score is an integer from 0 to 7. Below are some guidelines for checking. Naturally, compilers cannot foresee all cases. When evaluating a solution, one must proceed from whether the given solution is generally correct (although, perhaps, with shortcomings) - then the solution is scored at least 4 points. Or it is incorrect (although, perhaps, with significant progress) - in this case, the score should not be higher than 3 points. Problem 1. Correct solution – 7 points. The monomial was found correctly, the correct solution with one error (misprint) when squaring – 4 points. A monomial is found, but it is not explained why it is a solution – 2 points. Problem 2. Correct solution – 7 points. The areas of the included figures are partially found – 2 points. The only answer is 0 points. Task 3. a) Any correct example of a number – 2 points. b) Correct reasoned decision – 5 points. Partially correct reasoning with an incorrect answer - up to 1-2 points. The only correct answer is 0 points. Problem 4. Complete solution – 7 points. It was found how far Kolya ran, but the distance between Sasha and Kolya was not found - 5 points. There has been significant progress, for example, the speed ratios of runners have been found to be 2-3 points. The only answer is 0 points. Task 5. One correct example of a sweep – 7 points. No justification required. Problem 6. Complete solution – 7 points. The correct answer received on specific example– 2 points.

All-Russian Olympiad for schoolchildren 2013-2014 in Moscow Typical tasks of the I (school) stage of the Olympiad in mathematics, grade 9. Brief solutions. 1. Replace the asterisk (*) in the expression 2324 *)()3( xx) with a monomial so that after squaring and bringing similar terms, you get four terms. Solution: Replace the asterisk (*) with 3x:   2324)3()3(xxx  24648 9696 xxxxx 99 268  xxx 2. What is the ratio of the area of the shaded part to the white part? (The vertices of all squares except the largest are in the middles of the corresponding sides.) Answer 5:3. Solution. Consider the “quarter” of this figure (the upper right “quarter” is taken in the figure). Divide the shaded area into equal triangles as shown in the figure. The shaded area consists of five equal triangles, and the white area - of three such same equal triangles. Area ratio: 5:3. 3. Let's call a number mirror if from left to right it is “read" in the same way as from right to left. For example, the number 12321 is a mirror number. How many five-digit mirror numbers are there that are divisible by 5? Answer 100. Solution: A number that is divisible by 5 must end in 5 or 0. A mirror number cannot end in 0, since then it must begin with 0. So, the first and last digits are 5. The second and third digits can be anything - from the combination 00 to the combination 99 - there are 100 options in total. Since the fourth digit repeats the second, there will be 100 different numbers in total. 4. Vasya thought of two numbers. Their sum is equal to their product and equal to their quotient. What numbers did Vasya have in mind? Answer. 2 1, 1. Solution. Let's denote the numbers x and y. Then, according to the conditions of the problem: y x xyyx  . From the equation y x xy  it follows that either 0x and 0y, or 12 y, and x is any. At 0x, it follows from the equation xyyx  that 0y, a contradiction. From equation 12 y we obtain that either 1y or 1y. For 1y, the equation xyyx  has no solutions, and for 1y, from the equation xyyx  we obtain 2 1 x .

5. Petya cut a 2x1 paper parallelepiped along its edges and obtained a development. Then Dima cut off one square from this development, and nine squares remained, as in the picture. Where could the cut square be? Draw a full development and mark the cut square on it. It is enough to give one correct version of the scan. Solution. There are 5 options for where the cut square could be: 6. Each of the 10 gnomes either always tells the truth or always lies. It is known that each of them loves exactly one type of ice cream: cream, chocolate or fruit. First, Snow White asked those who like ice cream to raise their hands, and everyone raised their hands, then those who like chocolate ice cream - and half of the dwarfs raised their hands, then those who like popsicles - and only one dwarf raised their hand. How many of the gnomes are truthful? Answer. 4. Solution. The gnomes who always tell the truth raised their hand once, and the gnomes who always lie raised their hand twice. A total of 16 hands were raised (10+5+1). If all the dwarves told the truth, 10 hands would be raised. If one truthful dwarf is replaced by one liar, then the number of raised hands will increase by 1. Since 6 “extra” hands were raised, then 6 dwarves lied, and 4 told the truth.

9th grade. Recommendations for verification. Each task is scored out of 7 points. Each score is an integer from 0 to 7. Below are some guidelines for checking. Naturally, compilers cannot foresee all cases. When evaluating a solution, one must proceed from whether the given solution is generally correct (although, perhaps, with shortcomings) - then the solution is scored at least 4 points. Or it is incorrect (although, perhaps, with significant progress) - in this case, the score should not be higher than 3 points. Problem 1. Correct solution – 7 points. The monomial was found correctly, the correct solution with one error (misprint) when squaring – 4 points. A monomial is found, but it is not explained why it is a solution – 2 points. Problem 2. Correct solution – 7 points. The areas of the included figures are partially found – 2 points. The only answer is 0 points. Task 3. Correct reasoned decision – 7 points. Partially correct reasoning with an incorrect answer - up to 2-3 points. The only correct answer is 0 points. Problem 4. Complete solution – 7 points. The solution is correct, one of the numbers is found correctly, an arithmetic error in the last action - 4 points. An answer that verifies the conditions of the problem, but without explaining why the pair of numbers is unique – 3 points. The only answer is 0 points. Task 5. One correct example of a sweep – 7 points. No justification required. Problem 6. Complete solution – 7 points. The correct answer obtained from a specific example is 2 points.

All-Russian Olympiad for schoolchildren 2013-2014 in Moscow Typical tasks of the 1st (school) stage of the Olympiad in mathematics, grades 10-11. Brief solutions. 1. If the number 10 100 is written as a sum of tens (10+10+10+...), then how many terms will there be? Answer. 19 10. Solution. 10 100 = 20 10 = 19 1010 . This means there will be 19 10 terms in total. 2. What is the ratio of the area of the shaded area to the white area? (The vertices of all squares except the largest are in the middle of the corresponding sides). Answer. 5:3. Solution. Let's look at the “quarter” of this figure (the upper right “quarter” is taken in the figure). Let's divide the shaded area into equal triangles as shown in the figure. The shaded area consists of five equal triangles, and the white area consists of three equal triangles. Area ratio: 5:3. 3. Vasya thought of two numbers. Their sum is equal to their product and equal to their quotient. What numbers did Vasya have in mind? Answer. 2 1, 1. Solution. Let's denote the numbers x and y. Then, according to the conditions of the problem: y x xyyx  . From the equation y x xy  it follows that either 0x and 0y, or 12 y, and x is any. At 0x, it follows from the equation xyyx  that 0y, a contradiction. From equation 12 y we obtain that either 1y or 1y. For 1y, the equation xyyx  has no solutions, and for 1y, from the equation xyyx  we obtain 2 1 x . 4. Each of the 10 dwarves either always tells the truth or always lies. It is known that each of them loves exactly one type of ice cream: cream, chocolate or fruit. First, Snow White asked those who like ice cream to raise their hands, and everyone raised their hands, then those who like chocolate ice cream - and half of the dwarfs raised their hands, then those who like popsicles - and only one dwarf raised their hand. How many of the gnomes are truthful? Answer. 4. Solution. The gnomes who always tell the truth raised their hand once, and the gnomes who always lie raised their hand twice. A total of 16 hands were raised (10+5+1). If all the dwarves told the truth, 10 hands would be raised. If one truthful dwarf is replaced by one liar, then the number of raised hands will increase by 1. Since 6 “extra” hands were raised, then 6 dwarves lied, and 4 told the truth.

5. Plot the graph of function 22)1()( xxy. Solution. Function 22)1()( xxy is defined at 0x. Let’s transform it to the form 1 xxy. At 1x 12  xy, with 10  x 1y. The graph is shown in the figure: 6. In a quadrilateral, the diagonals are perpendicular. A circle can be inscribed in it and a circle can be described around it. Can we say that this is a square? Answer: No. Solution Let us consider the diameter AC in a circle and the chord BD perpendicular to it, which does not pass through the center (see figure). Let us show that the quadrilateral ABCD satisfies the conditions of the problem. To do this, it is enough to prove that a circle can be inscribed in it. In a circle, the diameter divides the chord perpendicular to it in half, which means that in triangle BAD the height is the median and this triangle is isosceles: AB = AD. Similarly, CB = CD. Since the sums of the opposite sides of the quadrilateral ABCD are equal, a circle can be inscribed in it. Y X 1 1 A C D B

10-11 grade. Recommendations for verification. Each task is scored out of 7 points. Each score is an integer from 0 to 7. Below are some guidelines for checking. Naturally, compilers cannot foresee all cases. When evaluating a solution, one must proceed from whether the given solution is generally correct (although, perhaps, with shortcomings) - then the solution is scored at least 4 points. Or it is incorrect (although, perhaps, with significant progress) - in this case, the score should not be higher than 3 points. Problem 1. Correct solution – 7 points. Only answer - 2 points. Problem 2. Correct solution – 7 points. The areas of the included figures are partially found – 2 points. The only answer is 0 points. Problem 3. Complete solution – 7 points. The solution is correct, one of the numbers is found correctly, an arithmetic error in the last action – 4 points. An answer that verifies the conditions of the problem, but without explaining why the pair of numbers is unique – 2 points. The only answer is 0 points. Problem 4. Complete solution – 7 points. The correct answer obtained from a specific example is 2 points. Problem 5. Correct solution – 7 points. The signs of the radicals have been “removed” correctly, but the graph is partially incorrect – 2-3 points. In general, a correct graph obtained “point by point”, without justification for why the graph consists of combining a segment and a ray – 2 points. Problem 6. Complete solution – 7 points. A correct example of a figure, but one of the properties is not proven (inscribed, circumscribed or perpendicular to the diagonals) – 3 points. Correct “picture” – 1 point.

The museum has 16 halls located. I (school) stage of the All-Russian Mathematical Olympiad with solutions and criteria

History of the Russian state in the XIV-XVI centuries. Hall 13 Formation of a unified Russian state in the XIV-XVI centuries

Hall 14 Russian culture of the XIV - early XVI centuries. History of the Russian state in the XIV-XVI centuries

Hall 15 Russian city and Russian village in the 15th–17th centuries. Feudal relations in the Russian state. History of the Russian state in the XIV-XVI centuries

Hall 16. Craft and trade of the 16th–17th centuries. History of the Russian state in the XIV-XVI centuries

“Evaluation plus example Evaluation plus example is a special mathematical reasoning that is used in some problems on...”

History of the Russian state in the XIV-XVI centuries. Hall 13
Formation of a unified Russian state in the XIV-XVI centuries

Hall 14
Russian culture of the XIV - early XVI centuries. History of the Russian state in the XIV-XVI centuries

Hall 15
Russian city and Russian village in the 15th–17th centuries. Feudal relations in the Russian state. History of the Russian state in the XIV-XVI centuries

Hall 16.
Craft and trade of the 16th–17th centuries. History of the Russian state in the XIV-XVI centuries