*115*
*2*
*3MB*

*English*
*Pages [110]*
*Year 2022*

- Author / Uploaded
- Evdokimov M.A.

Library ofjournal the journal «Kvantik» Library of the «Kvantik» LIBRARY

OF THE JOURNAL KVANTIK

M. A. EVDOKIMOV

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ILLUSTRATED BY N. N. KRUTIKOV

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COLORS OF MATH

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Library of the journal «Kvantik» Release 1

Mikhail Evdokimov

A hundred colors of math Illustrated by N. N. Krutikov

Moscow MCCME

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Mikhail Evdokimov A hundred colors of math. — М.: MCCME, 2022. — 109 p.: ill. ISBN 978-4439-5-4631-3 The book contains hundred test-problems. All problems have answer options to choose from but correct answer are usually not those that come to mind first. Most of the problems were composed by the author and proposed at various mathematical olympiads or published in the journal «Kvantik». All tasks are illustrated with entertaining pictures. This book is for a wide range of readers. Illustrator N. N. Krutikov Cover: Yustas

Translation from the Russian language edition: Евдокимов М. А., Сто граней математики, Москва, МЦНМО, 2020, ISBN 978-4439-5-2916-3 Translation by the author, R.S. Bulavin, M.V. Nekrashevich.

ISBN 978-4439-5-4631-3

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М. А. Evdokimov (text), 2022 N. N. Krutikov (illustrations), 2022 MCCME, 2022 Yustas (cover), 2022

PREFACE The book contains hundred test-problems. All problems have answers options to choose from but correct answers are usually not those that come to mind first. The given problems are very diverse in the methods of their solution and have deep connections with various branches of mathematics. The level of difficulty increases with the growth of the number from relatively simple (1-20) to very difficult problems (80-100). Most of them were composed by the author and proposed at various mathematical olympiads or published in the journal «Kvantik» (vk.com/kvantik12 and facebook.com/kvantik12). These are № 2, 6, 10, 11, 12, 14, 17, 31, 44, 48, 60, 63 etc. There are also some great problems by other authors: A. Tolpygo (16), A. Romanov (18), S. Tokarev (19, 78), P. Kozhevnikov (29), R. Zhenodarov (68), I. Mitrofanov (93), M. Garber (95), M. Ostrovsky (98), A. Shapovalov (99), O. Kosukhin (100). Some problems are so-called «mathematical folklore» and their authorship is difficult to establish. I would like to thank the editors of Kvantik, the illustrator N. Krutikov and the MCCME publishing house for their help in preparing this book. I wish you to enjoy reading! August 2021

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Problems

No, Simon, you, of course, amazed me, but you didn’t answer the question

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Problems

1. Watermelons Watermelons weigh 180 kg and almost entirely consist of water (99%). Over time, watermelons dried out and the water content dropped by one percentage point (to 98%). How much do the watermelons weigh now?

Dan, where’s the watermelon? Dried out …

Answer options: A. Approx. 178 kg B. Approx. 176 kg C. 162 kg D. 90 kg E. The correct answer is different 5 Powered by TCPDF (www.tcpdf.org)

Problems

2. Conversation At a conference Alex, Ben and Charles represent one of two competing firms Megasoft and Gamesoft each. Representatives of the same company always tell the truth to each other and lie to their competitors. Alex told Ben: «Charles is from Megasoft.» Ben replied, «Me too.» Where does Alex work?

Answer options: A. Megasoft B. Gamesoft C. Microsoft D. There is no definite answer E. Alex is unemployed 6 Powered by TCPDF (www.tcpdf.org)

Problems

3. A sheet of paper Suppose that a large sheet of paper 0.1 mm thick was folded in half. Then it was folded in half once again, and so on, 50 times. What is the approximate thickness of the folded sheet?

I decided to take more paper. Just in case the task will be difficult

Answer options: A. 5 mm B. 5 cm C. 5 m D. 5 km E. Approximately the distance from the Earth to the Sun 7 Powered by TCPDF (www.tcpdf.org)

Problems

4. Clock hands How many times during the day do the hour and minute hands form a right angle?

Man, and just where’s the angle in here?

Answer options: A. 22 B. 24 C. 44 D. 48 E. I have a different answer 8 Powered by TCPDF (www.tcpdf.org)

Problems

5. Table Tennis Kurt, Michael and Arnold played several games of table tennis: the player who loses the game gives way to the player who did not participate in it. It turned out that Kurt played 8 games, and Michael played 17. Who lost the fifth game?

Answer options: A. Kurt B. Michael C. Arnold D. Not enough information for a definite answer E. The one who plays the worst 9 Powered by TCPDF (www.tcpdf.org)

Problems

6. How many children are in the family? In a large family, every child was asked: «How many brothers do you have?» Each one named a positive integer, and the sum of all these numbers was equal to 35. How many children were in the family, if all of them answered correctly?

Jim, What’s the problem? The question is simple: «How many brothers do you have?»

Answer options: A. 5 B. 7 C. 8 D. Not enough information for a definite answer E. This is impossible! Someone made a mistake 10 Powered by TCPDF (www.tcpdf.org)

Problems

7. Island There are 250 people living on the island. Some of them always lie, while the rest always tell the truth. Every one of them worships one of the gods, namely the god of the Sun, the god of the Moon or the god of the Earth. All of them were asked three questions: 1. Do you worship the Sun god? 2. Do you worship the Moon god? 3. Do you worship the Earth god? The first question was answered affirmatively by 140 people, the second one, by 120 people and the third, by 110 people. How many liars are there on the island?

And you, from what I can tell, worship the god of Food?

Answer options: A. 140 B. 130 C. 120 D. 110 E. Not enough information to answer 11 Powered by TCPDF (www.tcpdf.org)

Problems

8. Engineer goes to work An engineer arrives by train at the station at 8 a.m. every day. At exactly 8 o’clock, a car drives up to the station and takes the engineer to the factory. Once an engineer arrived at the station at 7 a.m. and walked to the factory. He met the car on his way, got into it and arrived at the factory 20 minutes earlier than usual. At what time did the engineer meet the car?

Answer options: A. 7:30 a.m. B. 7:40 a.m. C. 7:50 a.m. D. The correct answer is different E. Not enough data for a definite answer 12 Powered by TCPDF (www.tcpdf.org)

Problems

9. Calculation The Robot Kvantik found the product of digits for each twodigit integer, and then calculated the sum of all the products. What number did he get in the end?

Let’s integrate the complex function along the contour and use the Ramanujan formula

Answer options: A. 1225 B. 2125 C. 2215 D. The correct answer is different E. How do I know, I’m not a robot! 13 Powered by TCPDF (www.tcpdf.org)

Problems

10. Unusual match After the soccer match (each team had 10 players) between the team of liars (always lie) and the team of truth-tellers (always tell the truth), each player was asked: «How many goals did you score?» As a result, some participants of the match answered «one», Bill said «two», many answered «three», and the rest said «five». Is Bill lying if the truth-tellers won with a score of 20:17 (there were no own goals in that match)?

And how did you score 37 goals if your team won with a score of 20:17?

Oh, I’ve also scored 17 more own goals!

Answer options: A. Bill lies B. Bill tells the truth C. It is not clear D. I have a different answer E. Where on Earth have you seen such a match?! 14 Powered by TCPDF (www.tcpdf.org)

Problems

11. Ice cream In the city of Catchfools, coins of value of 1, 2, 3, ... 19 and 20 soldi are used (no others). Pinocchio had one coin. He bought an ice cream and received one coin in change. Then he bought a same ice cream and received change in three different coins. He wanted to buy a third ice cream, but he didn’t have enough money. How much does the ice cream cost?

No ice cream, Malvina! We’ll bury the coins and tomorrow we’ll buy three times more

Answer options: A. 4 soldi B. 5 soldi C. 6 soldi D. 7 soldi E. Not enough data for a definite answer 15 Powered by TCPDF (www.tcpdf.org)

Problems

12. Specify the card All 36 cards of a deck are laid face down forming a 6 × 6 «square», as shown in the figure. Each turn, a player can choose 9 cards that form a 3 × 3 «square», and find out their suits and values (without specifying where exactly each card is). What is the least number of turns you need to identify a corner card?

Answer options: A. 2 B. 3 C. 4 D. 5 E. This cannot be done! 16 Powered by TCPDF (www.tcpdf.org)

Problems

13. Conversation The following conversation took place between colleagues Alex, Ben and Charles. Alex: «Our boss has read more than 10 books.» Ben: «No, he has read less than 10 books.» Charles: «He has definitely read at least one book.» How many books did the boss read if exactly one of the three statements is true?

Are you sure that he can read at all?

Answer options: A. None B. One C. Ten D. The correct answer differs from answers A, B, C E. Do they work there or just read books?! 17 Powered by TCPDF (www.tcpdf.org)

Problems

14. Electronic display Ben arrived at the airport, looked at the electronic display, which showed the time (hours and minutes), and noticed that four different digits were on the display. The next time he looked at the display, four other different digits were there. What is the minimal possible time period between these two times?

Answer options: A. 24 min B. 36 min C. 1 h 12 min D. 1 h 47 min E. The correct answer is different 18 Powered by TCPDF (www.tcpdf.org)

Problems

15. Chessboard In one move, you can select several rows and columns (perhaps only some rows or some columns) of a chessboard and change the color of all selected squares (black to white and white to black). What is the minimal number of moves necessary to make chessboard white?

What are you painting?

Answer options: A. 2 B. 4 C. 8 D. 16 E. The correct answer is different 19 Powered by TCPDF (www.tcpdf.org)

Problems

16. The greedy brothers There are 4 apples weighing 600 g, 400 g, 300 g and 250 g on a plate. Two brothers are going to eat them. The elder brother has the right to choose; he takes one apple and begins to eat it. Immediately after him, the younger brother takes one of the remaining apples and begins to eat it. The brothers eat apples equally fast and the eating time is proportional to the weight of apples. Anyone who has eaten his apple has the right to take the next of the remaining apples. What apple should the elder brother take in the beginning in order to eat as much as possible?

Answer options: A. The 600 g apple B. The 400 g apple C. The 300 g apple D. The 250 g apple E. The whole plate, because he is stronger 20 Powered by TCPDF (www.tcpdf.org)

Problems

17. Runaway cell The figure depicted in the picture (a 6 × 6 square, in which the top row is moved by 1 square), was cut along the grid lines into several identical parts which could be put together to form a square 6 × 6 (the parts are allowed to be turned over). What is the minimal possible number of such identical parts?

Cut, please

Answer options: A. 2 B. 3 C. 4 D. 5 E. 6 21 Powered by TCPDF (www.tcpdf.org)

Problems

18. Volleyball circle There are twelve children in a volleyball circle. For each game, the coach divides them into two teams of 6 people. He wants to hold several games, so that everyone plays with everyone else in the same team. What is the minimal number of games to do this?

Great volleyball player! And, most important, he doesn’t stop in his professional growth

Answer options: A. 3 B. 6 C. 12 D. The correct answer is different E. It will require too many games! 22 Powered by TCPDF (www.tcpdf.org)

Problems

19. Knights and liars There are knights (who always tell the truth) and liars (who always lie) on the island. The traveler met three inhabitants of the island and asked each of them: «How many knights are among your companions?» The first answered: «Not a single one.» The second said: «One.» What did the third say?

And who of you is a knight?

Guess who?

Answer options: A. «None» B. «One» C. «Two» D. Not enough information E. «I don’t know these people» 23 Powered by TCPDF (www.tcpdf.org)

Problems

20. Trap When the criminal passed 3/8 of a bridge, he noticed a police car approaching him at a speed of 60 km/h. If he runs back, he will meet the car at the beginning of the bridge. If he runs forward, the car will catch him at the end of the bridge. How fast is the criminal running?

Answer options: A. 12 km/h B. 15 km/h C. 18 km/h D. 20 km/h E. I have a different answer 24 Powered by TCPDF (www.tcpdf.org)

Problems

21. Three hunters Three hunters cooked porridge. The first gave two mugs of cereal, the second gave one, the third gave none, but he paid with seven gun cartridges. How should the first two hunters share them if everyone ate equally?

Answer options: A. 4 to the first and 3 to the second B. 5 to the first and 2 to the second C. 6 to the first and 1 to the second D. All cartridges to the first E. 4 to the first, 2 to the second, and shoot in the air with the remaining cartridge 25 Powered by TCPDF (www.tcpdf.org)

Problems

22. Average speed The road between the two mountain villages A and B goes sometimes uphill, sometimes downhill. The old bus, which had an average speed of 30 km/h uphill and 60 km/h downhill, drove from A to B and back. What was its average speed on all the way?

Are you still sure it will be faster than on a bus?

Answer options: A. 40 km/h B. 45 km/h C. 50 km/h D. Not enough data for a definite answer E. The old bus will definitely break down on the road 26 Powered by TCPDF (www.tcpdf.org)

Problems

23. New Year promotion, or fooled buyers In the store that buys Santa Claus caps from the supplier in wholesale and sells them in retail, there is a New Year’s promotion: when you buy two caps, you get a 20% discount, and when you buy three caps, you get a 30% discount on the purchase price. What is the store’s margin in percent (by how much is the retail price higher than the wholesale price) if the store has the same profit from each promotion sale?

Two caps — 20% discount, three caps — 30%... Thus, if I take three thousand caps, you have to pay me then…

Answer options: A. 25% B. 50% C. 75% D. 100% E. The correct answer is different 27 Powered by TCPDF (www.tcpdf.org)

Problems

24. Peasants and potatoes Three peasants went into the inn to rest and dine. They ordered the hostess to cook potatoes, and fell asleep. The hostess did not wake up the guests, but put the food on the table and left. One of peasants woke up, ate his share and fell asleep again. Then the second peasant woke up, counted the potatoes, ate the third part and fell asleep. Then the third peasant woke up, counted the potatoes and ate the third part. Then all of them woke up and saw that there were still 8 potatoes left in the cup. How many potatoes should each of them eat now so that they all eat the same number of potatoes?

Answer options: A. The first takes one, the second takes three, and the third takes the rest B. The second takes two, and the third takes the rest C. The second takes three, and the third takes the rest D. I have a different answer E. The strongest will take everything 28 Powered by TCPDF (www.tcpdf.org)

Problems

25. Soccer ball The soccer ball is made of leather parts: black pentagons and white hexagons (see the figure). John easily counted that there are exactly 12 black pentagons. And how many white hexagons are there? (You can only use the picture below to answer).

Answer options: A. 16 B. 18 C. 20 D. 24 E. I do not like soccer 29 Powered by TCPDF (www.tcpdf.org)

Problems

26. Card game There is a deck of 52 cards on the table. Alex, Ben and Charles are allowed to take either one or two cards from the deck. Whoever draws the last card wins the game. Alex goes first, then Ben, then Charles, then Alex again and so on in a circle. Does any of the players have a winning strategy, even if other two agreed to coordinate their actions?

And where is Alex?

There he is

Answer options: A. Alex has B. Ben has C. Charles has D. No one has E. I don’t know, I don’t play card games 30 Powered by TCPDF (www.tcpdf.org)

Problems

27. Nothing at all The expert has a bag with golden sand, two-cup scales and the only one 1-gram measuring weight. What is the minimal number of weighings to measure out exactly 100 grams of golden sand?

Can I borrow your weights? I need to weigh something

Answer options: A. 7 B. 8 C. 10 D. 100 E. The correct answer is different 31 Powered by TCPDF (www.tcpdf.org)

Problems

28. Volleyball The volleyball tournament was held in one round (each team played with each other just once; draws are impossible in volleyball). 20% of all teams did not win a single game. How many teams participated in this tournament?

How many times do I have to repeat? You need to hit the ball, not the head!

Answer options: A. 20 B. 10 C. 5 D. Not enough data for a definite answer E. I don’t like volleyball 32 Powered by TCPDF (www.tcpdf.org)

Problems

29. Renovation Aunt Stacy bought a roll of wallpaper with a 15-cm radius on a 5-cm radius coil (i.e., the roll thickness on the coil was 10 cm). After she covered with wallpaper half of the walls in the room, the roll thickness on the coil became 5 cm (i.e., the radius of the roll itself became 10 cm). «Well, half a roll is used, just enough for the second half is left» — thought aunt Stacy. What part of the room will the remaining part of the roll really cover?

Answer options: A. 25% B. 30% C. 33% D. 40% E. Aunt Stacy is right! 33 Powered by TCPDF (www.tcpdf.org)

Problems

30. Marked numbers Someone marked several natural numbers from 1 to 100 so that for any two marked numbers neither their sum nor their product is divisible by 100. How many numbers maximum could be marked?

What’s wrong, the Thinker? Is it a difficult problem?

Answer options: A. 50 B. 48 C. 45 D. 44 E. The correct answer is different 34 Powered by TCPDF (www.tcpdf.org)

Problems

31. Who will get the prize? Two players have a cubic cardboard box which has a prize inside. Players take turns choosing one of the edges of the box and then cut the box along the chosen edge. If it’s possible to take the prize out of the box after the cut, the person who made the last cut wins. Does any of the players have a winning strategy in such a game? The box opens if it’s cut along three edges of the same face.

I will win next time for sure!!!

Answer options: A. The first person has (the one who goes first) B. The second person has C. No one has D. We didn’t study it yet E. Everything ends up as it’s shown on the picture 35 Powered by TCPDF (www.tcpdf.org)

Problems

32. Ten friends Jake and his nine friends live on the shore of a round lake in 10 houses which are located every 100 meters along the shore as shown in the picture. One day Jake has decided to bring his friends together. He chooses a house where he has not been yet, goes there and takes a friend with him, then he chooses the next house, etc. Jake always follows the shortest route between the two houses, but the sequence of visits can be random. What maximum distance could have Jake passed by the moment when all of them gathered together at Jake’s last friend’s house?

I guess, it will be faster to bring friends together by swimming across the lake

Answer options: A. 4 km B. 4.1 km C. 4.2 km D. 4.5 km E. 5 km 36 Powered by TCPDF (www.tcpdf.org)

Problems

33. The chest with golden coins Three pirates shared a chest with golden coins. The first one took 30% of all coins, then the second one took 40% of the rest, then the third one took 50% of the rest. Then the pirates discovered that there were still 63 coins left in the chest. How many coins were there at the beginning?

I think it’s fair. The smartest one gets everything

Answer options: A. 300 B. 600 C. 650 D. 900 E. My answer is different 37 Powered by TCPDF (www.tcpdf.org)

Problems

34. Where should the firefighters go? All the residents of town A always tell the truth, all the town B residents always lie and the town C residents can either tell the truth or lie, but if they make two statements, one of them is always true and the other is false. Someone called the fire department which serves all three towns: «There is a fire in our town, please come as soon as possible!» «Where is the fire?» — asked a firefighter. «The town C!» — replied someone. What should the firefighters do if the fire is real?

Answer options: A. Go to the town A B. Go to the town B C. Go to the town C D. Not enough information to answer E. Do nothing. Firefighters have no time for puzzles! 38 Powered by TCPDF (www.tcpdf.org)

Problems

35. An escalator The bully Bob ran down the moving escalator in the mall and counted 20 stairs. When he ran up the escalator with the same speed relative to the escalator, he counted 80 stairs. How many stairs did Bob count while he was going down the still escalator along with the security guard?

Did you count all the stairs? Now let’s count how much your parents have to pay as a fine

Answer options: A. 32 B. 36 C. 40 D. 50 E. He certainly didn’t count stairs at that moment 39 Powered by TCPDF (www.tcpdf.org)

Problems

36. Cutting into equal triangles Terry cuts an equilateral hexagon into identical triangles. What minimal number of triangles can Terry get?

You got no triangles at all for some reason

Answer options: A. 2 B. 3 C. 4 D. 5 E. 6 40 Powered by TCPDF (www.tcpdf.org)

Problems

37. On some island 40% of adult men and 60% of adult women on the island are single. What part of the adult population of the island is single (polygamy and same-sex marriage are prohibited on the island)?

As a sign of your love, please exchange your nose rings Cheers!

Cheers!

Answer options: A. 48% B. 50% C. 52% D. 68.5% E. It’s too early for me to think about marriage 41 Powered by TCPDF (www.tcpdf.org)

Problems

38. The city scheme There is a road map of a certain city on the figure below: there are two ring roads (two circles with the same center) and 6 roads that come to the center at equal angles. Dale thinks how to get from the point A to the point B: either use the external ring road or the internal one. Which of these two ways is shorter?

And what is there to discuss? Just turn on the navigator and go

Answer options: A. They are of the same length B. Way through the internal ring road C. Way through the external ring road D. The answer depends on the ratio of radii E. The subway is faster anyway! 42 Powered by TCPDF (www.tcpdf.org)

Problems

39. The meeting Two brothers Rob and Pete were walking towards each other at a speed of 5 km/h each. When the distance between the brothers became 1 km, the dog Buddy walking along with Rob, noticed Pete and rushed to him at a speed of 20 km/h. When Buddy reached Pete, he turned back and ran towards Rob, and so on, until the brothers met. What distance did Buddy run?

Answer options: A. 1 km B. 2 km C. 3 km D. 4 km E. Need to sum up an infinite sequence 43 Powered by TCPDF (www.tcpdf.org)

Problems

40. Where do we build a school? Three villages A, B and C are located along the road as shown in the figure (B is between A and C: 1 km away from A and 2 km away from C). IIt was decided to build a school minimizing the total distance that all children walk from villages to school and back. Where should the school be built if there are 80, 50 and 30 children who need to go to school in villages A, B and C respectively?

The school is pretty far from our village. This is the only way we can get there

Answer options: A. Village A, as the most children live there B. In the middle between A and B C. In any point between A and C D. The correct answer is different E. Village C, since the principal’s children live there 44 Powered by TCPDF (www.tcpdf.org)

Problems

41. Holiday Mom was baking pancakes for a holiday. After some time, her husband and two sons came into the kitchen and started to eat pancakes, so there were no pancakes left in half an hour (while mother continued to bake pancakes). If only the two sons had come, then the pancakes would be eaten in an hour. How much time would it take to eat all the pancakes if only the father had come (the speed of eating pancakes by the father is the same as that of each of his sons)?

I wonder when would pancakes end if Charlie and Luna also came?

Answer options: A. 1.5 hours B. 2 hours C. 2.5 hours D. 3 hours E. The pancakes will never end 45 Powered by TCPDF (www.tcpdf.org)

Problems

42. Crossing the bridge The night falls. A boy, his dad, mom and granny are on the same bank of the river and they want to cross the bridge to the other bank. They only have one flashlight. Two people maximum can walk along the bridge at the same time and they have to have the flashlight with them. Dad is able to cross the bridge walking alone in 1 minute, the boy in 2 minutes, mom in 5 minutes, granny in 10 minutes. What is the minimal time for all of them to cross the bridge?

Answer options: A. 13 minutes B. 15 minutes C. 17 minutes D. 19 minutes E. My answer is different 46 Powered by TCPDF (www.tcpdf.org)

Problems

43. 100 boxers There are 100 boxers who participate in the Olympic tournament. A boxer who lost a boxing match does not participate in other matches. A certain tournament grid (match schedule) was made for this tournament. How many matches will it take to reveal the final winner?

Answer options: A. 99 B. 100 C. 127 D. It depends on the tournament grid E. My answer is different 47 Powered by TCPDF (www.tcpdf.org)

Problems

44. Tick-Tac-Toe What is the largest number of crosses that can be placed on a 5 × 5 table so that no three crosses stand «in a row» horizontally, vertically or diagonally?

Seems like a difficult task with crosses again

Answer options: A. 15 B. 16 C. 17 D. 18 E. The correct answer is different 48 Powered by TCPDF (www.tcpdf.org)

Problems

45. Measuring weights What is the minimal number of measuring weights necessary to measure out 1 g, 2 g, ..., 39 g, 40 g of golden sand on a two-cup scale (any integer from 1 to 40) in one weighing? Note: the measuring weights may be different.

Please weigh me a kilogram of gold! I’m in a real hurry!

Answer options: A. 3 B. 4 C. 5 D. 6 E. The correct answer is different 49 Powered by TCPDF (www.tcpdf.org)

Problems

46. Julia is having fun Julia invited Jake and Pete on her birthday party and baked a cake in the shape of a regular hexagon (top view). She cut the cake as shown on the figure (each cut goes through the vertex and the middle of the respective side) and gave one of the selected pieces to Jake (triangular) and the other one to Pete (quadrangular). Which one of the Julia’s guests got the largest piece of cake?

I think I know the correct answer

Answer options: A. Pieces are the same B. Pete C. Jake D. The one she likes the most! E. Julia’s father, since he had to eat up the rest of the cake 50 Powered by TCPDF (www.tcpdf.org)

Problems

47. Ghosts One school has 1000 school lockers with numbers from 1 to 1000 which are locked up at night. There are 1000 ghosts in this school. Exactly at midnight the first ghost opens all the lockers. After that the second one locks up all the lockers with numbers divisible by 2. Then the third one changes lockers’ state (if the locker is open the ghost closes it and if it is locked the ghost opens it) if the locker number is divisible by 3, etc. Finally, the 1000th ghost changes the state of the locker with the number 1000, after which all the ghosts disappear. How many lockers will remain open?

Could you please clean up a bit faster? I still need to deal with the lockers

Answer options: A. 20 B. 31 C. 42 D. 50 E. I need a computer to calculate this 51 Powered by TCPDF (www.tcpdf.org)

Problems

48. Four logicians and cards Four logicians A, B, C and D sit at the round table in this exact order (if you move clockwise). They were shown nine cards of the same suit (six, seven, ..., king, ace) and then the cards were mixed and given out to the logicians (each logician got 1 random card) so that everyone could see only his card. Then they were asked the same question in turn: «Is your card higher than your neighbor’s on the right?». After that A, B, C and D took turns saying «I don’t know». What card does D have?

Answer options: A. Nine B. Ten C. Jack D. Queen E. There is no definite answer 52 Powered by TCPDF (www.tcpdf.org)

Problems

49. Airlines The airline system of a country is designed in such a way that any city is connected by an airline route to no more than three other cities, and one can fly from any city to any one other stopping at no more than one other city. What is the largest number of cities in this country?

Whatever, I’ll fly by myself

Answer options: A. 6 B. 8 C. 10 D. 12 E. The correct answer is different 53 Powered by TCPDF (www.tcpdf.org)

Problems

50. Horse racing Three horses participate in a race: Alla, Bella and Viola. Bets on their victory are taken in the ratio of 1:1, 1:2 and 1:6, respectively. This means that if, for example, you bet on Bella and she comes in first, then you get back your money plus the doubled initial bet. Otherwise, you lose your money. Holmes has 205 pounds in his pocket. Can he win a positive sum for sure? If «yes», what amount?

Bella, what is he talking about And I say that Bella will come first

Answer options: A. No B. Yes, 1 pound C. Yes, 5 pounds D. Yes, 10 pounds E. I don’t gamble 54 Powered by TCPDF (www.tcpdf.org)

Problems

51. Continents and oceans There are 6 continents and several oceans on a sphereshaped planet. Moreover, from any point of the continent (or ocean) you can get to any other point of this continent (or ocean) without leaving it, that is, continents and oceans do not consist of separate parts. It turned out that every ocean borders with every continent on this planet. What is the largest possible number of oceans on this planet?

This planet’s name starts with «E», right?

Answer options: A. 1 B. 2 C. 3 D. 4 E. As many as you like 55 Powered by TCPDF (www.tcpdf.org)

Problems

52. The lonely king The king stands on one of the squares of the first horizontal line (the lower one on the picture). Two players take turns moving the king one square to the right, up, or right and up diagonally. The one who puts the king on the upper right square (marked with a cross on the picture) wins. Who has a strategy to win this game?

8 7 6 5 4 3 2 1

12345678

I give up...

Answer options: A. The one who goes first B. The one who goes second C. There will be a draw D. It depends on the initial position of the king E. The one who is stronger 56 Powered by TCPDF (www.tcpdf.org)

x

Problems

53. Yandex.Traffic Yandex.Traffic service shows street traffic. The green, yellow or red section between two adjacent intersections shows that it takes 1, 2 or 3 minutes respectively to pass through this section. What minimal time will it take to drive from point A to point B, moving only north or east?

B

A

Traffic jams are everywhere. Get out of the car and walk. After 6 km, turn right and go forward 15 km

Answer options: A. 16 minutes B. 15 minutes C. 14 minutes D. 13 minutes E. I use Google 57 Powered by TCPDF (www.tcpdf.org)

Problems

54. Rare numbers Harry told his friends that he had «a rare phone number, since all the digits are different». Friends began to laugh. Is Harry right? In other words, what share of all possible sevendigit numbers is «rare»?

Can’t remember my phone number at all. You see, all the digits are different...

Answer options: A. Almost all B. A bit more than a half C. About a third D. About 15% E. About 6% 58 Powered by TCPDF (www.tcpdf.org)

Problems

55. Grapefruit Peter bought a grapefruit of 10 cm diameter, 1cm of which is the thickness of its peel. What portion of the grapefruit is edible if the peel is not edible?

Answer options: A. About 90% B. About 80% C. About 70% D. About 50% E. 0%, for me it’s all not edible! 59 Powered by TCPDF (www.tcpdf.org)

Problems

56. Guess the number Bob chose a two-digit number and told the product of its digits to Pete only, and the sum of its digits to Alex only. There was a dialogue between the boys. Pete: «I will guess the chosen number in three attempts, but two attempts may not be enough for me». Alex: «In that case, four attempts are enough for me, but three may be not enough». What is the number reported to Alex?

Let’s continue. Attempt № 3375

Answer options: A. 6 B. 10 C. 12 D. The correct answer is different E. There cannot be exactly one definite answer 60 Powered by TCPDF (www.tcpdf.org)

Problems

57. Paper square A paper square ABCD of side of 1 is bent along a straight line so that the vertex A coincides with the midpoint of side CD. What is the area of the resulting hexagon?

Actually, you bent the square in pretty unique way

Answer options: A. 3/4 B. 5/8 C. 11/16 D. 61/96 E. I don’t understand where the hexagon is 61 Powered by TCPDF (www.tcpdf.org)

Problems

58. Vacation trips The total cost of a weekly tour of Spain consists of fixed costs (for a family staying in the same room) and variable costs which depend on the number of people in the family (food, air travel, etc.). The first travel agency sets prices in such a way as to have a certain fixed profit from each vacation package sold, and the second travel agency gets a certain fixed profit (possibly another amount) from each tourist. It turned out that the cost of trips for two and three people in both travel agencies were the same. In which travel agency is it cheaper to buy a vacation ticket for one person?

And Jackie will be checked in as a hand luggage. It will be cheaper

Answer options: A. The first one B. The second one C. The costs are the same D. Not enough data for a definite answer E. I don’t have a vacation! 62 Powered by TCPDF (www.tcpdf.org)

Problems

59. Strange sales guys «How much for this watch?» — asked Charlie from the sales assistant. «1200 dollars», — said the sales assistant. The second immediately approached him. «You know, my partner gives numbers that are 3 times the true numbers. Otherwise he is absolutely correct», — said the second sales guy. «So this watch costs 400 dollars?» — asked Charlie again. «You know, my partner understates all the numbers 12 times. Otherwise he is absolutely correct», — said the first sales guy. So how much does the watch cost?

Answer options: A. 200 dollars B. 2400 dollars C. 4800 dollars D. My answer is different E. Those guys are scammers! Run away! 63 Powered by TCPDF (www.tcpdf.org)

Problems

60. What angle? We took an arbitrary right triangle and built a rectangular projection of its inscribed circle onto the hypotenuse (see the figure). At what angle is this projection visible from the vertex with the right angle?

?

Answer options: A. 30° B. 45° C. 60° D. It depends on the size of the acute angles E. The correct answer is different 64 Powered by TCPDF (www.tcpdf.org)

Problems

61. Pirates and ducats Nine pirates had a chest with silver and golden coins in denominations of 1 and 10 ducats respectively, with both of them in equal quantities. The chest had no more than 1000 ducats. When the pirates started to count how many ducats are due to everyone, there were 7 extra ducats left. A dispute arose over who should get the extra share, and as a result, two pirates were killed. After that, pirates counted how many ducats are due to everyone again and 3 extra ducats were left. How many coins were in the chest?

It seems like there are no excess ducats, only excess pirates

Answer options: A. 62 B. 124 C. 126 D. 682 E. The correct answer is different 65 Powered by TCPDF (www.tcpdf.org)

Problems

62. Department of Logic The staff of the department of logic consists of truth-tellers who always tell the truth, and liars who always lie. Once, each of the employees made two statements: 1) no ten people at the department work more than I; 2) at least twenty people in the department have higher salaries than mine. It is known that the workload on all employees is different and the salaries are also different. How many people work in the department of logic?

And I never lie

I always tell the truth

Answer options: A. 20 B. 21 C. 30 D. 31 E. Not enough data for a definite answer 66 Powered by TCPDF (www.tcpdf.org)

Problems

63. Cutlet surprise Thomas bought a pack of frozen cutlets called «Surprise». The cutlets were all the same, round, and their diameter was only half the diameter of the pan. However, after a few minutes, two cutlets on a pan were fried, retaining a round shape (the ice melted and the water evaporated), and Thomas was able to additionally place the two remaining cutlets from the pack on the pan. What portion of the cutlets (at least) was ice?

Roasted up...

Answer options: A. 33% B. 50% C. 55% D. 67% E. Where have you seen such cutlets?! 67 Powered by TCPDF (www.tcpdf.org)

Problems

64. House James lives in a house with no more than 1000 apartments. Each section of the house has the same number of floors and each floor has 4 apartments. James noted that in his section of the house, the number of apartments with a double-digit number on the door is exactly 10 times the number of sections. How many apartments are in this house?

And here is another puzzle…

This problem has: A. 2 solutions B. 3 solutions C. 4 solutions D. 5 solutions E. No solution at all 68 Powered by TCPDF (www.tcpdf.org)

Problems

65. Basketball The basketball match between school teams A and B lasted for 45 minutes. At the end of each minute, one of the teams made either 2 or 3 points. It turned out that both teams were leading half of the time throughout the match. What is the biggest possible difference in the score at the end of the match?

Got these two cool kids to our basketball team. Now we will definitely win!

Answer options: A. 3 B. 33 C. 68 D. My answer is different E. I don’t like basketball 69 Powered by TCPDF (www.tcpdf.org)

Problems

66. GPS navigator Pete was going to Moscow from his house in Moscow region. After Pete drove through 3/4 of the way, his navigator showed that the estimated travel time to Moscow is 15 minutes (the navigator determines that the average speed on the remaining part of the way will be equal to the average speed since the beginning). However, immediately after this, the traffic slowed down and the speed remained constant throughout the rest of the way to Moscow. As a result, after 15 minutes, the navigator showed that the estimated time to Moscow is 15 minutes again. What estimated time will the navigator show in another half an hour?

Turn left. If you increase the speed to 130 km / h, you will reach Moscow in 15 minutes

Answer options: A. 10 minutes B. 15 minutes C. 30 minutes D. Pete will reach Moscow by that moment E. Pete won’t stand it and will go back 70 Powered by TCPDF (www.tcpdf.org)

Problems

67. Angle Points M and N are selected on the diagonal BD = 1 of the square ABCD so that BM = 1/3 and DN = 1/4. What is the angle MAN?

Well, both of us could not solve the angle problem

Answer options: A. 45° B. 30° C. 60° D. Another answer E. «MAN» — is not an angle, it’s just a word in English 71 Powered by TCPDF (www.tcpdf.org)

Problems

68. Two parties There was a party of liars and a party of truth-tellers at the united conference. 32 people were elected to the presidium. These people were seated in 4 rows of 8 people in each row. During the break, each member of the presidium said that among his neighbours there were representatives of both parties. It is known that liars always lie, and truthtellers always tell the truth. What is the minimal possible number of liars in the presidium? (Two presidium members are neighbours if one of them sits on the left, right, front, or behind the other.)

So, do you claim yourself to be a liar? But if you are a liar, then this is not true, and you are truth-teller. How confusing...

Answer options: A. 6 B. 8 C. 10 D. 12 E. 16 72 Powered by TCPDF (www.tcpdf.org)

Problems

69. Billiards The larger side of a rectangular billiard table is twice the smaller side. There are pockets in the middle of the large sides and in the corners, and the ball is in the center of the table. At what angle to the larger side do you need to strike the ball so that it will hit each of the four sides exactly once and fall in the pocket?

Well, looks like your partner hit the ball at completely wrong angle

Answer options: A. ≈ 28° B. ≈ 30° C. ≈ 32° D. ≈ 45° E. It’s impossible! 73 Powered by TCPDF (www.tcpdf.org)

Problems

70. Tram, student and professor A professor and his student live in the same building, not far from the tram line. They leave the house at the same time to go to a lecture. The student runs to the nearest tram stop at a speed of 12 km/h, and the professor walks along the tram line to another stop at half the speed of the student. However, the student is late for the lecture (though he does not stop anywhere on his way), and the professor arrives on time. What is the highest possible tram speed if it is known that the tram speed is constant and expressed as an integer km/h?

Answer options: A. 18 km/h B. 23 km/h C. 30 km/h D. There is no correct answer here E. What nonsense? It’s impossible! 74 Powered by TCPDF (www.tcpdf.org)

Problems

71. Computer tables The store buys computer tables in bulk from the manufacturer and sells them retail. Demand for the tables is stable, uniform throughout the year and amounts to 130 pieces per week. To store the goods, the store rents a certain part of a warehouse (rental period is 1 year) and the storage cost is 10 dollars per year per table. The cost of processing and delivery of one wholesale order is 100 dollars and it does not depend on the size of the order. How many times during the year should a store make bulk purchases in order to get the most profit?

Mr. Johnson, there are visitors here. They told that they are from tax police…

Answer options: A. 12 times B. 24 times C. 26 times D. 52 times E. Not enough data to answer 75 Powered by TCPDF (www.tcpdf.org)

Problems

72. Crossroads Two cars located 1 km away from the same crossroad (roads intersect at right angle) were moving on different roads at speed 60 km/h and 90 km/h respectively towards the crossroad. Both cars passed the crossroad without stopping. What was the minimal straight line distance between the cars (the car size can be neglected)?

Have you read the conditions of the problem at all? It clearly states: «Cars pass without stopping ...»

Answer options: A. Not more than 250 m B. About 275 m C. A little more than 300 m D. Approximately 333 m E. Different answer

76 Powered by TCPDF (www.tcpdf.org)

Problems

73. Subway A city has 40 subway lines. Any two lines intersect at a single point forming a station with transfer from one line to another. Moreover, any station is the intersection of exactly two lines. For what maximum k can any k subway stations be closed for repairs, so that it is still possible to get from any working station to any other working station (closing a station closes the transfers but not the whole subway line)?

How long will this last?

Until repair is finished…

Answer options: A. 40 B. 75 C. 80 D. 85 E. The correct answer is different 77 Powered by TCPDF (www.tcpdf.org)

Problems

74. Radius and chord OA is the radius of a circle of center O and 15 cm diameter. The chord BC of this circle is parallel to OA and cuts off a segment of height 1 cm from the circle (the smaller of the arcs with ends A and B contains point C). At what distance from the center of the circle do the straight lines OB and AC intersect?

Actually, you have to draw a circle

Answer options: A. 30 cm B. 72 cm C. About 3 m D. About 34 m E. It is obvious that they do not intersect 78 Powered by TCPDF (www.tcpdf.org)

Problems

75. Birthdays What is the chance that in a class of 30 students at least two have the same birthday?

Ms. Becker, you and I have a birthday today. I suggest — cake and bowling are on you and I will call our entire class

Answer options: A. About 5% B. 50% C. About 70% D. About 95% E. The correct answer is different 79 Powered by TCPDF (www.tcpdf.org)

Problems

76. Four cities Four major cities of the country Abdulia are located in the desert at the vertices of a square with a 100 km side. King Abdul wants to connect them with a system of roads so that one can get to any city from any other one along the roads. The cost of building one kilometer of a road is 1 million dinars. What is the minimal cost to build such a road system?

Answer options: A. 263 million dinars B. 273 million dinars C. 283 million dinars D. 300 million dinars E. Half will be stolen anyway 80 Powered by TCPDF (www.tcpdf.org)

Problems

77. Hikers Hikers are walking through the forest. They are 5 km away from a village and 3 km away from the straight road which goes to the village. Hikers can move through the forest at an average speed of 3 km/h and move along the road at an average speed of 5 km/h. What is the minimal time for them to reach the village?

It feels like our average speed is more than 3 km/h

Answer options: A. 1 hour 30 minutes B. 1 hour 32 minutes C. 1 hour 36 minutes D. 1 hour 40 minutes E. If they are in such a hurry they could have moved faster 81 Powered by TCPDF (www.tcpdf.org)

Problems

78. How many friends? Jake has 20 classmates. Every two out of the 20 have a different number of friends in this class. How many friends does Jake have?

It seems like Jake has no friends at all, only enemies

Answer options: A. 1 B. 10 C. 20 D. The correct answer is different E. Not enough data for definite answer 82 Powered by TCPDF (www.tcpdf.org)

Problems

79. «Horns & Hooves» The «Horns & Hooves» company recorded its expenses in dollars under 40 budget items, receiving a list of 40 numbers (each number has no more than two digits after the decimal point). Each accountant of the company took a copy of the list and found the approximate amount of expenses, acting as follows. First, he chooses any two numbers from the list, adds them up, drops the digits after decimal point of the sum (if any), and writes the result instead of the two selected numbers. Then he does the same procedure with the resulting list of 39 numbers, and so on, until a single integer remains in the list. As a result it turned out that all accountants obtained different numbers at the end. What is the largest number of accountants that could work in this company?

Have you tried a calculator?

Answer options: A. 20 B. 21 C. 39 D. 40 E. Don’t they have enough money for a computer?! 83 Powered by TCPDF (www.tcpdf.org)

Problems

80. Business meeting Alex and Ben agreed to meet in a cafe today between 2:00 p.m and 2:30 p.m. Each of them can come with equal probability in any of the minute time intervals between 2:00 p.m and 2:30 p.m. Each of them then waits for 10 minutes and then leaves. What is the chance that Alex and Ben will meet today in the cafe?

You ordered the thirty-fifth cup of coffee already. Are you sure your friend will come?

Answer options: A. 1/2 B. 2/3 C. 3/4 D. 5/9 E. The correct answer is different 84 Powered by TCPDF (www.tcpdf.org)

Problems

81. Black squares An artist had a wooden cube 7 × 7 × 7. He divided each face into 49 unit squares and painted each of them in one of three colours: black, white or red so that there were no squares of the same color, adjacent by side. What is the minimal possible number of black squares? (Squares with a common side are considered adjacent even if they are on different faces of the cube.)

Did you get Rubik’s cube again?

Answer options: A. 24 B. 26 C. 28 D. 30 E. My answer is different 85 Powered by TCPDF (www.tcpdf.org)

Problems

82. 12 chairs Ostap Bender has 12 chairs. The diamonds are hidden in one of the chairs. One chair has the gold inside (it can be the same chair as the one with diamonds). Ostap opens each chair in turn and checks all its content. Ostap knows about diamonds, but does not know about gold. Therefore, if Ostap finds the diamonds, he will not touch the remaining chairs. What is the probability that Ostap will find the diamonds, but will not find the gold?

Madam, could you please take this chair?

Answer options: A. 1/2 B. 5/12 C. 11/24 D. The correct answer is different E. No chance, I’ve seen the movie! 86 Powered by TCPDF (www.tcpdf.org)

Problems

83. Experiment A monkey goes up to one of the 100 floors of a skyscraper and throws down a coconut. The monkey is trying to figure out what is the lowest floor from which one can throw a coconut so that it crashes. What is the minimum number of attempts sufficient for this if the monkey has only two coconuts?

I’ve had enough of this monkey. It still didn’t solve the problem

Answer options: A. 10 B. 12 C. 14 D. 16 E. The correct answer is different 87 Powered by TCPDF (www.tcpdf.org)

Problems

84. Crossing the Alps A group of people are at the foot of a mountain of conical form with a 45° angle to the horizon. The height 45˚ B of the mountain is 2000 meters. The A group wants to get to the diametrically opposite point of the foot of the mountain (to go from point A to point B in the figure). What is the approximate length of the shortest route from A to B?

Do we really need this task? Isn’t it better to go down already?

Answer options: A. 4.5 km B. 5 km C. 5.5 km D. 6 km E. 6.5 km 88 Powered by TCPDF (www.tcpdf.org)

Problems

85. Cowboy Joe Cowboy Joe walks out of the bar into the middle of a 3 meter wide road that leads directly to his house which is 20 meters away from the bar. With each step, Joe, being drunk, moves forward by half a meter and also randomly deviates half a meter to the right or to the left (with probability of 1/2 to the right and 1/2 to the left). If Joe is on the edge of the road, he falls into the ditch and stays there until morning. What is Joe’s chance to get home tonight?

It seems Joe has no chances at all

Answer options: A. 15% B. 0.4% C. 0.2% D. 0.15% E. 50%, he will either reach the home or not 89 Powered by TCPDF (www.tcpdf.org)

Problems

86. Football tournament The football tournament of 16 teams was held in one round (every two teams play once). By the end of the tournament, every two teams scored a different number of points (victory — 3 points, draw — 1 point). It turned out that the Amkar team lost to all the teams that scored fewer points in the end. What is the best possible result for Amkar (indicate the place)?

Answer options: A. Fifth place B. Sixth place C. Seventh place D. Eighth place E. I support Barcelona! 90 Powered by TCPDF (www.tcpdf.org)

Problems

87. The old lady and 99 gentlemen The old lady enters the plane first and randomly selects her seat. Then the first gentleman comes in. If his seat is free, he takes it. If his seat is already taken, then he takes the first available seat. Then the second gentleman comes in and follows the same procedure, and so on. What is the chance that the last 99th gentleman will take his own seat as written in his ticket (there are 100 seats on this plane)?

He says it’s his seat...

Answer options: A. 1/2 B. 1/3 C. 1/99 D. 1/100 E. The correct answer is different 91 Powered by TCPDF (www.tcpdf.org)

Problems

88. Treasure Island Ten pirates on the island are going to share a treasure consisting of hundred identical gold coins. The procedure is the following: the senior pirate offers how to share the coins, and then each pirate agrees or disagrees with his proposal. If at least half of the pirates, including the one who proposed, support the plan, then they will share the coins as it was suggested by the senior pirate. If fewer than half of the pirates agree then they kill the senior pirate and start all over again. The senior pirate (of those who survived) offers a new plan, they vote for it according to the same rules, and then they either share the treasure as proposed or kill the senior pirate. This continues until some plan is adopted. What is the largest number of coins that the senior pirate could guarantee to get if all of them are greedy, think logically, don’t conspire (there is no group which coordinates its actions) and they all want to live?

With whom do I share the coins now?

Answer options: A. 10 B. 20 C. 96 D. The correct answer is different E. These are pirates. Nothing could be guaranteed 92 Powered by TCPDF (www.tcpdf.org)

Problems

89. Goat-metry. Part 1

fence post

ring

A mathematician has a goat in his garden. He took a rope 10 meters long, tied one end to the fence post, passed the rope through the ring on the goat’s collar and attached a ring to the other end of the rope so that it can slide on the wire along the fence. As a result, the goat ate all the grass in the area that it could reach. What curve is the boundary of this area?

I’m probably going to limit myself to a parabola today. I have no appetite at all

Answer options: A. Circle B. Parabola C. Ellipse D. Some other curve E. The mathematician is just crazy 93 Powered by TCPDF (www.tcpdf.org)

Problems

90. Goat-metry. Part 2 A mathematician has a goat in his garden. He took a rope 20 meters long, tied B one end to a tree, passed the rope through a ring that A l can slide on a wire along a ring straight fence, and tied the B´ other end to a goat collar. As a result, the goat ate all the grass that it could reach. What curve is the boundary of this area?

Answer options: A. Circle B. Parabola C. Ellipse D. Some other curve E. Mathematician, stop experiments with the poor animal!

94 Powered by TCPDF (www.tcpdf.org)

Problems

91. Football tournament A football tournament of 16 teams was held in one round (every two teams play once). It turned out that at some moment each team had played at least k matches, but no four teams had already played with each other. What is the largest possible value of k?

Do you remember the year we started playing?

Answer options: A. 6 B. 8 C. 10 D. 12 E. The correct answer is different 95 Powered by TCPDF (www.tcpdf.org)

Problems

92. Robinson explores the Island Robinson ended up on a desert island of circular shape. Once Robinson left his hut near the seashore. He traveled 3 km to the west and 4 km to the south and found himself on the seashore again. The next day Robinson left his hut, went southwest and was on seashore after 10 km. A day later, Robinson decided to go around the island along the coast. What distance will he need to travel?

You won’t go further this island anyway

Answer options: A. About 16 km B. About 31 km C. About 63 km D. About 113 km E. The correct answer is different 96 Powered by TCPDF (www.tcpdf.org)

Problems

93. Broken ATM An absent-minded mathematician has 50 bank cards with 1, 2, 3, ..., 50 dollars. The mathematician knows about this, but does not know how much money each of the cards has. The mathematician can insert any card into the ATM and request a certain amount of money. The ATM returns the required amount if it is available on the card, and does not return anything if the balance on the card is less than the amount required. In any case, ATM does not return the card and does not show how much money was on the card. What is the largest amount the mathematician can guarantee to get from all of his cards?

This ATM doesn’t like my cards at all

Answer options: A. 625 dollars B. 650 dollars C. 750 dollars D. The correct answer is different E. Nothing can be guaranteed

97 Powered by TCPDF (www.tcpdf.org)

Problems

94. Almost regular triangle The sides of the triangle are 2020, 2021 and 2022. What is the distance between the intersection point of the medians and the center of the inscribed circle of this triangle?

Is it problem with some triangle again?

Answer options: A. 1/3 B. 1/2 C. 1 D. The correct answer is different E. This can only be calculated on a computer 98 Powered by TCPDF (www.tcpdf.org)

Problems

95. Holiday 16 girls and 16 boys study in one class. Each boy called some girls from this class and congratulated them (no boy called the same girl twice). It turned out that there is only one way to make 16 pairs so that each pair consists of a girl and a boy who congratulated her. What is the largest total number of calls that girls could have received from boys that day?

Answer options: A. 64 B. 120 C. 128 D. 144 E. The correct answer is different 99 Powered by TCPDF (www.tcpdf.org)

Problems

96. Stupid recruits 100 recruits are standing in one line and facing their commander. On a command «To the left» everyone turns by 90° at the same time, but some turn left and others turn right. Exactly a second later, everyone who is now facing his neighbour turns «around» (by 180°). After another second, everyone who is now facing his neighbour turns around, and so on. What is the longest time for movements to continue?

I got totally confused...

Answer options: A. 1 min 39 sec B. 100 sec C. About 5 min D. Forever E. Until the commander stops it! 100 Powered by TCPDF (www.tcpdf.org)

Problems

97. Red ace A croupier mixes the deck of 52 cards and offers a player to bet on any number k from 1 to 52. Then the croupier begins to draw the cards from the deck one by one until the ace of one of the red suits appears for the first time. The player wins if this card is the k-th drawn. What number k should a player bet on so that his chance to win is as high as possible?

I didn’t want to play cards from the very beginning!

Answer options: A. 1 B. 26 or 27 C. Doesn’t matter, the chances are equal D. My answer is different E. There is no chance at all since the croupier clearly has a red ace up his sleeve 101 Powered by TCPDF (www.tcpdf.org)

Problems

98. Paid information Jake secretly writes a natural number from 1 to 55. Pete can choose any set of numbers and ask if the written number belongs to this set. Jake always answers the truth, but Pete must pay 2 dollars for the answer «yes» and 1 dollar for the answer «no». What is the minimal amount of money that Pete needs in order to find the written number for sure?

Dad, I need three hundred dollars to guess the number

Answer options: A. 16 B. 12 C. 9 D. 8 E. The correct answer is different 102 Powered by TCPDF (www.tcpdf.org)

Ask for five hundred

Problems

99. Careful weighing A collector knows that among several similar golden coins there is a fake one (it is lighter). He asked an expert to find this coin using cup scales without measuring weights, and demanded that each coin participate in weighings no more than twice. What maximum number of coins must the collector have so that the expert can determine the fake coin in six weighings?

Works the old fashioned way…

Answer options: A. 24 B. 36 C. 64 D. 73 E. 99 103 Powered by TCPDF (www.tcpdf.org)

Problems

100. Detective A detective is investigating a crime. The case involves 70 people, among whom one is the criminal, and one other is a witness (both are unknown). Every day, the detective can invite one or more of these 70 people, and if the witness but not the criminal is in this group, the witness will tell who the criminal is. What is the minimal number of days for a detective to determine the criminal?

There are some new facts. It turned out that three horses are also involved. Who will interrogate the horses?

Answer options: A. 8 B. 10 C. 12 D. The correct answer is different E. The criminal won’t stand it and will reveal himself earlier

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Answers

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Answers

1. D 2. A 3. E 4. C 5. C 6. C 7. C 8. C 9. D 10. A 11. D 12. B 13. D 14. B 15. A 16. D 17. A 18. A 19. B 20. B 21. D 22. A 23. D 24. C 25. C 26. D 27. A 28. C 29. B 30. C 31. B 32. B 33. A 34. A

35. A 36. A 37. C 38. B 39. B 40. D 41. E 42. C 43. A 44. B 45. B 46. A 47. B 48. B 49. C 50. C 51. B 52. A 53. C 54. E 55. D 56. B 57. D 58. C 59. A 60. B 61. B 62. C 63. C 64. D 65. C 66. A 67. A 68. B

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69. C 70. B 71. C 72. B 73. B 74. D 75. C 76. B 77. C 78. B 79. B 80. D 81. B 82. C 83. C 84. B 85. B 86. C 87. A 88. C 89. B 90. A 91. C 92. D 93. B 94. A 95. E 96. A 97. A 98. C 99. D 100. A

Contents Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

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www.kvantik.com [email protected] Лауреат IV Всероссийской премии «За верность науке» в номинации «Лучший детский проект о науке»

В журнале вы найдёте интересные статьи и задачи по математике, лингвистике, физике и другим естественным наукам, сможете принять участие в математическои и лингвистическом конкурсах!

КАК СКЛЕИТЬ ПИРАМИДКУ ИЗ ПРЯМОУГОЛЬНИКА? ЗАЧЕМ НУЖНЫ МАШИНКИ МОЛЕКУЛЯРНЫХ РАЗМЕРОВ? ПОЧЕМУ ЗАПОТЕВШИЕ ОЧКИ ДЛЯ БЛИЗОРУКОСТИ И ДЛЯ ДАЛЬНОЗОРКОСТИ ОТПОТЕВАЮТ ПО-РАЗНОМУ? НА КАКИХ ЯЗЫКАХ ГОВОРИТ СТАРИК ХОТТАБЫЧ? Все ответы знает «Квантик» — ежемесячный научно-познавательный журнал для школьников 5-8 классов Подписаться на журнал «Квантик» можно в любом отделении связи Почты России и через интернет. Приобрести электронную версию журнала «Квантик» в хорошем качестве можно в интернет-магазине МЦНМО «Математическая книга». Заходите по ссылке kvan.tk/e-shop

Журнал «Квантик» в социальных сетях:

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Mikhail Evdokimov A hundred colors of math *** Михаил Евдокимов Сто граней математики (на английском языке)

Издательство Московского центра непрерывного математического образования 119002, Москва, Бол. Власьевский пер., 11. Тел.: +7 (499) 241 08 04.

Книги издательства МЦНМО можно приобрести в магазине «Математическая книга»: 119002, г. Москва, Бол. Власьевский пер., д. 11. Сайт: biblio.mccme.ru/shop. E-mail: [email protected]. Тел.: +7 (495) 745 80 31.

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