The international kangaroo game is an affordable way for people to communicate. Kangaroo - math for everyone

Constructions and logical reasoning.

Task 19. winding coast (5 points) .
In the picture - an island on which a palm tree grows and several frogs sit. island restricted coastline. How many frogs are on the ISLAND?

Answer options:
BUT: 5; B: 6; AT: 7; G: 8; D: 10;

Decision
When solving this task on a computer, you can use the Fill tool. Now it is clearly seen that 6 frogs are sitting on the island.

You could do something similar to this fill with a pencil on a sheet of conditions. But there is another interesting way to determine whether a point is inside or outside a closed non-self-intersecting curve.

Let's connect this point (frog) with a point that we know for sure is outside the curve. If the connecting line has an odd number of intersections with the curve, then our point lies inside (i.e. on the island), and if it is even, then outside (on the water)

Correct answer: B 6

Task 20. Numbers on balls (5 points) .
Mudragelik has 10 balls, numbered from 0 to 9. He divided these balls among his three friends. Lasunchik got three balls, Krasunchik - four, Sonk about- three. Then Mudragelik asked each of his friends to multiply the numbers on the received balls. Lasunchik received a product equal to 0, Krasunchik - 72, and Sonyk about- 90. All the kangaroos correctly multiplied the numbers. What is the sum of the numbers on the balls Lasunchik got?

Answer options:
BUT: 11; B: 12; AT: 13; G: 14; D: 15;

Decision
It is clear that among the three balls that Lasunchik received, there is the number 0. It remains to find 2 more numbers. Krasunchik has as many as 4 balls, so it will be easier to first find which three numbers from 1 to 9 need to be multiplied to get 90, like Sonya a? 90 = 9x10 = 9x2x5. This will the only way represent 90 as a product of the numbers on the balls. After all, if Sonka a one of the balls was with one, then it would be required to break 90 into the product of two factors less than 10, which is impossible.

So Lasunchik has 0 and two other balls, Sonk a balls 2, 5, 9.
Four Krasunchik's balls give in the product 72. Let's first break 72 into the product of two factors, so that then each of these factors can be divided by 2 more:
72 = 1x72 = 2x36 = 3x24 = 4x18 = 6x12 = 8x9

From these options, we immediately exclude:
1x72 - because we can't split 1 into 2 different multipliers
2x36 - because 2 breaks only as 1x2, but Krasunchik definitely doesn’t have a ball with the number 2
8x9 - because 9 is broken like 1x9 (you can’t break it like 3x3, since there are no two balls with triples), and Krasunchik doesn’t have a nine either

Remaining options:
3x24 - splits into 4 multipliers as 1x3x4x6
4x18 - split into 4 multipliers as 1x4x3x6, that is, the same as the first option
6x12 - breaks like 1x6x3x4 (because, remember, there is no ball with a deuce).

So, for a set of Krasunchik's balls, there is only one option. He has balls 1, 3, 4, 6.

For Lasunchik, in addition to the ball with the number 0, there are balls 7 and 8. Their sum is 15

Correct answer: D 15

Task 21. Ropes (5 points) .
Three ropes are attached to the board as shown in the picture. You can attach three more to them and get a solid loop. Which of the ropes given in the answers will make it possible to do this?
According to groups "Kangaroo" VKontakte, only 14.6% of the participants of the Mathematical Olympiad from the third and fourth grades solved this problem correctly.

Answer options:
BUT: ; B: ; AT: ; G: ; D: ;

Decision
This problem can be solved by mentally applying the picture to the picture and carefully checking the connections. And you can do a little better. Let's renumber the ropes and write down the line 123132 - these are the ends of the loops on the figure given in the condition. Now, above the ends of the ropes in the answer options, we also sign these numbers.

Now it is easy to see that in the variant BUT rope 2 connects to itself. In the variant B rope 1 connects to itself. But in the variant AT all the ropes are connected to each other in one large loop.

Correct answer: B
Task 22. Elixir Recipe (5 points) .
To prepare an elixir, you need to mix five types of aromatic herbs, the mass of which is determined by the balance of the scales shown in the figure (we neglect the mass of the scales themselves). The healer knows that 5 grams of sage should be put into the elixir. How many grams of chamomile should he take?

Answer options:
BUT: 10 g; B: 20 g; AT: 30 g; G: 40 g; D: 50 g;

Decision
Basil should be taken as much as sage, that is, also 5 grams. There is as much mint as sage and basil together (we do not take into account the weight of the scales themselves). So, mint should be taken 10 grams. Melissa should be taken as much as mint, sage and basil, that is, 20g. And chamomile - as much as all the previous herbs, 40 g.

Correct answer: G 40g

Task 23. Unseen Beasts (5 points) .
Tom drew a pig, a shark, and a rhinoceros on the cards and cut each card as shown. Now he can stack different "animals" by connecting one head, one middle and one back. How many different fantasy creatures can Tom collect?

Answer options:
BUT: 3; B: 9; AT: 15; G: 27; D: 20;

Decision
This is classic problem to combinatorics. the good thing is that they can (and should) be solved not mechanically by applying the rules for calculating the number of permutations and combinations, but by reasoning. How much different options is for the head of an animal? Three options. And for the middle part? Also three. There are three options for the tail. This means that there will be 3x3x3 = 27 different options in total. We multiply these options because any body and any tail can be attached to each head, so that each segment of the animal increases the combination options exactly 3 times.

By the way, the condition contains the word "fantastic". But after all, by combining any heads, torsos and tails, we will get real pigs, sharks and rhinos. So the correct answer should have been 24 fantasy animals and three real ones. However, apparently fearing different interpretations of the condition, the authors did not include option 24 in their answers. Therefore, we choose the answer D, 27. And who knows, what if the drawings also depict a fantastic talking pig, a fantastic flying shark and a fantastic rhinoceros who proved Fermat's theorem? :)

Correct answer: G 27

Task 24. Kangaroo bakers (5 points) .
Mudragelik, Lasunchik, Krasunchik, Khitrun and Sonko baked cakes on Saturday and Sunday. During this time, Mudragelik baked 48 cakes, Lasunchik - 49, Krasunchik - 50, Khitrun - 51, Sonko - 52. It turned out that on Sunday each kangaroo baked more cakes than on Saturday. One of them baked twice as much, one - 3 times, one - 4 times, one - 5 times, and one - 6 times.
Which kangaroo baked the most cakes on Saturday?

Answer options:
BUT: Mudragelik; B: Lasunchik; AT: Krasunchik; G: Khitrun; D: Sonko;

Decision
Let's first think about what information the fact that someone baked exactly 2 times more cakes on Sunday than on Saturday gives us? If on Saturday the kangaroo baked some cakes, then on Sunday - so many and so many more. This means that in just two days he baked three times (1 + 2 = 3) more cakes than on Saturday.

So what? And the fact that, for example, he could not bake 49 or cakes, since these .

It turns out that the one who baked three times more cakes on Sunday than on Saturday, their total number should be whitened by 4 = 1 + 3. Some people have 5, some have 6 and some have 7.

The principle of solving this problem emerges. Here we have five numbers: 48, 49, 50, 51, 52. 2 numbers (48 and 51) are divisible by 3 of them and 2 numbers are also divisible by 4 (48 and 52). But only one number, 50, is divisible by 5. It turns out that the one who baked 50 pies on Sunday baked 4 times more of them than on Saturday.

Only one number is also divisible by 6, this is 48. It turns out that the kangaroo, who baked only 48 cakes, baked them like this: 8 on Saturday and 40 on Sunday. Well, then it's simple. We get that:
Mudragelik baked 48 cakes: 8 on Saturday and 40 on Sunday (5 times more)
Lasunchik baked 49 cakes: 7 on Saturday and 42 on Sunday (6 times more)
Krasunchik baked 50 cakes: 10 on Saturday and 40 on Sunday (4 times more)
Khitrun baked 51 cakes: 17 on Saturday and 34 on Sunday (2 times more)
Sonko baked 52 cakes: 13 on Saturday and 39 on Sunday (3 times more)

It turns out that Hitrun baked the most cakes on Saturday.

Correct answer: G Khitrun

Sometimes life brings pleasant surprises.

My youngest son won International Mathematical Olympiad "Kangaroo-2016" by earning 100 points. Absolute result.

It is believed that men numbers more important than feelings or emotions.

Therefore, as a man, I should immediately go to the statistics of the Olympiad, analysis of problems, analysis of solutions ...

A little bit later.

And now I will not dissemble and, like a man, with a restrained dryness, I will say:

I'm very pleased.

Who creates myths about "masculinity"?

"Majority", "gray mass", which, in the words of Franklin Roosevelt, 32 President of the United States,

"He can neither enjoy from the heart, nor suffer
because he lives in gray darkness,
where there is neither victory nor defeat.

Emotions are the essence human life. Contact with reality, with Life generates emotions. Those who do not feel do not experience emotions.

Such a person is either not alive, or an official.

Both my grandfather and my father, who went through the Second World War, happened to not hide their emotions when talking about it.

The athlete who won the hardest fight, standing on the pedestal, does not hide tears of joy.

Why should I be hypocritical? I am very pleased and I feel proud of my son.

School education has completely discredited itself.

The impact of school grades on the fate of the child is minimal or negative. Any Mark for me is no more significant than the opinion of any of the representatives of the "majority".

But the Olympics are a different reality. Here the child can really show his abilities, will, ability to overcome himself and the desire to win...

Therefore, for the development of the child, the formation of his self-esteem, the Olympiads have a completely different meaning ...

100 points is good and pleasant.

But even just participate in the Olympics, where there is nowhere to write off and no one to ask and ... to score these very points more than " average value“For a child, this is already a victory. milestone in its development. The first experience of victories. The seeds of success that will inevitably sprout in his adulthood.

To give the child the experience of such independence is closer to the concept of "Education" than the whole program. modern school, which stereotypes the child's thinking, kills his abilities in the bud and minimizes the chances of becoming a truly successful and happy person.

Therefore, when, a week after the announcement of the results of the Kangaroo Mathematical Olympiad, my son took second place in the boxing tournament, I was no less happy, and maybe even more.

Yes, he could not outplay on points an opponent who was older and more experienced. But the judging panel of the competition, among whose members were two world champions, awarded the son special prize: "For the will to win".

Self-confidence, and not fear of "bad evaluation" - this is what true education should be directed to. Because it is this quality that will allow the child to become successful in adult life, and not slide into " gray mass who knows neither victory nor defeat" ...

And it doesn't matter where this quality is formed: in math or boxing classes...

Or even chess...

Therefore, when it turned out that my son reached the final of the Grand Prix Cup of the Russian Chess School, I was also happy. This time in the final, he failed to take a prize. “But still,” I said to myself, “To reach the final after a six-month series of qualifying rounds is not so bad, what do you think? ..”

...Too early and too narrow specialization is the enemy of natural and effective development human.

Even in agriculture for. in order to avoid soil depletion and maintain its productivity for many years, the so-called. "Crop rotation", sowing different crops in one field...

Even if Vitali Klitschko, the world heavyweight champion, has a chess rank and is able to hold out with ex-world chess champion Garry Kasparov for 31 moves ... why can't an ordinary boy develop legs, arms and head at the same time - for the benefit of "everything yourself"?

What ordinary peasants have understood for thousands of years, unfortunately, is not understood by most teachers and parents ... Otherwise, we would live in a different society, more reasonable and happy.

And with fewer officials on one human soul.

Sometimes I hear: "Oh, what a capable child! .."

What are you all about?!

Remembering and paraphrasing Professor Preobrazhensky from " dog heart" I will tell:

What are your "Abilities"? teacher-educator kindergarten? School teacher with a diploma from a pedagogical university that has eroded the remnants of rationality and humanism? Yes, they do not exist at all! What do you mean by this word? This is what: if I, instead of educating and teaching every day own child I will leave it to the aforementioned "specialists" to do this - then after a while I will find out that he has a "lack of abilities". Therefore, "ability" is in your desire to raise your own child and in understanding how to do it correctly.

This is what I will talk about in a series of open summer webinars on school education.

Competition "Kangaroo" is an Olympiad for all schoolchildren from grades 3 to 11. The purpose of the competition is to captivate children with a decision math problems. The tasks of the competition are very interesting, all participants (both strong and weak in mathematics) find exciting tasks for themselves.

The competition was invented by Australian scientist Peter Halloran in the late 80s of the last century. "Kangaroo" quickly gained popularity among schoolchildren in different parts of the Earth. In 2010, more than 6 million schoolchildren from about fifty countries of the world participated in the competition. The geography of participants is very extensive: European countries, USA, countries Latin America, Canada, Asian countries. The competition has been held in Russia since 1994.

Competition "Kangaroo"

The Kangaroo Competition is an annual competition, it is always held on the third Thursday of March.

Students are asked to solve 30 tasks of three levels of difficulty. Points are awarded for each correctly completed task.

The Kangaroo competition is paid, but its price is not high, in 2012 it was necessary to pay only 43 rubles.

The Russian organizing committee of the competition is located in St. Petersburg. Participants of the competition send all forms with answers to this city. Answers are checked automatically - on the computer.

The results of the "Kangaroo" contest are delivered to schools at the end of April. The winners of the competition receive diplomas, and the rest of the participants receive certificates.

Personal results of the competition can be found out faster - in early April. To do this, you need to use a personal code. The code can be obtained at http://mathkang.ru/

How to Prepare for the Kangaroo Contest

Peterson's textbooks contain problems that were in previous years at the Kangaroo competition.

On the Kangaroo website, you can see problems with answers that were in previous years.

And also for better preparation you can use the books from the series "Library of the Mathematical Club "Kangaroo". In these books, fascinating form entertaining stories in mathematics are told, interesting math games. The tasks that were in the past years at the mathematical competition are analyzed, extraordinary ways of solving them are given.

Mathematical club "Kangaroo", issue No. 12 (grades 3-8), St. Petersburg, 2011

I really liked the book, which is called "The Book of Inches, Vershoks and Centimeters." It tells about how units of measurement arose and developed: pie, inches, cables, miles, etc.

Mathematical club "Kangaroo"

Here are a few entertaining stories from this book.

V.I. Dal, a connoisseur of the Russian people, has such a record “what a city, then faith, what a village, then a measure.”

For a long time, in different countries different measures were used. Yes, in ancient China for men and women's clothing various measures have been taken. For men, they used "duan", which was 13.82 meters, and for women they used "pi" - 11.06 meters.

AT Everyday life Measures varied not only across countries, but also across towns and villages. For example, in some Russian villages, the measure of duration was the time “until the boiler of water boils.”

Now solve problem #1.

Old clocks lose 20 seconds every hour. The hands are set to 12 o'clock, what time will the clock show in a day?

Task number 2.

In the pirate market, a barrel of rum costs 100 piastres or 800 doubloons. A pistol costs 250 ducats or 100 doubloons. For a parrot, the seller asks for 100 ducats, but how many piastres will that be?

Mathematical club "Kangaroo", children's mathematical calendar, St. Petersburg, 2011

In the Kangaroo Library series, a mathematical calendar is released, in which there is one task for each day. By solving these problems, you will be able to give excellent food to your brain, and at the same time prepare for the next Kangaroo competition.

Mathematical club "Kangaroo"

Ben chose a number, divided it by 7, then added 7 and multiplied the result by 7. It turned out to be 77. What number did he choose?

An experienced trainer washes an elephant in 40 minutes, and his son 2 hours. If they wash the elephants together, how long will it take them to wash three elephants?

Mathematical club "Kangaroo", issue No. 18 (grades 6-8), St. Petersburg, 2010

This edition features combinatorial problems from a branch of mathematics that studies various relationships in finite sets of objects. Combinatorial problems occupy most in mathematical entertainment: games and puzzles.

Kangaroo Club

Problem number 5.

Count how many ways there are to place a white and a black rook on a chessboard with the condition that they do not kill each other?

This is the most difficult task, so I'll give her solution here.

Each rook keeps under attack all the cells of that vertical and that horizontal on which it stands. And she occupies one more cell herself. Therefore, 64-15=49 free cells remain on the board, each of which can be safely placed with a second rook.

Now it remains to note that for the first (for example, white) rook, we can choose any of the 64 squares of the board, and for the second (black) - any of the 49 squares, which after that will remain free and will not be under attack. This means that we can apply the multiplication rule: total options for the required arrangement is 64*49=3136.

When solving this problem, it helps that the very condition of the problem (everything happens on a chessboard) helps to visualize possible options relative position figures. If the conditions of conception are not so clear, you should try to make them clear.

I hope you enjoyed getting to know mathematical competition "Kangaroo" .

Millions of children in many countries of the world no longer need to be explained what "Kangaroo", is a massive international math contest-game under the motto - Math for everyone!".

The main goal of the competition is to involve as many children as possible in solving mathematical problems, to show each student that thinking over a problem can be a lively, exciting, and even fun affair. This goal is achieved quite successfully: for example, in 2009 more than 5.5 million children from 46 countries participated in the competition. And the number of participants in the competition in Russia exceeded 1.8 million!

Of course, the name of the competition is associated with distant Australia. But why? After all, mass mathematical competitions have been held in many countries for more than a decade, and Europe, in which the new competition was born, is so far from Australia! The fact is that in the early 80s of the twentieth century, the famous Australian mathematician and teacher Peter Halloran (1931 - 1994) came up with two very significant innovations that significantly changed the traditional school olympiads. He divided all the problems of the Olympiad into three categories of difficulty, and simple tasks should be accessible to literally every student. And besides, the tasks were offered in the form of a multiple-choice test, focused on computer processing results The presence of simple but entertaining questions ensured a wide interest in the competition, and a computer check made it possible to quickly process a large number of works.

The new form of competition was so successful that in the mid-80s, about 500,000 Australian schoolchildren participated in it. In 1991 the group French mathematicians, based on the Australian experience, held a similar competition in France. In honor of the Australian colleagues, the competition was named "Kangaroo". To emphasize the entertainingness of the tasks, they began to call it a contest-game. And one more difference - participation in the competition has become paid. The fee is very small, but as a result, the competition ceased to depend on sponsors, and a significant part of the participants began to receive prizes.

In the first year, about 120,000 French schoolchildren took part in this game, and soon the number of participants grew to 600,000. This began the rapid spread of the competition across countries and continents. Now about 40 countries of Europe, Asia and America participate in it, and in Europe it is much easier to list countries that do not participate in the competition than those where it has been held for many years.

In Russia, the Kangaroo competition was first held in 1994 and since then the number of its participants has been growing rapidly. The competition is included in the program "Productive game contests» Institute for Productive Learning under the guidance of Academician of the Russian Academy of Education M.I. Bashmakov and is supported by Russian Academy education, the St. Petersburg Mathematical Society and the Russian State Pedagogical University them. A.I. Herzen. The Kangaroo Plus Testing Technology Center took over the direct organizational work.

In our country, a clear structure of mathematical Olympiads has long been established, covering all regions and accessible to every student interested in mathematics. However, these Olympiads, starting from the regional and ending with the All-Russian, are aimed at highlighting the most capable and gifted from the students who are already passionate about mathematics. The role of such Olympiads in shaping the scientific elite of our country is enormous, but the vast majority of schoolchildren remain aloof from them. After all, the problems that are offered there, as a rule, are designed for those who are already interested in mathematics and are familiar with mathematical ideas and methods that go beyond school curriculum. Therefore, the Kangaroo contest, addressed to the most ordinary schoolchildren, quickly won the sympathy of both children and teachers.

The tasks of the competition are designed so that every student, even those who do not like mathematics, or even are afraid of it, will find interesting and accessible questions for themselves. After all the main objective of this competition is to interest the children, instill in them confidence in their abilities, and its motto is “Mathematics for All”.

Experience has shown that children are happy to solve competition problems that successfully fill the vacuum between standard and often boring examples from a school textbook and difficult, demanding special knowledge and preparation, tasks of city and regional mathematical Olympiads.