Checksum - 2014
1. Looking at the family album, Vanya discovered that he had 4 great-grandmothers and 4
great-grandfathers. And how many great-grandmothers and great-grandfathers did his great-grandmothers and
great-grandfathers all together?
Solution:
Every person has 4 great-grandparents and 4 great-grandfathers. Because total great-grandparents
Vanechka had 8, then 8*4 = 32 great-grandmothers and 32 great-grandfathers the Vanichkins had
great-grandparents combined.
Answer: Vanichka’s great-grandmothers and great-grandfathers combined had 32 great-grandmothers and 32 great-grandfathers.
2. Two trains are moving towards each other. Their speeds are 105 km/h and 85 km/h.
How far are these trains from each other half an hour before they meet?
105 0.5 + 85 0.5 = 95 Answer: 95 km.
3. Find the value of the expression 12 log 9 27.
Solution: Because =1 and = for x 0 we have:
12 9 27 = 12 9(33) = 12 3 9 3 = 12 3 = 18 Answer: 18.
4. The centers of disjoint circles of radius 2 are located at the vertices of the triangle. What is the sum of the areas of the three shaded sectors?
Solution: It is known that the sum of all angles of a triangle is 1800. Because circles of the same radius, and the sum of the angles of the shaded sectors is equal to 1800, then the total area of the shaded sectors will be equal to half the area of the circle.
2 Answer: = 2
5. Solve inequality:
Solution:
1 6 + () = 2 6 + 6 2 = 0 Multiply by 6 (0) 62 + 1 2 6 = 0
Let's introduce the replacement = 6, then:
2 2 + 1 = 0 1,2 = 1
Let's go back to the replacement:
6 = 1 = 0 Answer: (, 0) (0, +).
6. Solve the tg equation. In your answer, write the smallest positive = root.
(6) 1 Solution: Let =. Then =, = 6 +,.
(6) = + = 7 + 6, x(k) is an increasing function of k.
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Let's find the x value for each y value:
2. y2=2 x=3 Answer: (2, 3), (3,2).
11. When publishing the book, 6949 numbers were required to number its pages. How many pages are in the book?
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12. In a round frying pan with a diameter of 30 cm, a pancake was baked in the shape of a flat convex figure with an area of 400 cm2. Prove that the center of the frying pan is covered with a pancake.
Proof:
We will consider the frying pan as a circle with a diameter of 30 cm, and the pancake as a convex figure located inside the circle.
Find the area of the frying pan:
2 = 152 = 225,706.86 cm2 We find that the area of the pancake is more than half the area of the frying pan.
From the properties of convex figures it follows that through any point inside the frying pan and outside the pancake, a straight line can be drawn that does not intersect the pancake.
Let's prove that the center of the frying pan is covered with a pancake. Let's prove it by contradiction:
Suppose the center is not covered, then we will draw such a straight line through it. Since the straight line does not intersect the pancake, and the pancake is completely on the frying pan, it turns out that the pancake lies completely on one half of the frying pan. But the area of the pancake is larger than the area of half the frying pan. We got a contradiction. Therefore, the center of the pan is covered with a pancake.
13. Mother goose lined up her 4 goslings in one line just as she did before, to go to the nearest lake to dive and swim.
On their way to the lake, the goslings rearranged and changed their original order.
Here's what we know about their new order:
1) Ha-Hee slowly rolls from one foot to the other, but now no one will step on her heels, as Hee-Ha did before.
2) Ha-Ha ran to another place because he does not like to go ahead of the “nippers” Ho-Ho.
3) Hee-Ha goes where he usually goes.
4) The first to come to the lake will be the gosling Ha-Ha, and not Ha-Hi, as it happened before.
What was the previous order of the goslings and what place will Ho-Ho be in now?
Solution:
Under the conditions that the gosling Ha-Ha will come to the lake first, and not Ha-Hi, as happened before, we know that Ha-hi became the first. And knowing that Ha-Hi slowly rolls from one foot to the other, but now no one will step on her heels, as Hi-Ha did before, we get that Ha-He now goes last. Ha-Ha ran to another place because he doesn’t like to go ahead of the “nippers” Ho-Ho, which means Ho-Ho is no longer second. From the fact that Hee-Ha goes where he usually goes, we understand that the second one. We get that in the previous order it was like this: Ha-Hi is the first, Hi-Ha is the second, Ha-Ha is the third, and Ho-Ho is the fourth.
Accordingly, in the new order it became like this: Ha-Ha - first (from condition 4), Hi-Ha - second (from condition 3), Ho-Ho - third, Ha-Hi - fourth (from condition 1).
Consequently, Ho-Ho became the third.
14. Many friends gathered at Anya’s birthday party. When the guests began to communicate, they noticed that the number of guests who knew an odd number of invitees was even. Anina’s best friend made the statement that this pattern is true for any company. Prove this to be true.
Solution:
Let us denote the number of friends who have an odd number of acquaintances in their company by k, and, accordingly, the number of acquaintances of these friends by a1, a2,…, ak. In addition, we denote the number of friends who know an even number of company members by n, and the number of acquaintances of these friends, respectively, by b1, b2, ..., bn. Based on this, then the total number of acquaintances is equal to (a1 + a2 +…+ ak + b1 + b2 +…+ bn)/ 2.
The sum b1 + b2 +…+ bn is even, since all its terms are even.
In order for this fraction to be equal to an integer, the sum a1 + a2 +…+ ak must be even. But all the terms of the last sum are odd, so the number k of terms of the sum can only be even.
15. The nimble pirates Captain Blood and Captain Hook, having dug up the entire uninhabited island, finally found a treasure chest. When they opened it, they saw 17 coins, 2 rings and 1 crown. All this wealth was divided among themselves in equal parts by Blood and Hook. Moreover, the crown went entirely to Hook. The coins and rings were also not cut into pieces. One coin is as much heavier than one ring as one coin is lighter than one crown. How many coins and rings does Blood have?
How many great-great-grandparents did all your great-great-grandparents have?
ANSWER
Each person has 2 parents, 4 grandparents, 8 great-grandparents, 16 great-great-grandparents. To find out how many great-great-grandmothers and great-great-grandfathers all of us had, we need 16 x 16. The result is 256. This result is obtained, of course, if we exclude cases of incest, i.e. marriages between different relatives.
If we take into account that one generation is approximately 25 years, then eight generations (which were discussed in the problem statement) correspond to 200 years, i.e. 200 years ago, every 256 people on Earth were related to each of us. Over 400 years, the number of our ancestors will be 256 x 256 = 65,536 people, i.e. 400 years ago, each of us had 65,536 relatives living on the planet. If we “unscrew” history a thousand years ago, it turns out that the entire population of the Earth at that time was relatives to each of us. This means that indeed all people, by and large, are brothers.
Each person has 2 parents, 4 grandparents, 8 great-grandparents.
281. Dialogue in a household goods store:
How much does one cost?
20 rubles,” the seller answered.
How much is 12?
40 rubles.
Okay, give me 120.
Please, 60 rubles from you.
What did the visitor buy?
Number for the apartment.
A bottle with a cork costs 1 rub. 10 kopecks. A bottle is 1 ruble more expensive than a cork. How much does a bottle cost and how much does a cork cost?
At first glance, it may seem that a bottle costs 1 ruble, and a cork costs 10 kopecks, but then the bottle is 90 kopecks more expensive than a cork, and not 1 ruble, as is the case. In fact, a bottle costs 1 rub. 05 k., and a cork costs 5 k.
Katya lives on the fourth floor, and Olya lives on the second. Rising to the fourth floor, Katya climbs 60 steps. How many steps does Ole have to go up to get to the second floor?
At first glance, it may seem that Olya walks 30 steps - half as many as Katya, since she lives half as low as her. Actually this is not true. When Katya goes up to the fourth floor, she climbs 3 flights of stairs between floors. This means there are 20 steps between the two floors: 60: 3 = 20. Olya rises from the first floor to the second, therefore, she climbs 20 steps.
How can you pour exactly half of a mug, ladle, pan or any other dish of regular cylindrical shape, filled to the brim with water, without using any measuring instruments?
Any dish of regular cylindrical shape, when viewed from the side, is a rectangle. As you know, the diagonal of a rectangle divides it into two equal parts. In the same way, a cylinder is divided in half by an ellipse. Water must be poured from a cylindrical container filled with water until the surface of the water on one side reaches the corner of the container, where its bottom meets the wall, and on the other side the edge of the container through which it is poured. In this case, exactly half of the water will remain in the dish:
Three hens lay three eggs in three days. How many eggs will 12 hens lay in 12 days?
You can immediately answer that 12 hens will lay 12 eggs in 12 days. However, it is not. If three hens lay three eggs in three days, then one hen lays one egg in the same three days. Therefore, in 12 days she will lay: 12: 3 = 4 eggs. If there are 12 hens, then in 12 days they will lay: 12 · 4 = 48 eggs.
Name two numbers whose number of digits is equal to the number of letters that make up the name of each of these numbers.
One hundred (100) and million (1,000,000)
I guarantee,” said the salesman at the pet store, “that this parrot will repeat every word he hears.” The delighted buyer purchased the miracle bird, but when he came home, he discovered that the parrot was as dumb as a fish. However, the seller did not lie. How is this possible? (The task is a joke.)
The parrot can indeed repeat every word it hears, but it is deaf and does not hear a single word.
There is a candle and a kerosene lamp in the room. What will you light first when you enter this room in the evening?
Of course, a match, since without it it is impossible to light a candle or a kerosene lamp. The question of the problem is ambiguous, because it can be understood either as a choice between a candle and a kerosene lamp, or as a sequence in lighting something (first a match, then everything else from it).
Half of half a number is equal to half. What number is this?
This number is 2. Half of this number is equal to 1, and half of half of this number (i.e., one) is equal to 0.5, i.e., also half.
Over time, man will definitely visit Mars. Sasha Ivanov is a person. Consequently, Sasha Ivanov will definitely visit Mars over time. Is this reasoning correct? If not, what mistake was made in it?
The reasoning is incorrect. It is not at all necessary that Sasha Ivanov will eventually visit Mars. The external correctness of this reasoning is created due to the use of one word (“man”) in two different senses: in the broad (abstract representative of humanity) and in the narrow (specific, given, this particular person).
They often say that you have to be born a composer, or an artist, or a writer, or a scientist. Is this true? Do you really have to be born a composer (artist, writer, scientist)? (The task is a joke.)
Of course, a composer, as well as an artist, writer or scientist, must be born, because if a person is not born, then he will not be able to compose music, draw pictures, write novels or make scientific discoveries. This joke problem is based on the ambiguity of the question: “Do you really have to be born?” This question can be taken literally: is it necessary to be born in order to engage in any type of activity; and this question can also be understood in a figurative sense: is the talent of a composer (artist, writer, scientist) innate, given by nature, or is it acquired during life through hard work.
You don't have to have eyes to see. Without the right eye we see. We also see it without the left one. And since we have no other eyes besides the left and right eyes, it turns out that not a single eye is necessary for vision. Is this statement true? If not, what mistake was made in it?
The reasoning is, of course, incorrect. Its external correctness is based on the almost imperceptible exclusion of one more option, which also needed to be considered in this argument. This is an option when no eye can see. It was he who was missed: “We see without the right eye, without the left one too, which means that the eyes are not necessary for vision.” The correct statement should be: “Without the right eye we see, without the left we also see, but without the two together we do not see, which means we see either with one eye, or with the other, or with both eyes together, but we cannot see without eyes, which, thus essential for vision.”
293. The parrot lived less than 100 years and can only answer “yes” and “no” questions. How many questions should he be asked to find out his age?
At first glance, it may seem that you can ask a parrot up to 99 questions. In reality, you can get by with a much smaller number of questions. Let’s ask him this way: “Are you over 50 years old?” If he answers yes, then his age is from 51 to 99 years; if he answers “no,” then he is from 1 to 50 years old. The number of options for his age after the first question is halved. The next similar question: “Are you over (you can ask, less than) 25 years old?”, “Are you over (less than) 75 years old?” (depending on the answer to the first question) reduces the number of options by four times, etc. As a result, the parrot needs to ask only 7 questions.
One man who was in captivity says the following: “My dungeon was in the upper part of the castle. After many days of effort, I managed to break out one of the bars in the narrow window. It was possible to crawl into the resulting hole, but the distance to the ground was too great to simply jump down. In the corner of the dungeon I found a rope forgotten by someone. However, it turned out to be too short to climb down. Then I remembered how one wise man lengthened a blanket that was too short for him by cutting off part of it from the bottom and sewing it on top. So I hastened to divide the rope in half and tie the two pieces together again. Then it became long enough, and I safely went down it.” How did the narrator manage to do this?
The narrator divided the rope not across, as most likely it might seem, but along it, making two ropes of the same length. When he tied the two pieces together, the rope became twice as long as it was at first.
I’m putting the family archive in order - scanning photographs and interviewing everyone who remembers what. I'll try to write the results here.
This is the oldest photograph of my mother's side of the family. Photo from the late 19th century. On it are my great-great-grandfather Grisha (Gottlieb) and great-great-grandmother Anyuta (Ita Aronovna) Pantel.
In our family they were called “Grandfather Grisha” and “Grandma Anyuta,” so I will call them the same - although they are my great-great-grandfather and great-great-grandmother.
Grandfather Grisha was from Belovezhskaya Pushcha. He was a Nikolaev soldier, demobilized from the army ahead of schedule - due to tuberculosis. And as someone who served in the Nikolaev army, he received permission to settle outside the Pale of Settlement. This is how he ended up in the city of Karachev.
Karachev is a small town 44 km from Bryansk, a very old Russian city. Arriving there, grandfather Grisha Pantel married grandmother Anyuta (Ita Aronovna Livshits).
Grandmother Anyuta, originally from Odessa, was an orphan. She was born in 1871. Her mother died in childbirth when her grandmother Anyuta was very young. And when she was 5 years old, her father died during a pogrom in Odessa, and she was sent to relatives on her father’s side. When she grew up, she studied at a seamstress and hat workshop. She got married with funds from the Jewish community.
Unfortunately, we know nothing about the family of Grisha’s great-great-grandfather. His daughter, my great-grandmother Fenya, recalled that his parents, her grandparents, came to see them once. She was little then, the only thing she remembered was that her grandmother was wearing a wig. His older brothers (and he was the youngest in the family) left for America.
He worked all his life as a shoemaker, he had his own workshop, and employed 2-3 apprentices. Grandmother Anyuta ran a sewing workshop and always had orphan girls to teach, and her daughters helped too. They didn’t have their own house, they were renting.
They had 17 children, and only seven lived to adulthood (or at least young age). Ten died in infancy and childhood.
And seven are Fedor (Fievel), born in 1898, he died in civilian life, the eldest. The third is Sonya (Sara), born in 1900, she lived in Bryansk all her life. I already remember her - we came to visit relatives in Bryansk when I was 10 years old, and there I saw my grandmother Sonya. The fourth is my great-grandmother Fenya (Feiga Leya), born in 1902, died in 1985. Then Sergei (Israel), born in 1904, he died a year or two after the revolution - he was shot at a post, he was a Red Army soldier. There was also Reuben, born in 1908 (died in the 60s), Efim, born in 1910 (missing in the Second World War), and daughter Frida, born in 1912. (she died at the age of 12: she was gored by a bull, she was seriously ill for a long time, was paralyzed and died some time later).
This photo is from around 1912. Grandma Anyuta has three younger children here - Reuben, Efim and little Frida.
On the passe-partout below you can see part of the inscription “Karachev”.
The year of this photograph is also not signed, so I date it around 1928. Grandma Anyuta is sitting in the center.
My great-grandmother Fenya is standing on the left, I think she is about 17 years old here. To her right is her brother Efim. The handsome young man sitting on the left is Brother Reuben. Little girls next to grandmother Anyuta - two granddaughters, Sonya's daughters (Fenya and Rosa - behind the barrier).
In 1915, his father’s brothers, Grisha’s grandfathers, sent a shift card to Fenya and Sonya so that they would move to live in America. They were prepared for the journey, but at the last moment Grandma Anyuta did not let her daughters go.
Ten of her children, as I already wrote, died in childhood and infancy. Several children died literally on the same day - one fell ill with diphtheria. There was never much money in the house, and on the advice of (sort of) neighbors, they put the little ones together - so that everyone would get sick at once, and so as not to call a paramedic to each one separately, because it’s expensive! So they buried everyone together.
In matters of raising children, apparently, they did not go far beyond their belts. My great-grandmother Fenya told how one day the nanny gave the girls a rag doll for the holiday. There were never many toys in the house, and the girls reveled in the gift. Well, the boys took the doll away and cut it open to see what was inside. The father ended up flogging everyone with a spandher - the boys for taking it away and cutting it, and the girls for crying, and the nanny got it for bringing the doll.
Grandmother Anyuta observed Jewish traditions. Therefore, for a long time she could not come to terms with the fact that her daughter - my great-grandmother - married a Russian, and because of this she did not communicate with her for many years. And when her husband, grandfather Grisha, died in 1921, she went to live not with my great-grandmother with her “Russian husband” Vasily Pervushov, but with her sister Sonya, who had the “right” husband - Yuda Livshits.
After the war, however, apparently over the years, the national issue ceased to be so acute, and until her death, grandmother Anyuta lived with my great-grandmother Fenya and her family, nursed her great-granddaughters - my mother and her sister.
She was very flexible and non-conflicting. Everyone in the house loved her and went to her for advice.
This photo is from 1950, Lviv. My mother is 7 months old, and she is held in the arms of her great-grandmother, Grandma Anyuta, who is 79 years old.
My mother remembers the last years of Grandma Anyuta’s life. I also got to see something - not the grandmother herself, of course, but her prayer book. An old, old Jewish prayer book from the 18th year of publication. I remember it from my childhood, it was upstairs in the closet. At first it didn’t interest me at all, but when I started going to the Jewish school at the synagogue and studying words in Hebrew, I saw familiar words in my great-great-grandmother’s prayer book.
Mom remembers that grandmother Anyuta always had a prayer book, and not only lay there, but was used all the time - she often prayed.
She also went to the synagogue in Lviv, where the whole family moved after the war. Grandmother Anyuta knew how to read prayers in Hebrew, and for the fact that she helped other women pray - she said the words out loud, and they repeated them after her - they bought her a place in the synagogue together.
She told my mother stories from the Torah, and in general she was happy to tell everyone who was ready to listen to her.
In addition to Russian and Hebrew (prayer), she spoke Yiddish well.
Mom remembers that Grandma Anyuta said blessings on food - she whispered a short prayer before eating anything. Before Passover there was matzo in the house - they bought local matzo in Lvov, and when they moved to Krasnodar, there was no matzo bakery or synagogue there, and her daughter Sonya from Bryansk sent matzo for Passover in a parcel.
She had a very small pension - she received it for her son Efim, who died in the Second World War. From this pension, she gave her daughter and granddaughter (my great-grandmother and grandmother) one crystal wine glass a year for their birthdays - all that she managed to save money for. She bought wine glasses that matched the color, and so over the course of several years she assembled a set of wine glasses :)
When she was already quite old, a television appeared in the house. And she watched TV shows until late at night, could not turn off the TV - she was afraid that she would offend the TV lady. My grandfather, my mother’s father, used to tell her: “Anna Efimovna, turn off the TV and go to bed!” And she always answered: “How can I turn it off when she looks at me and talks!” And only when the TV presenter said goodbye to the audience until tomorrow, Grandma Anyuta wished her good night and also went to bed :)
Before her death, her hands shook violently, and in order to somehow overcome this, she constantly crocheted. She died in 1962, at the age of 91. She was buried in the Jewish cemetery in Krasnodar. Since there was no Jewish funeral service in Krasnodar in those years, at her request, a person familiar with the traditions was found, he conducted her together with her relatives on the last day and recited Kaddish.
Every person has his own roots. Some people are proud of their ancestors. Some people don't know anything about them. Some people have their own genealogical tables going back a hundred or two years. Some people only know their mom and dad. Those who grew up in an orphanage often do not know about them.
However, for everyone without exception, both those who know and those who do not know, one can be confident in the same circumstance. Every person had these same ancestors. Moreover, they were along the entire chain, throughout the depths of centuries, before Adam and Eve. Without knowing them by name, we still know for sure that they definitely existed.
And then one day I thought about a very simple thing. How many were there in total? Asking this question, I knew for sure that there were a LOT of them.
And yet I decided to try to count. Perform purely arithmetic operations and simply find out their total number. Well, at least until the Nativity of Christ. In just two thousand years.
The result shocked me.
No, I didn’t count down to the planned times. I couldn't. But I reached even the more modest historical depths, completely crushed by the enormity of what was calculated.
I'm not a mathematician. Therefore, I simply don’t know the names of the orders of numbers following trillions and billions. And ten, to some extent, doesn’t mean much to me, as again, a layman in mathematics.
You can only define your feelings with this word. Space. The same finite infinity.
Naturally, we must take generations as objects of calculation. Father, mother - that's the first thing. Grandparents are second. Great-grandfathers are the third. And so on. I took the difference between generations to be 20 years. Someone can take another number, 25 or 30 - it doesn’t matter. Because the further you count, the more clearly you will understand that this does not affect the order of the numbers at all.
1st generation (father, mother) – 2 people.
2nd generation (grandfathers, grandmothers) – 4 people.
3rd generation (great-grandparents) – 8 people.
4th generation (great-great-grandfathers, great-great-grandmothers) – 16 people.
5th generation (we lower the degree of relationship further) – 32 people.
We have reached the end of the 19th century. As we can see, each of us had 62 ancestors in the 20th century.
I won't count further. You can take a pencil and do it yourself.
I'll just summarize.
In the 19th century (generations 6 to 10), I (and you) had one thousand nine hundred and eighty-four ancestors. The 10th generation alone produces 1024 ancestors.
I'll tell you right away. As you count, you will definitely notice that every 10 generations (or 200 years by my calculation) gives an increase in the number of about a thousand times. I didn't make a mistake. Not 1000 times more. But 1000 times more.
Here is the direct and first confirmation of this. The 5th generation, as we just saw, is 32 people. The 15th generation is 32 thousand 768 people.
And in just 15 generations - over 65 thousand people.
Please note. This is in just 300 years. We have only reached the era of Peter.
Another 200 years, or 10 generations. In total this will be five hundred years and 25 generations from this day. In total, during this time you had approximately 67 million ancestors. Only your direct ancestors. And only you have one.
In just one thousand years, from the times of Rurik and Svyatoslav (note, the time difference between them is no longer important here) to the present day, each of our contemporaries has one thousand trillion (or a million billion, as you like) ancestors.
But before that there were still centuries about which we know nothing. Times of the Goths-Huns, Scythians and Sarmatians. I'm not even talking about the Bronze Age, Paleolithics and so on.
Anyone who wants can calculate this space with their own hands.
Of course, all these calculations are wrong.
If at the time of Batu (somewhere in the 39th or 40th generation) you have about 500 or 1000 billion ancestors, this, of course, does not mean that then at least 500 or 1000 billion people lived on Earth. And, moreover, trillions or billions of people have never lived on our planet at the same time.
Moreover, if you remember that these astronomical numbers relate to just one person. But there is also humanity.
Humanity, as we see today, is not decreasing in number. On the contrary, it is growing.
During the time of the Roman Empire, if I am not mistaken, only a few million people lived in it. But this is almost all of present-day southern, central and western Europe, western Asia and northern Africa.
There are now more than six and a half billion inhabitants on Earth, and their number is growing all the time.
So, when we count our ancestors, it turns out that arithmetically everything here is flawless. But in life this cannot happen, because it can never happen.
The thing is that all these calculations do not take into account one, but very important factor.
Of course I know him. But I won’t voice it.
Because it is very important that every person understands this very factor himself. And I also came to the conclusions that follow from this factor myself.