© Abramson Math, 2022

Math problems

“You might think these problems are too complicated for a first or a fifth grader. Many parents can’t understand even the problems made for the 7-11th graders.

However, these problems can be easily solved in 10-30 minutes by the majority of ordinary school students, who received basic math knowledge in our classes.”

However, these problems can be easily solved in 10-30 minutes by the majority of ordinary school students, who received basic math knowledge in our classes.”

Primary school

Problems of the primary school

Who broke the cup? Could Oksana have broken the cup? What about Lena or Sonya?

Oksana said that it was Sonya who had broken the cup. Lena and Sonya both said who had broken the cup, but so quietly that no one heard them. It is known that only the girl who had broken the cup told the truth.

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How to cook three pancakes?

How to cook three pancakes (both sides) as fast as possible on a pan that can fit only two pancakes simultaneously? It takes 2 minutes to cook one side of one pancake.

Every pancake has 2 sides, there are 6 sides in total. In one minute, you can cook 2 bottom sides of 2 pancakes. If you cooked a maximum of two sides in a 2 minutes, you would be able to cook all 6 sides in 3 minutes.

There is one restriction: you can’t simultaneously cook two sides of the same pancake.

Message us to get a hint or to check your answer and get your free online class as a gift.

Every pancake has 2 sides, there are 6 sides in total. In one minute, you can cook 2 bottom sides of 2 pancakes. If you cooked a maximum of two sides in a 2 minutes, you would be able to cook all 6 sides in 3 minutes.

There is one restriction: you can’t simultaneously cook two sides of the same pancake.

Message us to get a hint or to check your answer and get your free online class as a gift.

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Problems for the second grade

Which player would have a winning strategy: the first one or the second one? What is the winning strategy?

There is a box with 120 chocolates on the table. Two players take chocolates from the box in turn. There are two rules: you can’t take all chocolates in your first turn, and you can’t take more chocolates in your turn than the other player took in theirs. The winner is the player who took the last chocolate. The first player is the one who makes the first move.

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Watch second graders solve problems in Yakov Abramson’s classes:

How many knights and knaves are there in the cabinet of ministers?

Is it possible to assert anything about the first minister? About the last one? About the second one?

Is it possible to assert anything about the first minister? About the last one? About the second one?

There are 16 people in the cabinet of ministers. All ministers are either knights or knaves. Knights always tell the truth, and knaves always lie.

The first minister said: “There are no knights among us!”

The second minister said: “There is no more than one knight among us!”

The third minister said: “There are no more than two knights among us!”

………………………..

The last minister said: “There are no more than 15 knights among us!”

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The first minister said: “There are no knights among us!”

The second minister said: “There is no more than one knight among us!”

The third minister said: “There are no more than two knights among us!”

………………………..

The last minister said: “There are no more than 15 knights among us!”

Message us to get a hint or to check your answer and get your free online class as a gift.

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Problems for the third grade

Find out to which barrel an evil pirate has added sleeping pills.

A good doctor Powderpill is delivering four barrels of medicines to Africa. An evil pirate managed to add and dissolve sleeping pills into one of the barrels. The sleeping draft kicks in in an hour. The good doctor has only two assistants: a parrot and a hamster. He can pour medicine from any barrel only once to his assistants, but he must do it simultaneously — he cannot give medicine to only one of the assistants and wait for results.

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Чтобы получить подсказку или проверить правильность своего ответа, напишите нам. Каждому написавшему онлайн-урок в подарок.

Prove that there is a regular triangle in the plane (possibly of another size and position) with vertices of the same color.

The plane is divided into regular triangles by three equidistant families of parallel lines. The vertices of these triangles are randomly painted in two colors.

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Problems for the fourth grade

How many markers were produced?

The factory produced a batch of marker pens. If this batch is laid out in boxes of 6 marker pens, then 3 marker pens will remain; if in boxes of 7, then 4 marker pens will remain; if in boxes of 8, then only 1 marker pen will remain. It is known that there are no more than 350 marker pens in this batch.

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“After our online math course, you will be able to find other solutions for these problems. The course is interesting for both children and adults. You can study in your own time. You will have access to the course for six months.”

Middle school

Problems that are easily solved by the fifth graders who study with Yakov Abramson

Prove that the point N, which is symmetric to the point H with respect to M, lies on the circle circumscribed about the triangle ABC.

Let AA1 and BB1 be the altitudes in the triangle ABC, H is the point of their intersection (orthocenter of the triangle ABC), M is the midpoint of the side AB.

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Message us to get a hint or to check your answer and get your free online class as a gift.

How long did the candles burn?

Two candles of the same length were lit at the same time: the thicker candle, burning down in 4 hours, and the thinner candle, burning down in 2 hours. After a while, both candles were extinguished. The stub of a thick candle turned out to be 3 times longer than the stub of a thin candle.

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Message us to get a hint or to check your answer and get your free online class as a gift.

Problems for the sixth graders

Find the taxi’s serial number, given that the distance between the train station and the airport is 10 km.

Taxis run between the train station and the airport. Each of the cars has its own serial number: 1, 2, and so on. The traffic is organized in such a way that the first taxi leaves the station at 6:00, and after it with an interval of 7 minutes, the rest of the cars depart according to their numbers.

Once the taxi begins its shift, it runs back and forth independently of other cars, making 5-minute stops after reaching each of the destination points.

It is known that each taxi spends 2 minutes on its first kilometer and 1/9 min less for each subsequent kilometer than the previous one. The passenger left the airport at 9.44 am.

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Once the taxi begins its shift, it runs back and forth independently of other cars, making 5-minute stops after reaching each of the destination points.

It is known that each taxi spends 2 minutes on its first kilometer and 1/9 min less for each subsequent kilometer than the previous one. The passenger left the airport at 9.44 am.

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Problems for the seventh grade

How many bees were in the swarm?

An old Indian problem.

The number of bees, equal to the square root of half of their entire swarm, sat on a jasmine bush, leaving eight-ninths of a swarm behind them. And only one bee from the same swarm is circling near the lotus, attracted by the buzzing of a friend who has inadvertently fallen into the trap of a sweet-smelling flower.

Message us to get a hint or to check your answer and get your free online class as a gift.

The number of bees, equal to the square root of half of their entire swarm, sat on a jasmine bush, leaving eight-ninths of a swarm behind them. And only one bee from the same swarm is circling near the lotus, attracted by the buzzing of a friend who has inadvertently fallen into the trap of a sweet-smelling flower.

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Problems for the eighth grade

Prove that the center of the circle lies on the first circle.

A circle is inscribed in triangle ABC.

It touches its sides AC and AB at points M and N, respectively.

Another circle is inscribed in triangle AMN.

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It touches its sides AC and AB at points M and N, respectively.

Another circle is inscribed in triangle AMN.

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Senior school

A problem that students solve in the ninth grade

How often do they meet there on average, say, during the academic year?

The director and their deputy come to the office at a random time every day between 9 and 10 a.m., wait for each other for 5 minutes, and then leave the office.

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Message us to get a hint or to check your answer and get your free online class as a gift.

The problem that is given in the freshman year at the Faculty of Mechanics and Mathematics of the Lomonosov Moscow State University and easily solved by tenth graders

The problem that is given in the sophomore year at the Faculty of Mechanics and Mathematics of the Lomonosov Moscow State University and easily solved by eleventh graders who study Abramson Math

Prove that EF||AD

ABCD is a parallelogram; rays BE and DF (CBE=CDF) are drawn from opposite vertices B and D inside it at equal angles to neighboring sides. From another pair of opposite vertices A and C, rays AE and CF (DAE=DCF) are drawn inside it at equal angles to neighboring sides. {E}=BEAE, {F}=DFCF.

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Learn how to solve complex and non-standard problems with Abramson Math

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