In one class there were 5 students in after school. One of them drew very well and quickly. The teacher gave him the task of drawing all those students who cannot draw themselves. The student began to reason and became confused: should he draw himself or not? If he can draw himself, then he should not draw himself, because... he can draw himself. But if he doesn’t draw himself, he must still draw himself. What should he do?
Answer
Two diggers need to dig a trench for 2 francs. The first one digs at the same speed as the second one throws out soil. The second one digs four times faster than the first one throws out soil. How should they divide the money received after completing the work?
Answer
A man and a woman are sitting next to each other. “I am a woman,” says the man with dark hair. “I am a man,” says the man with blond hair. Of these, at least one definitely told a lie. Who exactly do you think told the lie? Or are they both lying?
Answer
Three couples (husbands and wives) received a weekly wage of £1,000 between them. The wives received a total of £396. Diana received 10 pounds more than Katya, and Maria received 10 pounds more than Diana. Dmitry Smirnov received the same amount as his wife, Georgy Sidorov received one and a half times as much as his wife, Timofey Ivanov received twice as much as his wife. Think and answer, who is married to whom, who received and how much money?
Answer
One gentleman left an inheritance of £1,320 to his four sons. If the third brother received a share from the fourth, he would receive the same amount as the first and second brothers combined. If the fourth brother's share went to the second son, he would receive twice as much as the first and third sons combined. How much money did each of the four sons receive?
Answer
One man, traveling through the Amazon forest, was accidentally captured by the local aborigines. The Aborigines were a cruel tribe and informed him that he would be executed, but in what way was up to him. If he tells a lie, he will be thrown off a cliff, and if he tells the truth, he will be hanged. What should a traveler say to stay alive?
Answer
Two businessmen decided to open a joint business. The first businessman invested 1.5 times more money than the second. Later they decided to invite a third businessman to their business, but the amount of total contributions remains unchanged. The third businessman contributed £2,500. This amount must be divided between two other businessmen so that the contributions of all three businessmen are then the same. How should £2,500 be divided between three people?
Answer
One rich man left a will, according to which two of his nephews inherit 200,000 francs. If the third part of the amount received by the first nephew is subtracted from the quarter of the amount received by the second nephew, then 22,000 francs remain. How much money did each of the nephews receive, according to the will?
Answer
There is a seven-liter vessel filled with water to the brim (i.e., seven liters of water). There are also 2 empty vessels for 3 and 4 liters. How to make it so that in a 3-liter vessel there are 2 liters of water in 4 transfusions?
Answer
One man bequeathed an inheritance of just under £1,500. The amount was divided among his five children, and a small part went to the notary. The square root of the first son's share, half of the second's inheritance, the third's share minus 2 pounds, the fourth's share plus 2 pounds, the double share of the fifth son, and the squared amount for the notary were all equal. Each of the sons and the notary received a whole number of pounds. What was the amount left as an inheritance?
Answer
One person came to the store. He spent half the money on groceries and bought chewing gum with 5 cents. Next, he bought a recipe book for half the remaining amount plus 10 cents. Of the remaining amount, half went to buy a calendar, and with 15 cents he bought a hot dog. In the end, he only had 5 cents left. How much cash did he initially have in the store before making purchases?
Answer
There are 9 identical coins, but one of them is lighter than the others. It is necessary to find this coin in two weighings on the scales. Scales - ordinary with two cups, i.e. lever
Answer
Nine gambling enthusiasts once gathered in their small circle. They decided to play a game of distributing money. The first one gives each other as much money as each already had. Then the second one does the same, i.e. distributes money to the other eight, as much as they each have. And so on, all 9 players do it in turn. In the end it turns out that all players have the same amount of money. How much money did each player initially have?
Answer
One person constantly asks another to buy a piano from him. He initially asked £1,024 for the piano; when he was refused, he asked for a reduced price of £640. Having been refused again, he now asked for 400 pounds. After another refusal, he asked for 250 pounds. What do you think, if the seller receives a new refusal, what new amount, judging by the dependence of the reduction, will the piano seller ask for?
Answer
There are two buckets. One contains 5 liters of water, the other contains the same amount of alcohol. 0.5 liters was taken from a bucket of water and poured into a bucket of alcohol. After thorough mixing, 0.5 liters of the mixture was taken from a bucket of alcohol and 0.5 liters of water and poured into a bucket of water. Which of the following statements do you think is true:
A) There is more alcohol in a bucket of water than in a bucket of alcohol in water.
B) There is more water in a bucket of alcohol than there is alcohol in a bucket of water.
C) There is as much water in a bucket of alcohol as there is alcohol in a bucket of water.
D) There is no correct option.
Answer
One person purchasing goods for a company bought in one household appliances store: a certain number of refrigerators for 344 pounds and a certain number of televisions for 265 pounds. The cost of all refrigerators is £33 more than the cost of all televisions. What is the smallest number of refrigerators and televisions he could purchase?
Answer
One merchant bought a shipment of denim trousers for a total of 6,000 francs. He kept 15 jeans for himself, but he sold the rest in his boutique for a total of 5,400 francs. After the sale, the entrepreneur received 10 francs in profit from each piece of jeans sold. How many denim trousers did the entrepreneur initially buy?
Answer
There are four people in line. Semyon is between Boris and Masha. Masha stands in front of two other people, Dima takes a place in front of Masha. Who is first, second, third and fourth in line?
Answer
One man was addicted to hoarding one-dollar bills, 50-cent coins, and 25-cent coins. One day he had a sufficient amount of them, and all 3 types of money were in equal quantities. The man decided to put them into 8 bags so that each one contained the same amount of each of the 3 types of money. The next day, the man put the same money into 7 bags. The next day he put the same money into 6 bags. A day later, he tried to put it into 5 bags according to the same rules, but it didn’t work out for him. What is the smallest amount of dollars that this person could put into bags?
Answer
Two street vendors were selling plums, one for 2 and the other for 3 for one cent. Both traders expected to sell 25 cents worth of plums jointly. When each of them had 30 unsold plums left, they left for lunch, but left a third for the two of them. He began selling plums for 2 cents per 5 pieces. After both traders returned from lunch, all the remaining plums were sold by the third seller. The two merchants were surprised that the total revenue was not 25 cents, as they had planned, but only 24 cents. Where did the one cent go?
Law on the Protection of Consumer Rights No. 2300-1 of February 7, 1992 (Articles 18 and 25) gives the right to a consumer who discovers any defects in the purchased product to return the product to the seller or exchange the product for another, and in the case when The purchased product is not suitable based on external features (color, style, etc.), it can be exchanged for another similar product. This rule applies to any payment method for purchased goods. That is, when goods are purchased with borrowed bank funds under a consumer lending agreement, the rule will work.
But, in connection with the emergence of new obligations under the concluded loan agreement, some additional points appear. An additional complicating point may be the concluded collateral agreement to secure the loan; in this case, all actions to dispose of the purchased goods, including its return, are agreed upon with the bank, in accordance with the Civil Code of the Russian Federation (clause 1 of Article 334.1, clause 2 of Article 346 ). Let's figure out how to return or exchange goods purchased on credit in accordance with the Consumer Protection Law.
Terms of the loan agreement and agreement on pledge of goodsThe consumer lending agreement may contain a provision on the procedure for returning and exchanging goods. If there is such a provision in the agreement, in it, according to paragraphs 4, 9 and 10 of Art. 5, art. 11 of Law No. 353-FZ of December 21, 2013, there may be conditions for terminating the loan agreement on the basis of termination of the purchase and sale agreement, as well as a period for notifying the bank about the early repayment of the loan amount.
The Civil Code of the Russian Federation (Article 346, paragraph 2) describes the legal procedure for the alienation of purchased goods and the return of the deposit.
You must demand to terminate the contract for the purchase and sale of goods, return the goods or exchange them for a similar product or for a product of the same brand.
When expressing a demand to return the goods upon termination of the contract, you must indicate in the claim the details of your bank account for the transfer of funds for the return to pay for the loan for the goods. If a defective product is returned, interest under the consumer loan agreement and other payments under the agreement will be transferred to this account. According to the Consumer Protection Law No. 2300-1 (Article 22, paragraph 6 of Article 24), the amount must be returned within ten days from the date of receipt of your claim.
The claim is made in writing in two copies, one of which, with the seal and signature of the person who accepted the claim from you, you keep for yourself. You can submit your claim in person or send it by registered mail with acknowledgment of receipt and a list of the contents.
If the seller refuses to accept the claim, it can be submitted in the presence of two witnesses. Witness testimony that the seller refused to accept the claim will have weight in the future.
If a low-quality product is returned, the seller must, at his own expense, conduct an examination of its quality (see Law No. 2300-1, paragraph 5 of Article 18).
The act of returning the goods together with the seller’s representative and receiving money for the goodsA document confirming to the bank that the goods have been returned to the seller is an act of return of goods.
If the goods are not returned but exchanged, the seller must issue you new documents for the goods.
If a return is made, the seller must return to the buyer part of the price of the product, which the buyer made as a down payment. The money is returned to the buyer’s bank account (if the initial payment was paid by card), or transferred to him from the store’s cash register.
If the buyer returns a low-quality product, then the seller will transfer interest and payments under the loan agreement to the buyer after receiving payment documents that confirm the fact that the consumer has paid bank interest and payments under the loan agreement.
When a consumer returns a product purchased on credit, only the sales contract is terminated. The loan agreement has not yet been terminated, so payments under it continue until termination, otherwise penalties will be imposed by the bank, according to Art. 811 of the Civil Code of the Russian Federation.
We contact the bank to terminate the loan agreement and collateral agreementYou must write an application to the bank, listing the following facts:
When returning a purchase to the bank, you must proceed according to the following procedure:
If a consumer exchanges a low-quality product or replaces it with a similar one, he must submit the following documents to the bank:
Please note that if you repay the loan early, you must notify the bank 30 calendar days in advance, unless otherwise specified in the consumer loan agreement itself (see Parts 3, 4, Article 11 of Law No. 353-FZ). However, if less than 30 days have passed since the loan was opened, this rule does not apply and you can close the loan early without notice.
6th grade
6.1.
6.2.
cutting example).
6.3.
6.4.
boxes
6.5.
Why did she decide this?
7th grade
7.1. Find some natural number such that if you add the sum to it
its digits, it will be 2222.
7.2. Mom bought 10 large cakes, 7 medium ones and 4 small ones. Small
a cake weighs half as much as a medium one, and a large one weighs three times as much as a small one. How
mother divide them between six children so that the total weight of the cakes received
everyone, was the same, if she doesn’t want to cut the cakes?
7.3. The train, moving at a constant speed, had covered 1.2 times the distance by 17:00,
than by 16:00. When did the train leave?
7.4. How to cut a 6x6 checkered square into four equal ones
perimeter figures 16 each, if you can only cut on the sides of the cells?
The side of the cell is 1.
7.5. Twenty-seven classmates ate candy during the first and second breaks,
Moreover, at the second break, everyone ate one more candy than at the first. Peter
said that he counted the total number of candies eaten and received the answer 210.
Did he count correctly? Explain your answer.
6th grade
6.1. Find all three-digit numbers whose second digit is four times larger than the first.
and the sum of all three digits is 14.
6.2. A central 1x1 square was cut out of a 5x5 checkered square. Cut
the remaining figure into 4 equal checkered pieces. (Give one
cutting example).
6.3. From the apple box they took half of the total number of apples, then another half
the remainder, then half of the new remainder, and finally half of the next remainder.
After this, there are 10 apples left in the box. How many apples were in the box at the beginning?
6.4. Three boxes contain Christmas balls: one contains two red ones, the other contains a red one.
and blue, in the third there are two blue balls. The boxes say: "Two red", "Red
and blue", "Two blue". It is known that none of the inscriptions are correct.
How can you determine which box contains which balls by pulling out just one ball?
Indicate which box it should be taken from and how to determine the contents later
boxes
6.5. Three friends brought candy to school. The second brought twice as much
sweets than the first, and the third - three times more than the first. They added it all up
candy together. After the friends ate 3 candies, the first one left, and the second
I divided the remaining candies equally. The third told the second that she was mistaken.
Why did she decide this?
8th grade
8.1.
What is the wholesale price of the pen?
8.2.
8.3. a and b satisfying the equality
a 2 +b=b 2 + a
8.4.
8.5.
9th grade
9.1. Find the area of a square whose vertices all lie on two straight lines:
x+ y= 0 and x+ y= 2 .
9.2. On a small island, 2/3 of all men are married and 3/5 of all women are married.
How many people on the island are married if there are 1,900 people living there?
9.3. On a circle with diameter AB and center O, point C is chosen so that
the bisector of the angle CAB is perpendicular to the radius OC. In what respect is direct CO
divides angle ACB?
9.4. Find the number of three-digit numbers whose decimal notation involves
exactly one digit 3.
9.5. Mom wants to punish Petya for failing in math. They agreed on
next. Petya thinks of a two-digit number with different digits and reports it
mom. After this, mom tells Petya her two-digit number. Petya adds
mom’s number to your number, then to the amount received, then to the amount received again
amount, etc. until he gets a sum ending in two
the same numbers. Will mom be able to stop Petya from playing football that day?
8th grade
8.1. The merchant bought a batch of pens at the wholesale market and offers customers either
one pen for 10 rubles, or three pens for 20 rubles. Moreover, in both cases he
receives the same profit (the difference between buying a product and selling it).
What is the wholesale price of the pen?
8.2. In a right triangle, the bisector of an acute angle is equal to one of two
the segments into which she divided the opposite side. Prove that she
twice as long as the second of these segments.
8.3. Find the sum of two different numbers a and b satisfying the equality
a 2 +b=b 2 + a
8.4. Three students A, B and C were competing in the 100m race. When A reached the finish line, B
was 10m behind him, also when B finished, C was 10m behind him.
How many meters was A ahead of C at the finish line?
8.5. At Masha’s birthday party, each of the 10 guests had an equal amount of
sweets During the tea party, the first ate one candy, the second - two, the third - three, and
etc., tenth – 10 candies. Masha wanted to shift the
candies so that again everyone has an equal amount of candy in front of them, but dad,
without looking at the table, he said that she couldn’t do it. Why did he decide this?
Preview:6th grade
6.1. Answer. 149 and 284.
If the first digit is not less than 3, then the second is not less than 12, which is impossible. Means,
6.2. One example is shown in Figure 1. This example is not the only one.
Rice. 1
6.3. Answer. 160 apples.
When half of the apples are taken from the box, half of the apples remain in it.
the amount that was before. This means that before this there were twice as many apples.
Therefore, at the beginning there were 10x2x2x2x2 = 160 apples in the box.
6.4. Answer. From the Red and Blue box.
It follows from the condition that this box contains either two blue balls or two red ones. Taking out
one ball, we will know the contents of this box. If there are two blue balls in it, then
the one on which it says “Two red” will have multi-colored balls, since it does not contain
two red (according to condition) and not two blue (they are in the first box). In box with
The inscription “Two blue” means two red balls. If we took out a red ball, then
similarly, in the “Two Blue” box there are multi-colored balls, and in the “Two Red” box
- blue balls.
6.5. Answer. Because the number of remaining candies must be odd.
The total number of candies brought is even. This can be explained this way: second
the girl brought an even number of sweets - this follows from the condition. And the first and
the third is the number of candies of the same parity (because triple an odd number is odd, and triple an even number is even). This means the total is an even number of candies. Otherwise - algebraically. The number of sweets brought is x 2 x 3 x 6 x 2 3 x – an even number. The girls ate 9 candies during recess - an odd number. Therefore, they must have an odd number of candies left, and they will not be able to divide it equally.
7th grade
7.1. Answer. 2209.
2209 + (2 +2 + 0 + 9) = 2222.
7.2. Answer. For example, like this: give five people two large cakes and one
the middle one, and the sixth - two medium ones and all four small ones.
Let m be the weight of a small cake, then the medium one weighs 2 m, and the large one weighs 3 m.
The total weight of all cakes is: 4 m 7 2 m 10 3 m 48 m, so one child
you should get cakes with a total weight of 8 m.
7.3. Answer. At 11:00.
If the distance covered by the train by 16:00 is S, then by 17:00 it has covered distance 1, 2 S.
This means that in the last hour the train has traveled 0.2 S, that is, it covers a path of length S in 5
hours. The initial movement time is 16 – 5 = 11 (hours).
7.4. The answer is shown in Figure 2.
Rice. 2
7.5. Answer.
7.5. Answer. He made a mistake.
The sum of two consecutive numbers is the sum of two numbers of different parities, and
therefore it is odd. This means that each of the classmates ate an odd number of candies.
Classmates are an odd number (27), and the sum of an odd number of odd
numbers is odd and cannot equal 210.
8th grade
8.1. Answer. 5 rub.
If x is the wholesale price of a pen, then when selling one for 10 rubles. the seller receives
profit 10 – x (rub.). Selling three pens for 20 rubles. he makes a profit of 20 – 3 x
(rub.). According to the condition, 10 – x = 20 – 3 x, whence x = 5 (rub.).
8.2. Let AL be the bisector of the acute angle CAB of the right triangle ABC
(ACB 90) and, by convention, AL BL. Then if CAB 2, then LAB, and, therefore, ABL. The sum of the acute angles of triangle ABC is 3, hence 30.
Then in the right triangle ACL the leg opposite the angle of 30 is equal to
half of the hypotenuse, from where
C.L.A.L. The statement has been proven.
8.3. Answer. 1.
Let's transform this equality: a 2 b 2 (a b ) 0 or (a b )(a b 1) 0 . By
condition, these numbers are different. Therefore the first parenthesis is not zero. Means,
a b 1 0, whence a b 1.
8.4. Answer. At 19 m.
It follows from the condition that the student’s speed is B A , and
student C's speed is 0.9 of student B's speed. It follows that
student C's speed is 0.81 of student A's speed. So, when A runs
100m, student C will run 81m.
8.5. Answer. Because the number of remaining sweets was odd, that is, it could not
divide by 10.
At the beginning, the number of candies was even, since it was divisible by 10. The total number of candies eaten at the beginning is 1 + 2+ 3 + ... + 10 = 55 - an odd number.
Therefore, the number of remaining candies is odd, as the difference between even and odd
numbers.
9th grade
9.1. Answer. 2.
The length of the side of this square is the distance between the lines x y 0 and x y 2, so
as on each of the straight lines there are two vertices of the square. And this distance is equal
the distance from the origin to the straight line x y 2 intersecting the coordinate axes at a distance of 2 from the origin. This means that the required distance is the height in
an isosceles right triangle with legs of length 2, which is equal to 2.
9.2. Answer. 1200 people.
Let x be the number of men and y the number of women on this island. From the condition
follows that
x y, in addition, x y 1900. Solving this system, we get:
x 900, y 1000 . Hence the number of married men is equal to
900 600
And the general
the number of married people is 1200.
9.3. Answer. 2:1.
The bisector of angle CAO is the altitude of triangle CAO, so CA AO. But
OA OC are like radii, which means triangle CAO is equilateral. Then
ACO 60. Moreover, in an isosceles triangle OCB (OC OB)
COB 120, so OCB 30 (otherwise this can be obtained by using
the fact that ACB - based on the diameter, is equal to 90).
9.4. Answer. 225.
If a three-digit number has 3 in the first place, then the other two digits are
arbitrary, different from 3. This means that any of the 9 others can take second place
digits, and on the third - any of the 9 other digits - a total of 9x9 = 81 options. If three
is in second place, then any number except 3 and 0 can be in first place, and on
the last - any, except three. In total we get 8x9 = 72 options. As many
we will get options if the three is in last place. Total: 81 + 72 +
72 = 225 options.
9.5. Answer. Will be able.
If Petya thinks of a number with two digits of different parities, then his mother needs to name,
for example, the number 20. Then the parity of each of the last two digits after each
the increase will continue, and these numbers will never coincide. If the numbers
Petya's numbers will be of the same parity, then mom just needs to name the number 50. After
every two additions, the last two digits will be repeated, i.e. will not
coincide, and after the first (third, fifth, etc.) addition these numbers will be
have different parities, i.e. won't match either.
