The seller has purchased a batch. Mathematical Olympiads and Olympiad Problems

Sell \u200b\u200bme this pen - this is the question asked to newbie candidates who are interviewed. This is a standard technique that is very effective. Within a minute, the second recruiting agent (well, or who conducts interviews there?) Will be able to understand whether the candidate is suitable. The reaction of people to this question is as follows:

  1. The first category of people is lost and do not understand what to do. They were taken by surprise and have no idea what to say or how to sell this fucking pen (sorry for the jargon).
  2. The second category of candidates is using templated expressions like: this is a very good pen that will never fail. The ink is of high quality, the pen itself is of high quality and reliable! Buy it and you will definitely not regret it.
  3. The third category is creative and creative personalitieswho eventually get the job. Their answers to questions like "sell me a pen" are non-standard and original. They make even the most experienced recruiting agents, who have heard the most varied versions, admire.

A simple example?

Understand the point: in order to sell a pen to a person, first of all, he must be motivated to buy, i.e. you should make you want to buy it. Here's a simple example:

- sell me this pen (the recruiting agent will tell you and at the same time he will definitely offer his pen - you will have to sell it);

(here you need to motivate him to buy)

- well. Will you give me your autograph? (tell you)

- Yes, but I do not have a pen (the interlocutor will answer. He gave it to you).

- then buy mine (you will answer and offer an item of trade).

Your interlocutor will have no choice but to buy a pen, because he promised to give you an avotgraph.

Here's a simple example - a video from the excellent The Wolf of Wall Street movie (I recommend watching it):

It is this method that is very effective at the moment, but in reality, sooner or later it will become hackneyed.

Another example of selling a pen in a job interview

The big boss was hiring a young man. As expected, he asked to sell him a pen. It was a very cool pen (most likely a valuable gift), expensive and with a gold outline.

Naturally, the candidate agreed. He tried to praise her and offer her in various ways to the boss, but he refused, explaining his position by the fact that he already had a pen.

- well, if you don’t want to buy, you don’t need to, - answered the young man;

- you gave up quickly, - the chief answered;

- then it's time for me to leave (gets up and leaves), - the candidate answers;

- wait! - shouts the chief - you have to give me my pen;

- yeah, I see that you have changed your mind. You're in luck, I already agree to sell the pen at half price;

- stop joking, I need my pen! Bring her back now. Otherwise, I'll call security;

- take it. But keep in mind: I was able to motivate you - the young man answered, gave the pen and left.

The boss thought for a while and realized that this was the best candidate of all with whom he was able to interview. Later it was he who got the job.

Instead of conclusions

In order to sell a pen in an interview, you need to motivate the other person to buy. The easiest way to do this is to ask for an autograph or to do as the young man did. It makes no sense to praise her corny - this is not what recruiting agents are waiting for. Preference is given to people who were able to show creativity and ingenuity.

Content

Task 1. (7 points)

We added the sum of its digits to the number and got 2017. Give an example of such a number.

Examples:2012, 1994. Other numbers don't fit.

Criteria.Any of these numbers are given: 7 points.

Any other number is given: 0 points.

Task 2. (7 points)

The girl replaced each letter in her name with its number in the Russian alphabet. It turned out the number 2011533. What is her name?

Answer:Tanya.

There may be other sequences of letters, but only one of them is a name.

Criteria.Received the name Tanya: 7 points.

Any other sequence of letters received: 0 points.

Task 3. (7 points)

The seller bought a batch of pens and sold them. At the same time, some buyers bought one pen for 10 rubles, and some bought 3 pens for 20 rubles. It turned out that the seller received the same profit from each purchase. Find the price at which the seller purchased the pens.

Answer:5 rubles.

Decision.Let the purchase price of the handle x... Then the profit per handle is 10 x, for 3 handles 20 3x... Solving equation 10 x= 20 3x, we get x= 5.

Criteria.

It is not substantiated that the purchase price of the pen should be 5 rubles, but it is verified that in this case the condition is fulfilled: 4 points.

Only the correct answer without any explanation: 2 points.

Task 4 (7 points)

The student spider pulled a web between 11 points so that its webs did not intersect anywhere, and at the end returned to the starting point. The spider-teacher praised him, and only noticed that real masters observe the same conditions, but stretch the “correct” web, in which no segments lie on one straight line. Help the spider connect all 11 points in the picture with the "correct" cobweb.

Decision.One of the possible ways is shown below.

Criteria.Any true example: 7 points.

An example in which some line segments lie on one straight line, but none neighboringthe segments do not lie on one straight line: 4 points.

Example with adjacent line segments: 0 points.

Task 5. (7 points)

The square window opening is formed by two rectangular frames. Inside each of them, a number was written equal to the perimeter of the frame. Write down what is the square side of the entire window opening and explain how you got it.

Answer:5.

Decision.Let the side of the square be aand the width of the left rectangle is b... Then the width of the right rectangle is ab.

The left rectangle gives the ratio 2 a+ 2b\u003d 14, and the right rectangle gives the ratio 2 a+2a2b\u003d 16. Adding these two ratios with each other, we get 6 a\u003d 30, whence a= 5.

Criteria.Correct solution in any way: 7 points.

Logically correct solution, but due to an arithmetic error, the answer is incorrect: 3 points.

An example was selected where the rectangles really have the required perimeter, but it was not explained how the lengths of their sides were obtained: 2 points.

Only the correct answer without explanation: 1 point.

It is not your fault that the product was not bought and its expiration date has expired.
It is your responsibility not to trade in expired goods! You can be disciplined for this - your employer. And Rospotrebnadzor, if detected, impose an administrative fine.

But nowhere is your obligation to pay the employer the cost of expired goods... Neither in the job description nor in the agreement on the mat. responsibility, nor in the laws.
If you are obliged to pay, this suggests that you are thereby forced to trade in expired goods that threaten the life and health of the population!

You need to file a complaint with Rospotrebnadzor, state. labor inspection. And submit an application to the employer - the head. The manager may give you a written answer or not answer at all, it is enough to sign in the copy of the application or the number of the outgoing number confirming your appeal.

Labor Code - Labor Code of the Russian Federation - Chapter 39. MATERIAL RESPONSIBILITY OF THE EMPLOYEE

Article 238. Material liability of an employee for damage caused to the employer. The employee is obliged to compensate the employer for the direct actual damage caused to him. Unearned income (lost profits) are not subject to collection from the employee. Direct actual damage means a real decrease in the employer's cash assets or deterioration of the specified property (including property of third parties held by the employer, if the employer is responsible for the safety of this property), and also the need for the employer to make costs or unnecessary payments for the acquisition, restoration of property or for compensation for damage caused by the employee to third parties. Part three is no longer valid. - Federal Law of June 30, 2006 N 90-FZ.

Article 239. Circumstances precluding material liability of the employee. Material liability of the employee is excluded in cases of damage due to force majeure, normal economic risk, extreme necessity or necessary defense, or the employer's failure to fulfill the obligation to ensure proper conditions for the storage of property entrusted to the employee.

Article 240. The right of the employer to refuse to recover damages from the employee. The employer has the right, taking into account the specific circumstances in which the damage was caused, to completely or partially refuse to collect it from the guilty employee. The owner of the organization's property can restrict the specified right of the employer in cases provided for by federal laws, other regulatory legal acts Russian Federation, laws and other regulatory legal acts of the constituent entities of the Russian Federation, regulatory legal acts of local governments, constituent documents of the organization.

Article 241. Limits of the employee's material liability. For the damage caused, the employee bears material responsibility within the limits of his average monthly earnings, unless otherwise provided by this Code or other federal laws.

Article 242. Full material responsibility of the employee. The full material responsibility of the employee consists in his obligation to compensate the direct actual damage caused to the employer in full. Material liability in the full amount of the damage caused can be imposed on the employee only in cases provided for by this Code or other federal laws. Employees under the age of eighteen bear full material liability only for intentional damage, for damage caused in a state of alcoholic, drug or other toxic intoxication, as well as for damage caused as a result of a crime or an administrative offense.

Article 243. Cases of full financial liability. Material liability in full amount of the damage caused shall be borne by the employee in the following cases:
1) when, in accordance with this Code or other federal laws, the employee is fully liable for damage caused to the employer in the performance of the employee's job duties;
2) shortage of values \u200b\u200bentrusted to him on the basis of a special written contract or received by him under a one-time document;
3) deliberate infliction of damage;
4) causing damage in a state of alcoholic, drug or other toxic intoxication;
5) damage caused as a result of criminal actions of an employee, established by a court verdict;
6) infliction of damage as a result of an administrative violation, if such is established by the relevant state body;
7) disclosure of information constituting a secret protected by law (state, official, commercial or other) in the cases provided for by federal laws;
8) damage caused not during the performance of the employee's labor duties. Material liability in full amount of damage caused to the employer can be established by an employment contract concluded with the deputy heads of the organization, the chief accountant.

Article 244. Written agreements on full material liability of employees Written agreements on full individual or collective (brigade) material liability, that is, on compensation to the employer for the damage caused in full for the shortage of property entrusted to employees, may be concluded with employees who have reached the age of eighteen and directly servicing or using monetary, commodity values \u200b\u200bor other property. Lists of jobs and categories of workers with whom these contracts may be concluded, as well as standard forms of these contracts are approved in the manner established by the Government of the Russian Federation.

Article 245. Collective (brigade) material liability for damage. When employees jointly perform certain types of work related to storage, processing, sale (vacation), transportation, use or other use of the values \u200b\u200btransferred to them, when it is impossible to delineate the responsibility of each employee for causing damage and conclude an agreement with him on compensation for damage in full, collective (brigade) material liability may be introduced. A written agreement on collective (brigade) material liability for damage is concluded between the employer and all members of the team (brigade). Under the agreement on collective (brigade) material liability, the values \u200b\u200bare entrusted to a predetermined group of persons to which full financial responsibility for their shortage is imposed. In order to be exempted from material liability, a member of the team (team) must prove that he is not guilty. In case of voluntary compensation for damage, the degree of guilt of each member of the team (team) is determined by agreement between all members of the team (team) and the employer. When recovering damage in court, the degree of guilt of each member of the team (brigade) is determined by the court.

Article 246. Determination of the amount of damage caused. The amount of damage caused to the employer in the event of loss and damage to property is determined by actual losses calculated based on market prices in effect in the area on the day of damage, but not lower than the value of the property according to accounting data, taking into account the degree of wear and tear of this property. a special procedure should be established for determining the amount of damage subject to compensation caused to the employer by theft, deliberate damage, shortage or loss of certain types of property and other valuables, as well as in cases where the actual amount of damage caused exceeds its nominal amount.

Article 247. The duty of the employer to establish the amount of damage caused to him and the reason for its occurrence. Before making a decision on compensation for damage by specific employees, the employer is obliged to conduct an inspection to establish the amount of damage caused and the reasons for its occurrence. To carry out such an inspection, the employer has the right to create a commission with the participation of relevant specialists. Requiring a written explanation from the employee to establish the cause of the damage is mandatory. In case of refusal or evasion of the employee from providing the specified explanation, an appropriate act is drawn up. (Part two as amended by Federal Law of June 30, 2006 N 90-FZ) The employee and (or) his representative have the right to get acquainted with all the materials of the inspection and appeal them in the manner established by this Code.

Article 248. Procedure for recovery of damage. The recovery from the guilty employee of the amount of damage caused, not exceeding the average monthly earnings, is carried out by order of the employer. The order can be made no later than one month from the date of the final determination by the employer of the amount of damage caused by the employee. If the monthly period has expired or the employee does not agree to voluntarily compensate the damage caused to the employer, and the amount of damage caused to be recovered from the employee exceeds his average monthly earnings, then the recovery can only be carried out by a court. If the employer does not comply with the established procedure for recovering damage, the employee has the right to appeal against the employer's actions in court. An employee who is guilty of causing damage to the employer may voluntarily compensate it in whole or in part. By agreement of the parties to the employment contract, compensation for damage with payment by installments is allowed. In this case, the employee submits to the employer a written commitment to compensate for the damage, indicating the specific timing of payments. In the event of the dismissal of an employee who gave a written commitment to voluntarily compensate for damage, but refused to compensate for the specified damage, the outstanding debt is recovered in court. With the consent of the employer, the employee can transfer to him for compensation for the damage caused by equivalent property or fix the damaged property. Compensation for damage is made regardless of bringing an employee to disciplinary, administrative or criminal liability for actions or omissions that have caused damage to the employer.

Article 249. Reimbursement of expenses related to employee training. In the event of dismissal without good reason before the expiration of the term stipulated by the employment contract or the training agreement at the expense of the employer, the employee is obliged to reimburse the costs incurred by the employer for his training, calculated in proportion to the time actually not worked after the end of training, unless otherwise provided by the employment contract or learning agreement.

Article 250. Reduction by the labor dispute settlement body of the amount of damage to be recovered from the employee The labor dispute resolution body may, taking into account the degree and form of fault, the financial situation of the employee and other circumstances, reduce the amount of damage to be recovered from the employee. from an employee is not made if the damage was caused by a crime committed for personal gain.

6th grade

6.1.

6.2.

cutting example).

6.3.

6.4.

boxes.

6.5.

Why did she decide that?

7th grade

7.1. Find some natural number such that if you add to it the amount

its numbers, you get 2222.

7.2. Mom bought 10 large cakes, 7 medium and 4 small ones. Small

the cake weighs half the average, and the large cake weighs three times the small. how

mom to 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. Train moving with constant speed, by 17:00 traveled 1.2 times longer,

than by 16:00. When did the train leave?

7.4. How to cut a 6x6 checkered square into four of the same size

perimeter figures 16 each, if you can cut only on the sides of the cells?

The side of the cell is 1.

7.5. Twenty-seven classmates ate candy at first and second breaks,

and 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 a response of 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 piece into 4 equal checkered pieces. (Bring one

cutting example).

6.3. They took half of the total number of apples from the box with apples, then another half

the remainder, then half of the new remainder, and finally half of the next remainder.

After that, there are 10 apples left in the box. How many apples were in the box at the beginning?

6.4. There are Christmas balls in three boxes: in one there are two red, in the other red

and blue, in the third - two blue balls. The boxes say: "Two red", "Red

and blue ”,“ Two blue ”. None of the inscriptions are known to be correct.

How, after pulling out just one ball, determine which box contains which balls?

Indicate from which box you want to take it and how then to determine the content

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 put it all together

candy together. After the friends ate 3 candies each, the first one left, and the second

divided the remaining candies equally. The third told the second that she was wrong.

Why did she decide that?

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 \u003d b 2 + a

8.4.

8.5.

Grade 9

9.1. Find the area of \u200b\u200ba square all of whose vertices lie on two lines:

x + y \u003d 0 and x + y \u003d 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 1900 people living there?

9.3. On a circle with a diameterAB and point C is chosen as center O so that

angle bisectorCAB perpendicular to radiusOC ... In what respect is the directCO

divides the angle ACB?

9.4. Find the number of three-digit numbers in the decimal notation of which is involved

exactly one digit 3.

9.5. Mom wants to punish Petya for a deuce in mathematics. They agreed on

next. Petya thinks of a two-digit number with different numbers and informs him

mom. After that, mom tells her two-digit number to Petya. Petya adds

mom's number to its number, then to the amount received, then to the newly received

amount, etc. until he has a sum ending in two

identical numbers. Will mom be able not to let Petya play football that day?

8th grade

8.1. A merchant has bought a batch of pens on 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-angled 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 numbersa and b satisfying the equality

a 2 + b \u003d b 2 + a

8.4. Three students A, B and C were running 100 meters.When A ran to the finish line, B

was 10 m behind him, also when B finished, C was 10 m behind him.

how many meters at the finish line was A ahead of C?

8.5. At Masha's birthday party, each of the 10 guests had an equal number of

sweets. During tea drinking, the first one ate one candy, the second - two, the third - three, and

etc., the tenth - 10 candies. Masha wanted to shift the

sweets so that again in front of everyone there is an equal number of sweets, but dad,

without looking at the table, said that she could not do it. Why did he decide that?

Preview:

6th grade

6.1. Answer. 149 and 284.

If the first digit is at least 3, then the second is at least 12, which is impossible. Means,

6.2. One example is shown in Figure 1. This example is not the only one.

Fig. one

6.3. Answer. 160 apples.

When half of the apples are taken from the box, then half of that remains in it.

the amount that came before. This means that before that there were twice as many apples.

Therefore, in the beginning there were 10x2x2x2x2 \u003d 160 apples in the box.

6.4. Answer. Out of the box "Red and Blue".

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 in

the one on which is written "Two red", there will be multi-colored balls, since it does not

two red (by condition) and not two blue (they are in the first box). In a box with

the inscription "Two blue" - two red balls. If we took out the red ball, then,

similarly, in the box "Two blue" there are colored balls, and in the box "Two red"

- blue balls.

6.5. Answer. Because the number of candies left must be odd.

The total number of sweets brought is even. This can be explained as follows: the 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 the triple odd number is odd, and the triple even number is even). This means that the total is an even number of sweets. Otherwise - algebraically. The number of sweets brought isx 2 x 3 x 6 x 2 3 x - even number. The girls ate 9 candies at recess - an odd number. Therefore, they must have an odd number of sweets, and it will not be possible 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 of them two large cakes and one

the middle one, and the sixth - two middle ones and all four small ones.

Let m - the weight of a small cake, then the average weighs 2m, and large - 3 m.

The total weight of all cakes is: 4m 7 2 m 10 3 m 48 m so one child

should get cakes with a total weight of 8m.

7.3. Answer. At 11:00.

If the distance traveled by the train by 16:00 isS , then by 17:00 he passed the way1, 2 S.

So, in the last hour, the train passed0, 2 S , that is, path lengthS it takes 5

hours. The initial movement time is 16 - 5 \u003d 11 (hours).

7.4. The answer is shown in Figure 2.

Fig. 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 parity, and

therefore it is odd. This means that each of the classmates ate an odd number of sweets.

Odnoklassniki - an odd number (27), and the sum of an odd number of odd

numbers are odd and cannot equal 210.

8th grade

8.1. Answer. RUB 5

If x - 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 - 3x

(rub.). By condition 10 -x \u003d 20 - 3 x, whence x \u003d 5 (rub.).

8.2. Let AL - bisector of an acute angleCAB right triangleABC

(ACB 90) and, by condition, AL BL. Then if CAB 2, then LAB, and, therefore, ABL ... The sum of the acute angles of a triangleABC is equal to 3, whence 30.

Then in a right triangleACL the leg, which lies opposite an angle of 30, is equal to

half of the hypotenuse, whence

CL AL ... The statement is proven.

8.3. Answer. one.

We 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 nonzero. Means,

a b 1 0, whence a b 1.

8.4. Answer. At 19 m.

It follows from the condition that the student's speedB A, and

student speedC is 0.9 times the student's speedB ... It follows that

student speedC is 0.81 of the student's speedA. So when A runs through

100 m, student C will run 81 m.

8.5. Answer. Because the number of remaining candies was odd, that is, it could not

divided by 10.

In the beginning, the number of candies was even, since it was divided by 10. The total number of candies eaten at the beginning is 1 + 2+ 3 +… + 10 \u003d 55 - an odd number.

Therefore, the number of remaining candies is odd, as the difference between even and odd

numbers.

Grade 9

9.1. Answer. 2.

The side length of this square is the distance between the straight linesx 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

distance from origin to linex y 2 intersecting the coordinate axes at a distance of 2 from the origin. Hence, the required distance is the height in

an isosceles right-angled triangle with legs of length 2, which is 2.

9.2. Answer. 1200 people.

Let x - the number of men,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

900 600

And the general

the number of people who are married is 1200.

9.3. Answer. 2: 1.

Angle bisectorCAO is the height of the triangleCAO, therefore CA AO. But

OA OC - as radii, so a triangleCAO - equilateral. Then

ACO 60. Also, in an isosceles triangleOCB (OC OB)

COB 120, so OCB 30 (otherwise it can be obtained by using

the fact that ACB - based on the diameter is 90).

9.4. Answer. 225.

If a three-digit number has the digit 3 in the first place, then the other two digits are

arbitrary, different from 3. Hence, in the second place can be any of 9 other

digits, and on the third - any of the other 9 digits - a total of 9x9 \u003d 81 options. If the three

is in second place, then any number, except 3 and 0, can be in the first place, and on

the last - any, except for the three. In total, it turns out 8x9 \u003d 72 options. As many

we will get variants if the triple is in the last place. Total: 81 + 72 +

72 \u003d 225 options.

9.5. Answer. Will be able.

If Petya comes up with a number with two digits of different parity, then mom needs to tell,

for example, the number 20. Then the parity of each of the last two digits after each

additions will persist and these numbers will never match. If the numbers

The pet's numbers will be of one parity, then it is enough for mom to tell 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 parity, i.e. will not match either.