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Option 1

1.    The three divisions A, B and C of the trading firm were striving to maximize profits for the year. Economists have made the following assumptions:

1)    both division A will make a profit, and division C;

2)    department C will make a profit, and one of the two departments A or B will also make a profit;

3)    making profit A is tantamount to the fact that making a profit by department C will not be a sufficient reason for making a profit by department B.

Determine which divisions can get the most profit if one of the three assumptions is known to be false at the end of the year.

2.       There are two segments on the number line: P = [37; 60] and Q = [40; 77]. Indicate the smallest possible length of a segment A such that the formula

is identically true, that is, takes the value 1 for any value of the variable x.

1)    10 2) 30 3) 20 4) 25

3.    The number line contains two segments: P = [14,34] and Q = [24, 44]. Choose a segment A such that the formula

(x Î A) → ((x Î P) º (x Î Q))

is identically true, that is, takes the value 1 for any value of the variable x. If there are several such segments, indicate the one that is longest.

              1) [15, 29] 2) [25, 29] 3) [35.39] 4) [49.55]                                       

Additionally:

By the summer cottage season, Vasiliev decided to fulfill his old dreams - to repaint a country house. In the closet, he found four containers of paint - white, blue, yellow and black. The volumes of these paints are 2, 3, 4 and 5 liters. The containers were completely different - a bottle, a bucket, round and flat cans. The blue paint was not in the bucket or bottle. The least amount of paint was in the flat can, and this paint was yellow. Vasiliev mixed yellow and white paints, and it turned out 7 liters, which fit only in the bucket, where the white paint was. The volume of black paint was greater than the volume of blue. How much of each paint was there, and where was it?

Option 2

1. Which friend (Ivan, Peter, Alexey, Nikolay or Boris) collects stamps, if it is known that:

 - if Boris collects stamps, then Ivan and Nikolay collect them;

 - if Ivan collects them, then Peter also collects them;

 - of two friends (Petra and Aleksey), only one of them collects stamps;

 - Alexey collects stamps only if Nikolay collects them;

 - at least Nikolay or Boris collects stamps

2. There are three segments on the number line: P = [10, 25], Q = [15, 30] and R = [25,40]. Choose a segment A such that the formula

((x Î Q) → (x Ï R)) / \ (x Î A) / \ (x Ï P)

is identically false, that is, takes the value 0 for any value of the variable x.

1) [0, 15] 2) [10, 40] 3) [25, 35] 4) [15, 25]                                                  

3. Three intervals are given on the number line: P = (10, 15), Q = [5, 20] and R = [15,25]. Choose a segment A such that the expressions

(x Ï A) → (x Î P) and (x Î Q) → (x Î R)

take different values ​​for any x.

1) [7, 20] 2) [2, 15] 3) [5,12] 4) [20, 25]                                        

Additionally:

During the trip, five friends - Anton, Boris, Vadim, Dima and Grisha, got acquainted with a fellow traveler. They asked her to guess their names, and each of them made one true and one false statement:

Dima said: "My surname is Mishin, and Boris's surname is Khokhlov." Anton said: "Mishin is my surname, and Vadim's surname is Belkin." Boris said: "Vadim's last name is Tikhonov, and my last name is Mishin." Vadim said: "My surname is Belkin, and Grisha's surname is Chekhov." Grisha said: "Yes, my surname is Chekhov, and Anton's surname is Tikhonov."

What is the last name of each of your friends?