Pure maths, physics, logic (braingames.ru): non-trade-related brain games - page 73

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(1) If a positive answer to a question can be verified quickly (in polynomial time) (using some auxiliary information called a certificate), is it true that the answer itself (together with the certificate) to that question can also be found quickly?
 
aharata:

(1) If a positive answer to some question can be quickly (in polynomial time) checked (using some auxiliary information called a certificate), is it true that the answer itself (together with the certificate) to this question can also be quickly found?
No, the simplest and widely known counterexample is multiplication of a large number being the product of two prime numbers
 
alsu:
No, the simplest and widely known counterexample is multiplication of a large number being the product of two prime numbers.
Although, of course, there is a quantum Shor algorithm, so in the context of this problem, this example might not work
 
This seems to be one of the unsolved problems of mathematics. Or am I confused about something.
 
Mathemat:
This seems to be one of the unsolved problems of mathematics. Or maybe I got it wrong.
Shh... someone's going to solve it and then we'll get a million quid... :-)
 
aharata:
Shh... someone will decide, and then we'll take the million quid... :-)
The American mathematician George Danzig, as a university graduate student, was once late for class and mistook the equations written on the board for his homework. It seemed more difficult than usual, but a few days later he was able to complete it. It turned out that he had solved two "unsolvable" problems in statistics that many scientists were struggling with. =)
 
aharata:
Shh... someone will decide, and then we'll take the million quid... :-)
What's there to solve, difficulty 1, let's do it now))
 

Most of all I was pleased with this problem (the classes are P and NP).

Nowadays, most mathematicians believe that these classes are not equal. According to a survey conducted in 2002 among 100 scientists, 61 think the answer is "not equal", 9 think it is "equal", 22 find it difficult to answer and 8 think the hypothesis is not deductible from the current system of axioms and thus cannot be proved or disproved.
 
Mathemat:

(4) Brainiac is shaped like a right triangle. The inner boundary divides it into two states of equal area. Describe the shape and location of the boundary if it is known to be continuous and of the shortest possible length.

Obviously, whatever the division, at least one part is a corner of the original triangle cut off by a curve (or straight line) from the rest. It is somewhat tedious, but easy enough to show that the shortest length while keeping the area 1/2 will be the segment that divides 2 sides of the triangle in the ratio 1:sqrt(2) each (i.e. cutting off the smaller equilateral triangle from the original).
 
alsu:
Obviously, whatever the division is, at least one of the parts is a corner of the original triangle cut off by a curve (or straight line) from the rest of the triangle. It is somewhat tedious, but easy enough to show that the shortest length while keeping the area 1/2 will be the segment that divides 2 sides of the triangle in the ratio 1:sqrt(2) each (i.e. cutting off the smaller equilateral triangle from the original).
IMHO it will not be a straight line there =) and you can prove it without being tedious at all
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