[Archive!] Pure mathematics, physics, chemistry, etc.: brain-training problems not related to trade in any way - page 347

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Whole cells inside a polygon can be excluded from consideration. This leaves only segments coming out of vertices, such that no other segments come out of a given vertex (otherwise it would be an integer cell).
 
What about the segments that intersect the sides of a polygon?
 
Mathemat >>:
На бумаге "в клеточку" нарисован выпуклый многоугольник, так что все его вершины находятся в вершинах клеток и ни одна из его сторон не идёт по вертикали или горизонтали. Докажите, что сумма длин вертикальных отрезков линий сетки, заключённых внутри многоугольника, равна сумме длин горизонтальных отрезков линий сетки внутри многоугольника.

Draw horizontal lines L(i) and vertical lines H(j) through each cell (height=1)

The area is equal to:

S=L(1)/2+(L(1)+L(2))/2+...+(L(n-1)+L(n))/2+L(n)/2=L(1)+..+L(n)

S=H(1)/2+(H(1)+H(2))/+...+(H(m-1)+H(m))/2+H(m)/2=H(1)+..+H(m)

L(1)+..+L(n)=H(1)+..+H(m)

 
Yeah, ihor. It's all about squares.

Here's another one - just for those who play "on the money":

7 cards were taken out of the deck, shown to everyone, shuffled and dealt 3 cards each to Grisha and Lyosha, and the remaining card
a) hidden;
b) given to Kolya.
Grisha and Lyosha can take turns communicating aloud any information about their cards. Can they tell each other their cards so that Kolya cannot find out the location of any of the cards he cannot see? (Grisha and Lyosha did not agree on any special way of communicating; all negotiations are in plain text).
 
Here's what to solve:
A $1 million prize for solving each of the seven maths problems

http://nature.web.ru/db/msg.html?mid=1176827&s=120200000
 
Sergey, hi. Of all these problems, there is only one in my area of understanding - the Riemann problem.
 
Mathemat >>:
Сергей, привет. Из всех этим проблем в области моего понимания оказалась только одна - Римана.

Alexei Hi. Of all these problems in my area of understanding, none. :о) Well maybe a few where I just understand what it's about, or at least I think I do. :о) By the way, do I understand Cook's problem correctly, shifting to cryptography, that decryption with the presence of a key should be more time consuming than the encryption itself, including the whole development?

 
Well, it looks like it, although I don't understand the problem myself.
 

I don't mean to be rude, but all these problems are total ujdyj.
The real problems are: the cancer and AIDS problem, fusion, high temperature superconductivity, electricity and light storage, etc. These are the problems that need to be paid for.

 
Nobody's paying anybody any money here, Richie. Just having fun.
P.S. So, no one's ever paid you for your money, huh?
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