[Archive!] Pure mathematics, physics, chemistry, etc.: brain-training problems not related to trade in any way - page 213

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alsu >>:
Я вот не удержался, на RSDN сходил. Там получили машинное решение 25, но аналитического таки нет

Can I admire the maximum number?

 
Mathemat >>:

Спасибо, Андрей, но все же надеюсь, что можно будет как-то обойтись без этой каши :)

Ок, уж эта-то точно решается без индукции:

Доказать, что из n заданных натуральных можно всегда выбрать несколько (минимум одно) таких, что их сумма делится на n.

P.S. Пардон, задача тривиальна.

P.P.S. Нет, нетривиальна.

No, it's still trivial:)


you can consider n sums X1=x1, X2=x1+x2, X3=x1+x2+x3, ..., Xn=x1+x2+...+xn. If at least one of them is divisible by n, then the problem is solved. If not, you can find at least one pair that has the same residue of division by n (since there are exactly n-1 residues, excluding 0.) This means that the difference of these two sums, representing by itself the sum of the numbers included in one sum and not in the other, is divisible by n.

 

For brain training and may be useful for trading: http://www.chess.com/members/view/AIS1

 
Ais >>:

For brain training and may be useful for trading: http://www.chess.com/members/view/AIS1

Yeah, like if you can beat the market at chess, you can beat the market at orchestra

 

Yes

Attack rules

 

alsu писал(а) >>

n sums X1=x1, X2=x1+x2, X3=x1+x2+x3, ..., Xn=x1+x2+...+xn. If at least one of them is divisible by n, then the problem is solved. If not, you can find at least one pair that has the same residue of division by n (since there are exactly n-1 residues, excluding 0.) And that means that the difference of these two sums, representing by itself the sum of those numbers which are included in one sum and not in the other, is divisible by n.

:)))

Shit. I've been turning over and over the lists of residuals of the original numbers themselves, my head is fractured... Didn't bother to look at the rest of the sums... :)

Well done, Alexey!

 
Ais >>:
Yes

OBHSS

Why are you so English?

 
alsu >>:

ОБХСС

Че это вас на английский пробило?


+10)))


Sounds like a form of protest

 

Ask the administration to install PT Sans support on the forum, so that members can express themselves in their native Tatar, if they wish, and not descend into primitive English

:)))))))

 

Alsu, yes, the solution of the n-number problem is almost primitive. I have figured out the remainders of numbers, but I haven't figured out the remainders of sums.

OK, just to keep the thread alive, so it won't die for lack of progress:

Which polygon inscribed in the given circle has maximal sum of squares of its sides?

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