[Archive!] Pure mathematics, physics, chemistry, etc.: brain-training problems not related to trade in any way - page 306

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alsu писал(а) >>

But the difficulty of the problem is clearly for eighth-graders, not lower than the regional Olympiad.

Regional. More like regional.) I wasted 4 hours on it, it's hard to be a non-mathematician :)

 
Really you are rich for the time!!! Cool!!! If I could afford it, I would. :-)
 
alsu >>: Но задачка по сложности явно если и для восьмиклашек, то уровнем не ниже областной олимпиады.

All-Union :)

I try to choose not the most difficult ones.

But that last one didn't come from there.

 
coaster писал(а) >>
Really you are rich for the time!!! Cool!!! If I could afford it, I would. :-)


"The happiest people are those who can freely manage their time without any fear of consequences..."
© Max Otto von Stirlitz :)

 
Another one is a follow-up (9th):

First option: cross out all numbers less than the root of 1982 (from 2 to 44). There are 43 numbers in all. The one can be crossed out because the problem statement says "to the product of the other two".
Proof: If a number is found which is equal to the product of two others, then at least one of them is not greater than 44. But all numbers up to and including 44 are already crossed out.
Which one is less? Is it possible to cross out less than 43 numbers?
P.S. Sort those two out - remind me of 337.
 
Mathemat >>:
Еще одна - вдогонку (9-й):

Первый вариант: вычеркиваем все числа менее корня из 1982 (с 2 до 44). Всего 43 числа. Единичку можно не вычеркивать, т.к. в условии задачи указано "произведению двух других".
Доказательство: если находится число, равное произведению двух других, то хотя бы одно из них не больше 44. Но все числа до 44 включительно уже вычеркнуты.
Кто меньше? Можно ли вычеркнуть менее 43 чисел?
P.S. Разберемся с этими двумя - напомните мне о 337.

I may be wrong, but all the prime numbers will remain.

 
I don't get it. We're crossing out the natural ones, of course. Why do they all stay?
 
Mathemat >>:
Не понял. Мы вычеркиваем натуральные, конечно. Почему они все останутся?

If we keep only prime numbers, then none of the remaining numbers is equal to the product of the other two (except for one).

 
And how to find the minimum number of crossed out: make a multiplication table up to 1982, all results that do not fit in the table will be sought (assuming that only prime numbers remain)
 
sanyooooook >>:

если оставить только простые числа, то ни одно из оставшисля не будет равно произведению двух других(из оставшихся, кроме единицы)

You'll have to cross out a lot more compounds than 43.

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