[Archive!] Pure mathematics, physics, chemistry, etc.: brain-training problems not related to trade in any way - page 206
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9th grade, though:)))
>> I keep thinking how smart school kids are.)
Вот я всё думаю, какие умные у нас школьники :)
If only they were a little bit stupider, we'd all live better lives. Otherwise, such a schoolboy grows up, becomes some kind of manager and, by virtue of his natural talent and outstanding intellect, immediately figures out how to quickly and painlessly take away everything that is not right :(
alsu, cheers!
I have a different parametrization (essentially the same):
x = ( 1 + 1/a )^a
y = ( 1 + 1/a )^( a + 1 )
These are really all solutions - in the regular domain of definition of the function f(x)=x^(1/x). The proof follows from the fact that f has only one extremum. This means that for 1/a > -1, "a" itself can be any real number.
But here we can construct solutions which do not fall there. For example, when a = -1/5 all is tip-top:
x = (-4)^(-1/5),
y = (-4)^(4/5).
alsu, зачод!
У меня другая параметризация (по сути та же):
х = ( 1 + 1/a )^a
y = ( 1 + 1/a )^( a + 1 )
Это действительно все решения - в регулярной области определения функции f(x)=x^(1/x). Доказательство вытекает из того, что у f есть только один экстремум.
Но тут можно конструировать и решения, которые туда не попадают. Например, при а = -1/5 все тип-топ:
x = (-4)^(-1/5),
y = (-4)^(4/5).
well, since the set of negative rational numbers is countable, we can assume that solutions outside the regular domain of definition are "almost none
The beauty of this problem is that y is not expressed in elementary functions on x, and the parametrization is elementary.
The next one is for those who are awake:
There are digits 1, 2, 3, ..., 9 in random order (each digit occurs once). Every three digits in a clockwise direction form a three-digit number. What is the sum of all nine numbers? Please give all possible answers.
Since each digit occurs exactly once in the units, tens and hundreds, S=(1+2+3+4+5+6+7+8+9)*(1+10+100)=4995
What are the other options, I don't even know:)
I don't see them myself.
1. There were five numbers written on the board. Adding each of these numbers to each one gave 10 sums: 0, 2, 4, 4, 6, 8, 9, 11, 13, 15. Which numbers were written on the board?
2. Prove that if in a quadrilateral each angle is greater than 89°, then each angle is less than 93°.
I'll try to get more serious from now on. Because I see some people here are getting bored with 9th grade problems... Here's the first one:
Prove that 2x+3u and 9x+5u divide by 17 with the same integers x and y.
(This is the way in the original. I suggest that you understand the condition yourself. The error is excluded: the problem book was published in Soviet times and must have been very carefully checked.)
Дальше буду стараться подбирать более серьезные. А то, вижу, кое-кому тут откровенно скучно с задачками для 9-х классов... Вот первая:
Доказать, что 2х+3у и 9х+5у делятся на 17 при одних и тех же целых х и у.
(Так в оригинале. Предлагаю самостоятельно понять условие. Ошибка исключена: задачник был издан в совковое время и наверняка был очень тщательно проверен.)
:)
if to denote
2x+3y=a
9x+5y=b,
solving this system with respect to x and y, we obtain
x=(-5a+3b)/17, y=(9a+2b)/17.
Thus, if a is divisible by 17, in order for x and y to be integers, we must require that b is also divisible by 17. Similarly, if b%17=0, we have to require that a%17=0. So, for any fixed integer values of x,y, both expressions can only be divisible by 17 at the same time.
I won't touch the first two:)
Here's a simple task for you (animal lovers and especially children will love it):
Which floor is the best place to throw a cat?
Вот вам простенькая задачка (понравится любителям животных, и особенно детям):
С какого этажа лучше бросать кошку?
These are all misconceptions. It's better not to throw it at all, because even if it does land on its paws, a hard blow won't do any good anyway.
Conversations in a style of "a cat from XX floor has fallen down, and nothing, and at the acquaintance from X floor has crashed" - are admissible on any auto forum. But here people understand what probability is %)