[Archive!] Pure mathematics, physics, chemistry, etc.: brain-training problems not related to trade in any way - page 227
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This is a physics puzzle question. Out of three groups of students, only three students answered it correctly.
There is a capacitor consisting of two round copper plates of equal area, the dielectric is air.
The capacitor was charged to a voltage of 1000 volts, then the distance between its plates began to increase.
As they increased the distance between the plates, something strange happened: at a certain distance
the voltage of the capacitor dropped sharply to zero (the charge disappeared). Question: Explain why the charge disappeared.
The muzik flew between the plates and shorted out a thousand volts to itself.
The mucik flew between the plates and shorted out a thousand volts.
By the way, I still haven't found on the internet what a "mucik" is.
I was looking too. It's already been explained, "ze mandawaha :)".
Let's deal with at least one example (29 glasses with a grams and one glass with b grams), try to solve it in the general case. Let b = a + epsilon for certainty. Then after solving the problem positively there should be exactly a + epsilon/30 in each beaker.
On the other hand, how much milk can be in the glass after a finite number of steps? In the beginning it was like this:
a, a, a, ... a + epsilon
No matter how you combine the glasses in pairs, there can only be this much milk in the glass:
a + epsilon*sum(2^(-k_j))
(Put another way, the multiplier of the number epsilon is a finite binary fraction.) It is the binary notation that saves us here: if we add and divide in half two such different sums (in general, with different sets of powers), then the sum is of the same kind. OK, let's equate:
a + epsilon/30= a + epsilon*sum(2^(-k_j))
The number a is no longer quoted, we reduce and divide the remainder by epsilon. Well, the remaining equality is impossible with a finite sum on the right. It turns out that we do not get a + epsilon/30 in any cup. Where did I go wrong?
The most general case is probably very complicated, we can hardly do it. We can only argue that if the number of glasses is not equal to the power of two, then we can think of a case similar to ours, where the malcheg will fail. But that doesn't mean that all possible cases with that number of glasses are hopeless.
And, of course, it is obvious that for the number of glasses equal to a degree of two, nothing can be spoiled in any way, and the malscheg always can.
The next one (8, yes, yes, exactly 8... eh, how the poor eighth-graders are tortured): 252
.
Denote by a_n the integer closest to sqrt(n). Find the sum of 1/a_1 + 1/a_2 + ... + 1/a_1980.
P.S. It seems to be clear. OK, waiting for hypotheses.
Обозначим a_n целое число, ближайшее к sqrt(n). Найти сумму 1/a_1 + 1/a_2 + ... + 1/a_1980.
I can't vouch for that, as I got a C in maths. But: replace the sum by an integral (errors are more or less compensated for) and we get a good grade as 2 roots of 1979. Well, like that - count the legs and divide by 4.
No, no, imya, we need to find the exact solution and elementary (it's a problem for eighth-graders). What about integrals? I found the solution - it's really elementary.
Next (10th), as a follow-up, if someone has already solved the previous one: 460
The graph of the function y = f(x), defined on the entire number line, transforms to itself when rotated by an angle around the origin.
1) Prove that the equation f(x) = x has only one solution.
2) Give an example of such a function.
Honestly, I have no idea what kind of function this is yet.
We already have a trivial solution: it is any odd function (rotation angle equals Pi, i.e. it is centrally symmetric with respect to the origin). But for it the item. 1) is not necessarily satisfied (e.g. y = 5*sin(x) or a piece of a Taylor series up to the 5th degree for the same function).
It is probably assumed that this minimal angle is not a multiple of Pi.
Следующая (8-й, да-да, именно 8-й... эх, как же бедных восьмиклашек мучают): 252
Обозначим a_n целое число, ближайшее к sqrt(n). Найти сумму 1/a_1 + 1/a_2 + ... + 1/a_1980.
P.S. Кажись, понятно. ОК, ждем гипотез.
3/1 + 5/2+...89/44
88+1/1+1/2+...1/44
But I forgot how to calculate the sum of fractions...
Не-не, imya, надо точное решение найти и элементарное (это ж задачка для вось-ми-кла-шек). Какие уж там интегралы. Я нашел решение - оно и правда элементарно.
Следующая (10-й), вдогонку, если кто уже решил предыдущую: 460
График функции y = f(x), определенной на всей числовой прямой, переходит в себя при повороте на угол вокруг начала координат.
1) Доказать, что уравнение f(x) = x имеет только одно решение.
2) Привести пример такой функции.
Честно говоря, пока даже не представляю, что это за функция такая.
y=0*x
^))