[Archive!] Pure mathematics, physics, chemistry, etc.: brain-training problems not related to trade in any way - page 299

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Mathemat >>:
Доказать, что m*(m+1) не является степенью целого ни при каком натуральном m. 42

Well, it's between m^2 and (m+1)^2. It's between m^2 and (m+1)^2, i.e. between two adjacent squares, so why is there another one? Joker.

// Phew, man. I'm reading it wrong again. Do you mean ANY degree?

 
What do the squares have to do with it? Between the two adjacent squares 25 and 36 is cube 27. Got it?
P.S. No, of course not.
 
Mathemat >>:
А при чем тут квадраты-то? Между двумя смежными квадратами 25 и 36 находится куб 27. Уел?

You're caustic! I noticed it myself, but while I was writing.... :)

 
Well... m and m+1 are always mutually prime, so their product, if decomposed into prime factors, cannot contain them in repeating variants. That's all...
 
It's not all there yet, but it's clear. Why the difference between two identical degrees is greater than 1 is already straightforward.
 
Here's more (no numbers in the problem condition):

P.S. Um... 8th grade. No matinduction (if it could be applied) an eighth grader knows.
For three planets the proof is easy: there is a planet whose distance from any other planet is greater than the minimal one. But what's the next step?
 
Let's construct a system in which each planet is under observation.

Take a pair of planets (let's call them first and second) whose distance between them is minimal among all distances. Obviously, the astronomers on these planets observe each other.
Let us proceed with them as follows. If no one else observes any of the given planets, isolate them in some way from the others - for convenience. For example, circle them.
If at least one of them, e.g. the first one, is observed from the third planet, the distance from the third to the first is smaller than the distance from any other to the third. Since we want the third planet to be observed too, we have to find a fourth planet for this purpose, as the first and second are not suitable - they are already busy observing each other. Similarly, to "observe" the fourth one, we have to find the fifth one, and so on, until we come to the last one, which we cannot find an "observer" because the planets stock is exhausted. Therefore, to build a system with necessary for us property, at least the planets at minimum distance (the first and the second) should not be observed from other planets. Since we have isolated them, we can in the same way look at the system of the remaining planets: find the ones lying at minimum distance, etc. - and come to the same conclusion: two planets should be isolated. Obviously, we can construct a "fully observable" system if and only if all planets of the system can be split into such pairs. Hence, the number of planets must be even. If it is odd, this condition will never be fulfilled.
 
Zachod, alsu!
The next one (part b) will come later):
 
I propose that we make substitutions for the variables: use first-graders instead of eighth-graders and eleventh-graders instead of seventh-graders.
 
OK, let's replace them and reshuffle them. Just as long as they're the right height.
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