1)Let be real (or complex) distinct numbers where . Define . Can it be that ?
(For example, let then , , and but all three are not the same in this case).
The next problem is for the younger kids so please give them a chance.
2)Let and be functions which you can differenciate. Note that . Can you find a formula for where by means to preform differenciation times repeatedly.
Consider the case n = 3. Then the question is can
where all three are distinct?
Obviously not since .
Now, there's got to be something screwy here because I can generalize this argument to larger n and come up with similar results. But I can't believe it would be this easy...
I came up with problem #1 accidently when I playing around with polynomials. Here is my original solution, however it seems to me that this problem is easy even if approached directly.
Let be real (or complex numbers) and define . The key step is to note that by using the general product rule for derivatives. Now, thus, , this means that cannot attain the same values at (meaning ) because otherwise the situation is that a degree polynomial attains the same value times for distinct numbers. Which is impossible. Thus, cannot all be the same.