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- December 4th 2007, 09:26 PM #1

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## Problem 43

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.

- December 4th 2007, 10:28 PM #2

- December 5th 2007, 07:11 AM #3

- December 5th 2007, 09:51 AM #4

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- December 11th 2007, 09:06 AM #5
Since I know the way your mind works, there is something wrong with what I am about to say...

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...

-Dan

- December 11th 2007, 09:37 AM #6

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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.

**Proof:**

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.