Thread: Problem 43
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Old December 11th, 2007, 09:06 AM
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Originally Posted by ThePerfectHacker View Post
1)Let a_1,a_2,...,a_n be real (or complex) distinct numbers where n\geq 2. Define P_k = \prod_{j\not = k} (a_k - a_j). Can it be that P_1=P_2=...=P_n?
(For example, let a_1=1,a_2=2,a_3=3 then P_1 = (1-2)(1-3)=2, P_2 = (2-1)(2-3)=-1, and P_3=(3-1)(3-2)=2 but all three are not the same in this case).
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
a_1a_2 = a_1a_3 = a_2a_3
where all three are distinct?

Obviously not since a_1a_2 = a_1a_3 \implies a_2 = a_3.

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
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