Thread: Problem 43
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Old December 4th, 2007, 10:26 PM
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Default Problem 43
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).

The next problem is for the younger kids so please give them a chance.

2)Let f(x) and g(x) be functions which you can differenciate. Note that [f(x)g(x)]' = f'(x)g(x)+f(x)g'(x). Can you find a formula for [f(x)g(x)]^{(n)} where by ^{(n)} means to preform differenciation n times repeatedly.
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Last edited by ThePerfectHacker; December 6th, 2007 at 11:37 AM.
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