1)Let $\displaystyle a_1,a_2,...,a_n$ be real (or complex) distinct numbers where $\displaystyle n\geq 2$. Define $\displaystyle P_k = \prod_{j\not = k} (a_k - a_j)$. Can it be that $\displaystyle P_1=P_2=...=P_n$?

(For example, let $\displaystyle a_1=1,a_2=2,a_3=3$ then $\displaystyle P_1 = (1-2)(1-3)=2$, $\displaystyle P_2 = (2-1)(2-3)=-1$, and $\displaystyle P_3=(3-1)(3-2)=2$ 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 $\displaystyle f(x)$ and $\displaystyle g(x)$ be functions which you can differenciate. Note that $\displaystyle [f(x)g(x)]' = f'(x)g(x)+f(x)g'(x)$. Can you find a formula for $\displaystyle [f(x)g(x)]^{(n)}$ where by $\displaystyle ^{(n)}$ means to preform differenciation $\displaystyle n$ times repeatedly.