# 2013-23 Polynomials with rational zeros

Find all polynomials $$P(x) = a_n x^n + \cdots + a_1 x + a_0$$ satisfying (i) $$a_n \neq 0$$, (ii) $$(a_0, a_1, \cdots, a_n)$$ is a permutation of $$(0, 1, \cdots, n)$$, and (iii) all zeros of $$P(x)$$ are rational.

GD Star Rating
Let n be a positive integer and let Sn be the set of all permutations on {1,2,…,n}. Assume $$x_1+x_2 +\cdots +x_n =0$$ and $$\sum_{i\in A} x_i\neq 0$$ for all nonempty proper subsets A of {1,2,…,n}. Find all possible values of$\sum_{\pi \in S_n } \frac{1}{x_{\pi(1)}} \frac{1}{x_{\pi(1)}+x_{\pi(2)}}\cdots \frac{1}{x_{\pi(1)}+\cdots+ x_{\pi(n-1)}}.$