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.
Tag Archives: permutation
2012-2 sum with a permutation
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\[ \sum_{\pi \in S_n } \frac{1}{x_{\pi(1)}} \frac{1}{x_{\pi(1)}+x_{\pi(2)}}\cdots \frac{1}{x_{\pi(1)}+\cdots+ x_{\pi(n-1)}}. \]