IMC 2005 · Problem 4

Find all polynomials $P(x) = a_n x^n + a_{n-1}x^{n-1} + \dots + a_1 x + a_0$ ($a_n \ne 0$) satisfying the following two conditions:

(i) $(a_0, a_1, \dots, a_n)$ is a permutation of the numbers $(0, 1, \dots, n)$

and

(ii) all roots of $P(x)$ are rational numbers.