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Polynomials of the form P_n(x)=n!x^n+a_{n-1}x^{n-1}+...+a_1x+(-1)^n(n+1)

Source: 1979 Austrian-Polish Mathematical Competition p2

February 15, 2020
algebrapolynomial

Problem Statement

Find all polynomials of the form Pn(x)=n!xn+an−1xn−1+⋯+a1x+(−1)n(n+1)P_n(x)=n!x^n+a_{n-1}x^{n-1}+\dots+a_1x+(-1)^n(n+1) with integer coefficients, having nn real roots x1,…,xnx_1,\dots,x_n satisfying k≤xk≤k+1k \leq x_k \leq k+1 for k=1,…,nk=1, \dots,n.
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