IMO 1989 SL 5

Origin: COL

Problem

Consider the polynomial p(x) = xn+nxn−1+a2xn−2 +\cdot \cdot \cdot+an having all real roots. If r16 1 + r16 2 + \cdot \cdot \cdot + r16 n = n, where the rj are the roots of p(x), find all such roots.

Solution

According to the Cauchy–Schwarz inequality,

n  i=1 ai \leq

n  i=1 a2 i

n  i=1

= n

n  i=1 a2 i

. Since r1+\cdot \cdot \cdot+rn = −n, applying this inequality we obtain r2 1+. . .+r2 n \geqn, and applying it three more times, we obtain r16 1 + \cdot \cdot \cdot + r16 n \geqn, with equality if and only if r1 = r2 = . . . = rn = −1 and p(x) = (x + 1)n.