IMO 1967 LL CZS7

Origin: CZS

Problem

Find all real solutions of the system of equations x1 + x2 + \cdot \cdot \cdot + xn = a, x2 1 + x2 2 + \cdot \cdot \cdot + x2 n = a2, . . . . . . . . . . . . . . . . . . xn 1 + xn 2 + \cdot \cdot \cdot + xn n = an.

Solution

Let Sk = xk 1 + xk 2 + \cdot \cdot \cdot + xk n and let \sigmak, k = 1, 2, . . ., n denote the kth elementary symmetric polynomial in x1, . . . , xn. The given system can be written as Sk = ak, k = 1, . . . , n. Using Newton’s formulas k\sigmak = S1\sigmak−1−S2\sigmak−2+\cdot \cdot \cdot+(−1)kSk−1\sigma1+(−1)k−1Sk, k = 1, 2, . . . , n,

the system easily leads to \sigma1 = a and \sigmak = 0 for k = 2, . . . , n. By Vieta’s formulas, x1, x2, . . . , xn are the roots of the polynomial xn −axn−1, i.e., a, 0, 0, . . ., 0 in some order. Remark. This solution does not use the assumption that the xj’s are real.