IMO 1966 LL CZS26

Origin: CZS

Problem

(a) Prove that (a1 +a2 +\cdot \cdot \cdot+ak)2 \leqk(a2 1 +\cdot \cdot \cdot+a2 k), where k \geq1 is a natural number and a1, . . . , ak are arbitrary real numbers. (b) If real numbers a1, . . . , an satisfy a1 + a2 + \cdot \cdot \cdot + an \geq (n −1)(a2 1 + \cdot \cdot \cdot + a2n), show that they are all nonnegative.