IMO 1989 SL 16

Origin: ISR

Problem

The set {a0, a1, . . . , an} of real numbers satisfies the following conditions: (i) a0 = an = 0; (ii) for 1 \leqk \leqn −1, ak = c + n−1  i=k ai−k(ai + ai+1). Prove that c \leq 4n.

Solution

Define Sk = k i=0 ai (k = 0, 1, . . . , n) and S−1 = 0. We note that Sn−1 = Sn. Hence Sn = n−1  k=0 ak = nc + n−1  k=0 n−1  i=k ai−k(ai + ai+1) = nc + n−1  i=0 i  k=0 ai−k(ai + ai+1) = nc + n−1  i=0 (ai + ai+1) i  k=0 ai−k = nc + n−1  i=0 (Si+1 −Si−1)Si = nc + S2 n, i.e., S2 n −Sn + nc = 0. Since Sn is real, the discriminant of the quadratic equation must be positive, and hence c \leq 4n.