IMO 1973 SL 10

Origin: SWE

Problem

Let a1, a2, . . . , an be positive numbers and q a given real number, 0 < q < 1. Find n real numbers b1, b2, . . . , bn that satisfy: (1) ak < bk for all k = 1, 2, . . ., n; (2) q < bk+1 bk < 1 q for all k = 1, 2, . . ., n −1; (3) b1 + b2 + \cdot \cdot \cdot + bn < 1+q 1−q(a1 + a2 + \cdot \cdot \cdot + an).

Solution

Let bk = a1qk−1 + \cdot \cdot \cdot + ak−1q + ak + ak+1q + \cdot \cdot \cdot + anqn−k, k = 1, 2, . . . , n. We show that these numbers satisfy the required conditions. Obviously bk \geqak. Further, bk+1 −qbk = −[(q2 −1)ak+1 + \cdot \cdot \cdot + qn−k−1(q2 −1)an] > 0 ; we analogously obtain qbk+1 −bk < 0. Finally, b1 + b2 + \cdot \cdot \cdot + bn = a1(qn−1 + \cdot \cdot \cdot + q + 1) + . . . +ak(qn−k + \cdot \cdot \cdot + q + 1 + q + \cdot \cdot \cdot + qk−1) + . . . \leq(a1 + a2 + \cdot \cdot \cdot + an)(1 + 2q + 2q2 + \cdot \cdot \cdot + 2qn−1) < 1 + q 1 −q (a1 + \cdot \cdot \cdot + an).