IMO 2004 SL A7

Let a1, a2, . . . , an be positive real numbers, n > 1. Denote by

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IMO 2004 SL A7

Origin: IRE | Category: Algebra

Problem

Let a1, a2, . . . , an be positive real numbers, n > 1. Denote by gn their geometric mean, and by A1, A2, . . . , An the sequence of arithmetic means defined by Ak = a1+a2+\cdot\cdot\cdot+ak k , k = 1, 2, . . ., n. Let Gn be the geometric mean of A1, A2, . . . , An. Prove the inequality n n  Gn An

  • gn Gn \leqn + 1 and establish the cases of equality.

Solution

Let us set ck = Ak−1/Ak for k = 1, 2, . . ., n, where we define A0 = 0. We observe that ak/Ak = (kAk −(k −1)Ak−1)/Ak = k −(k −1)ck. Now we can write the LHS of the inequality to be proved in terms of ck, as follows: n  Gn An

n2 c2c2 3 \cdot \cdot \cdot cn−1 n and gn Gn

n n

k=1 (k −(k −1)ck).

By the AM −GM inequality we have n n2 1n(n+1)/2c2c2 3 . . . cn−1 n \leq1 n

n(n + 1) + n  k=2 (k −1)ck

= n + 1

  • 1 n n  k=1 (k −1)ck. (1) Also by the AM–GM inequality, we have n n

k=1 (k −(k −1)ck) \leqn + 1 −1 n n  k=1 (k −1)ck. (2) Adding (1) and (2), we obtain the desired inequality. Equality holds if and only if a1 = a2 = \cdot \cdot \cdot = an.