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Prove sum a_i/i >=a_n

Source: APMO 1999

March 18, 2006

logarithmsinequalitiesinequalities unsolvedinduction

Problem Statement

Let

a_1, a_2, \dots

be a sequence of real numbers satisfying

a_{i+j} \leq a_i+a_j

for all

i,j=1,2,\dots

. Prove that

a_1 + \frac{a_2}{2} + \frac{a_3}{3} + \cdots + \frac{a_n}{n} \geq a_n

for each positive integer

n