MathDB
Vietnam TST 1999 sum inequality

Source: Vietnam TST 1999 for the 40th IMO, problem 4

June 26, 2005
inequalitieslimitcalculusintegrationalgebra unsolvedalgebra

Problem Statement

Let a sequence of positive reals {un}n=1∞\{u_n\}^{\infty}_{n=1} be given. For every positive integer nn, let knk_n be the least positive integer satisfying: ∑i=1kn1i≥∑i=1nui.\sum^{k_n}_{i=1} \frac{1}{i} \geq \sum^n_{i=1} u_i. Show that the sequence {kn+1kn}\left\{\frac{k_{n+1}}{k_n}\right\} has finite limit if and only if {un}\{u_n\} does.
Back to ProblemsView on AoPS