Swedish shortlisted inequality
Source: IMO ShortList 1988, Problem 24, Sweden 2, Problem 74 of ILL
November 9, 2005
inequality systemInequalitySequenceLinear RecurrencesIMO Shortlist
Problem Statement
Let be a sequence of non-negative real numbers such that:
a_k \minus{} 2 a_{k \plus{} 1} \plus{} a_{k \plus{} 2} \geq 0
and \sum^k_{j \equal{} 1} a_j \leq 1 for all k \equal{} 1,2, \ldots. Prove that:
0 \leq a_{k} \minus{} a_{k \plus{} 1} < \frac {2}{k^2}
for all k \equal{} 1,2, \ldots.