MathDB
Combinatorial inequality

Source: Romanian TST 1 2006, Problem 4

April 19, 2006
inequalitiesinductioninequalities proposed

Problem Statement

The real numbers a1,a2,…,ana_1,a_2,\dots,a_n are given such that ∣ai∣≤1|a_i|\leq 1 for all i=1,2,…,ni=1,2,\dots,n and a1+a2+⋯+an=0a_1+a_2+\cdots+a_n=0.
a) Prove that there exists k∈{1,2,…,n}k\in\{1,2,\dots,n\} such that ∣a1+2a2+⋯+kak∣≤2k+14. |a_1+2a_2+\cdots+ka_k|\leq\frac{2k+1}{4}.
b) Prove that for n>2n > 2 the bound above is the best possible.
Radu Gologan, Dan Schwarz
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