MathDB

Old inequality in new clothing

Source: 6th German TST 2006, 21 may 2006, problem 3

December 29, 2006

inequalitiesinductioninequalities proposed

Problem Statement

Let

n

be a positive integer, and let

b_{1}

b_{2}

, ...,

b_{n}

n

positive reals. Set

a_{1}=\frac{b_{1}}{b_{1}+b_{2}+...+b_{n}}

and

a_{k}=\frac{b_{1}+b_{2}+...+b_{k}}{b_{1}+b_{2}+...+b_{k-1}}

for every

k>1

. Prove the inequality

a_{1}+a_{2}+...+a_{n}\leq\frac{1}{a_{1}}+\frac{1}{a_{2}}+...+\frac{1}{a_{n}}