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Minimizing a sum of inverses when variables have product 1

Source: Italian National Olympiad 2007, problem 6

May 13, 2007

inductionfunctioncalculusderivativeinequalitiesinequalities proposed

Problem Statement

a) For each

n \ge 2

, find the maximum constant

c_{n}

such that

\frac 1{a_{1}+1}+\frac 1{a_{2}+1}+\ldots+\frac 1{a_{n}+1}\ge c_{n}

for all positive reals

a_{1},a_{2},\ldots,a_{n}

such that

a_{1}a_{2}\cdots a_{n}= 1

. b) For each

n \ge 2

, find the maximum constant

d_{n}

such that

\frac 1{2a_{1}+1}+\frac 1{2a_{2}+1}+\ldots+\frac 1{2a_{n}+1}\ge d_{n}

for all positive reals

a_{1},a_{2},\ldots,a_{n}

such that

a_{1}a_{2}\cdots a_{n}= 1

.

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