IT-CESENATICO 2007 Problem 6

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