Question 23
For each integer $n \geq 2$, determine:
a. the largest real number $c_{n}$ such that $$
\frac{1}{1+a_{1}}+\frac{1}{1+a_{2}}+\ldots+\frac{1}{1+a_{n}} \geq c_{n}
$$ for each choice of positive real numbers $a_{1}, a_{2}, \ldots, a_{n}$ such that $a_{1} \cdot a_{2} \cdot \ldots \cdot a_{n}=1$;
b. The largest real number $d_{n}$ such that $$
\frac{1}{1+2 a_{1}}+\frac{1}{1+2 a_{2}}+\ldots+\frac{1}{1+2 a_{n}} \geq d_{n}
$$ for each choice of positive real numbers $a_{1}, a_{2}, \ldots, a_{n}$ such that $a_{1} \cdot a_{2} \cdot \ldots \cdot a_{n}=1$.
a. the largest real number $c_{n}$ such that $$
\frac{1}{1+a_{1}}+\frac{1}{1+a_{2}}+\ldots+\frac{1}{1+a_{n}} \geq c_{n}
$$ for each choice of positive real numbers $a_{1}, a_{2}, \ldots, a_{n}$ such that $a_{1} \cdot a_{2} \cdot \ldots \cdot a_{n}=1$;
b. The largest real number $d_{n}$ such that $$
\frac{1}{1+2 a_{1}}+\frac{1}{1+2 a_{2}}+\ldots+\frac{1}{1+2 a_{n}} \geq d_{n}
$$ for each choice of positive real numbers $a_{1}, a_{2}, \ldots, a_{n}$ such that $a_{1} \cdot a_{2} \cdot \ldots \cdot a_{n}=1$.