Prove that $\frac1{1+a_1}+\frac1{1+a_2}+...+\frac1{1+a_k}\ge\frac k{1+\sqrt[k]{a_1a_2...a_k}}$
Prove that $\frac{1}{1+x_1}+\frac{1}{1+x_2}+\cdots+\frac{1}{1+x_n} \geq \frac{n}{\sqrt[n]{x_1x_2\cdots x_n}+1}$