An Elementary Inequality Problem

By AM-GM $$\prod\limits_{i=1}^{n+1}\frac{x_{i}}{x_i+1}=\prod\limits_{i=1}^{n+1}\sum\limits_{k\neq i}\frac{1}{x_k+1}\geq\prod\limits_{i=1}^{n+1}\frac{n}{\sqrt[n]{\prod\limits_{k\neq i}(1+x_k)}}=\frac{n^{n+1}}{\prod\limits_{i=1 }^{n+1}(1+x_i)}$$ and we are done!

From last, let $\sqrt[n]{t_1t_2...t_{n+1}}=A$ with $AM-GM$: $$\frac{t_2+t_3+...+t_{n+1}}{n}\geq\sqrt[n]{t_2t_3...t_{n+1}}=\frac{A}{\sqrt[n]{t_1}}$$ so for the first $$t_2+t_3+...+t_{n+1}\geq\frac{nA}{\sqrt[n]{t_1}}$$ and for the rest take like this.