[TU Gifted Math#10]

Hint to all inequalities
1.For $x,y,z\in\mathbb{R}^+ ; xyz=1$ , prove that if $\frac{1}{x}+\frac{1}{y}+\frac{1}{z}\geq x+y+z$ Then $\frac{1}{x^2}+\frac{1}{y^2}+\frac{1}{z^2}\geq x^2+y^2+z^2$
2.algebraic manipulation and we must prove that $\frac{x_1}{n-1+x_1}+\frac{x_2}{n-1+x_2}+...+\frac{x_n}{n-1+x_n}\geq 1$ by C-S
3.delete $(a+b+c)$ from 2 sides and use weight AM-GM
4.Just expanding
5.Use AM-GM
Source
2.Romania
4.Hongkong 1997
5.CRUX 1994, vol 20, problem no 1907