Nguyen Le Phuong Thao

Chứng minh rằng:

$a)$ $\mid a \mid\ \leq\ 1,\ \mid b \mid\ \leq\ 1$ thì $\mid a+b \mid\ <\ \mid 1+ab \mid$

$b)$ $\mid x \mid\ + \mid y \mid\ + \mid z \mid\ \leq\ \mid x+y-z \mid + \mid y+z-x \mid + \mid z+x-y \mid$

$c)$ $x^2+4y^2=1$ thì $\mid x+y \mid\ \leq\ \frac{\sqrt{5}}{2}$

Cho $a,\ b,\ c$ là các số thực không âm, chứng minh rằng: 

$\frac{1}{a+3b}+\frac{1}{b+3c}+\frac{1}{c+3a}\geq \frac{1}{a+2b+c}+\frac{1}{a+b+2c}+\frac{1}{2a+b+c}$