Chủ đề này có 1 trả lời

[Khoa Linh]

Cho các số thực a, b, c thỏa mãn 

$(a+b+c)\left ( \frac{1}{a}+\frac{1}{b}+\frac{1}{c} \right )=10$

Chứng minh

$\left ( a^2+b^2+c^2 \right )\left ( \frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2} \right )\geq \frac{27}{2}$

[DOTOANNANG]

$$\boldsymbol{x= \frac{a}{b}+ \frac{b}{c}+ \frac{c}{a}, y= \frac{b}{a}+ \frac{c}{b}+ \frac{a}{c}}$$

$$\boldsymbol{x+ y= 7}$$

$$\boldsymbol{x^{2}= 2y+ \sum_{cyc}^{ }\frac{a^{2}}{b^{2}}; y^{2}= 2x+ \sum_{cyc}^{ }\frac{b^{2}}{a^{2}}}$$

$$\boldsymbol{P= x^{2}+ y^{2}- 2\left ( x+ y \right )+ 3\geq \frac{\left ( x+ y \right )^{2}}{2}- 2\left ( x+ y \right )+ 3= \frac{27}{2}}$$