Chủ đề này có 4 trả lời

[Khoa Linh]

Cho $a,b,c>0$. Chứng minh: 

$(a^2+b^2+c^2)^3\geq 9abc(a^3+b^3+c^3)$

[Khoa Linh]

Một lời giải trên mạng. 

[DOTOANNANG]

$$\left ( 9\,abc\sum\limits_{cyc}a^{3}  \right )^{2}\leqq 81\,a^{2}b^{2}c^{2}\sum\limits_{cyc}a^{2}\sum\limits_{cyc}a^{4}\leqq \left ( \sum\limits_{cyc}a^{2} \right )^{6}$$

 

$$\Leftrightarrow  81\,a^{2}b^{2}c^{2}\sum\limits_{cyc} a^{4}\leqq \left ( \sum\limits_{cyc}a^{2} \right )^{5}$$

 

$$\Leftrightarrow  81\,abc\sum\limits_{cyc} a^{2}\leqq \left ( \sum\limits_{cyc}a \right )^{5}$$

 

$$\Leftrightarrow  \sum\limits_{cyc} \left \{ \left ( b- c \right )^{2}\left ( 8\,a^{3}+ b^{3}+ \frac{3}{4}\, b^{2}c\right ) \right \}+ \sum\limits_{cyc} \left \{ 5\left ( b- c \right )^{4}a \right \}+ \sum\limits_{cyc} \left \{ \frac{5}{4}\left ( 2\,a- b- c \right )^{2}b^{2}c \right \}\geqq 0$$

 

Spoiler

https://diendantoanh...geq81xyzx2y2z2/

 

[DOTOANNANG]

$$\left ( 9\,abc\sum\limits_{cyc}a^{3}  \right )^{2}\leqq 81\,a^{2}b^{2}c^{2}\sum\limits_{cyc}a^{2}\sum\limits_{cyc}a^{4}\leqq \left ( \sum\limits_{cyc}a^{2} \right )^{6}$$

 

$$\Leftrightarrow  81\,a^{2}b^{2}c^{2}\sum\limits_{cyc} a^{4}\leqq \left ( \sum\limits_{cyc}a^{2} \right )^{5}$$

 

$$\Leftrightarrow  81\,abc\sum\limits_{cyc} a^{2}\leqq \left ( \sum\limits_{cyc}a \right )^{5}$$

 

$$\Leftrightarrow  \sum\limits_{cyc} \left \{ \left ( b- c \right )^{2}\left ( 8\,a^{3}+ b^{3}+ \frac{3}{4}\, b^{2}c\right ) \right \}+ \sum\limits_{cyc} \left \{ 5\left ( b- c \right )^{4}a \right \}+ \sum\limits_{cyc} \left \{ \frac{5}{4}\left ( 2\,a- b- c \right )^{2}b^{2}c \right \}\geqq 0$$

 

Spoiler

https://diendantoanh...geq81xyzx2y2z2/

 

[mạnh hơn]

 

$\left ( \sum\limits_{cyc}a^{2}  \right )^{2}\left ( \sum\limits_{cyc}a \right )^{2}\geqq 27\,abc\left ( a^{3}+ b^{3}+ c^{3} \right )$

 

$\Leftrightarrow  \sum\limits_{cyc} \frac{\left ( a- b \right )^{2}}{6}\left[ 27\,c^{4}+ \left ( a- b \right )^{2}\left ( 2\,c^{2}+ 16\,bc+ 16\,ab+ 3\,c^{2}+ 3\,a^{2}+ 8\,ca \right ) \right]\geqq K \geqq 0$

 

==============================================================================

 

$\left [ K= \sum\limits_{cyc} \left ( a- b \right )^{2}\left ( 6\,c^{4}+ 4\left ( a+ b \right )c^{3}- \frac{\left ( a+ b \right )^{2}c^{2}}{2}- \left ( a+ b \right )^{3}c+ \frac{5\left ( a+ b \right )^{4}}{16} \right ) \right ]$

[DOTOANNANG]

Cho $a,b,c>0$. Chứng minh: 

$(a^2+b^2+c^2)^3\geq 9abc(a^3+b^3+c^3)$

 

[khó hơn]

 

$$\left ( a^{3} b+ b^{3} c+ c^{3} a\right )\left ( a^{2}+ b^{2}+ c^{2} \right )\geqq 3\,abc\left ( a^{3}+ b^{3}+ c^{3} \right )$$