Chủ đề này có 4 trả lời

[DOTOANNANG]

$$\frac{(a-b)(b-c)}{ac}+\frac{(b-c)(c-a)}{ba}+\frac{(c-a)(a-b)}{cb} \le 0$$

 

(SCHUR)

[DOTOANNANG]

$$a\,b\,c\,\geq \, (\,a\,+\, b\,- \,c\,)\,(\,b\,+ \,c\,-\, a\,)\,(\,c\,+\, a\,-\, b\,)$$

 

$<\,=\,>$

 

$$a\,(\,a\,- \,b\,)\,(\,a\,-\, c\,)\,+\, b\,(\,b\,- \,c\,)\,(\,b\,- \,a\,)\,+ \,c\,(\,c\,-\, a\,)\,(\,c\,-\, b\,)\, \geq  \, 0$$

[DOTOANNANG]

$$\frac{a}{b}\,+\,\frac{b}{c}\,+\,\frac{c}{a}\,-\,\frac{3}{2}\,-\,\frac{a}{b+c}\,-\,\frac{b}{c+a}\,-\,\frac{c}{a+b}\,\geq \,\frac{(\,a-b\,)\,(\,b-c\,)}{ac}$$

[DOTOANNANG]

$$\sum  \, \,\frac{a}{b\,+\,c}\,\geq\, \frac{a^{\,3}\,+\,b^{\,3}\,+\,c^{\,3}\,-\, 3\,abc}{2\,(\,a+b\,)\,(\,b+c\,)\,(\,c+a\,)}\,+\, \frac{3}{2}$$

 

$<\,=\,>$

 

$$\sum \,a\,(\,a\,-\,b\,)\,(\,a\,-\,c\,)\,\geq \,0$$

 

 

 

https://diendantoanh...zx-ygeq-frac32/

[DOTOANNANG]

$$\sum\limits_{cyc} a^2\,(b+c)\,(a-b)\,(a-c) = \sum\limits_{cyc} a\,(a-b-c)^2\,(b-c)^2$$