Chứng minh các bất đẳng thức:
1)$a+\frac{1}{a}\geq \frac{10}{3}$ $(a\geq 3)$
2)$\frac{1}{a+b+1}+\frac{1}{b+c+1}+\frac{1}{c+a+1}\leq 1, a,b,c> 0$ và $abc= 1$
3)$a+\frac{4}{b(a-b)^{2}}\geq 2\sqrt{2}(a> b>0)$
4)$\frac{2a^{3}+1}{4b(a-b)}\geq 3(a\geq \frac{1}{2},\frac{a}{b}> 1 )$
5)$a+\frac{27}{2(a-1)(a+1)^{3}}\geq \frac{5}{2}(a> 1)$
6)$2a+\frac{1}{(a-b)(b-c)(c+1)}\geq 4 (a> b> c> 0)$
7)$(a^{3}+b^{3}+c^{3})(\frac{1}{a}+\frac{1}{b}+\frac{1}{c})\geq (a+b+c)^{2}, (a,b,c> 0)$
8)$(1+x)(1+\frac{y}{x})(1+\frac{9}{\sqrt{y}})^{2}\geq 256(x,y> 0)$
9)$\frac{a^{3}}{b(c+a)}+\frac{b^3}{c(a+b)}+\frac{c^3}{a(b+c)}\geq \frac{a+b+c}{2}(a,b,c> 0)$
10)Cho $a,b,c> 0,abc=1$. Chứng minh $\frac{a}{ab+1}+\frac{b}{bc+1}+\frac{c}{ac+1}\geq \frac{3}{2}$
11) Cho $a,b,c> 0$, $a+b+c=1$ Chứng minh $\frac{1}{a+bc}+\frac{1}{b+ca}+\frac{1}{c+ab}\geq \frac{27}{4}$
12) Cho $a,b,c>0$ thỏa $a^{3}c+b^{3}a+c^{3}b=abc$ Chứng minh $\frac{b}{a^{2}+ab}+\frac{c}{b^{2}+bc}+\frac{a}{c^{2}+ca}\geq \frac{9}{2}$
Bài viết đã được chỉnh sửa nội dung bởi NancyLe: 06-08-2014 - 14:05