MÓSP-Blue Group 2005

Question

cho a,b,c la các số thực thỏa mãn a.b.c=1 cm$\frac{a^{3}}{(1+b)(1+c)}+\frac{b^{3}}{(1+c)(1+a)}+\frac{c^{3}}{(1+b)(1+a)}\geq \frac{3}{4}$

score 8 · Answer 1

ÁD BĐT Cauchy:$\frac{a^{3}}{(1+b)(1+c)}+\frac{1+b}{8}+\frac{1+c}{8}\geq \frac{3a}{4}$

TT$\geq VT\geq \frac{3}{4}(a+b+c)-\frac{a+b+c}{4}-\frac{3}{4}\geq \frac{3}{4}$(do $a+b+c\geq 3\sqrt[3]{abc}=3)$

Nguyễn Nhung · Answer 2 · 8/5/2016 11:47:29 PM

ta có $\frac{a^{3}}{(1+b)(1+c)} +\frac{1+b}{8} +\frac{1+c}{8} \geq \frac{3}{4}a$

TT $\Rightarrow VT \geq \frac{1}{2}(a+b+c)-\frac{3}{4}\geq \frac{3}{2} \sqrt[3]{abc} -\frac{3}{4}=\frac{3}{4}$

Dấu "=" $\Leftrightarrow a=b=c=1$

Trả lời 05-08-16 11:47 PM

Nguyễn Nhung
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uk.hehe........ – Nguyễn Nhung 06-08-16 12:10 AM

uh, chỉ chờ đến h phút này :3 hahahahah – Confusion 06-08-16 12:09 AM

tui cũng z. hãy hết lượt oi – Nguyễn Nhung 06-08-16 12:07 AM

h ms vote đc :3 – Confusion 06-08-16 12:06 AM

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