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Question

Cho $a,b,c,d,e,f \in R$ sao cho $ab+bc+cd+de+ef\geq 1$. CMR:

\[ a^2+b^2+c^2+d^2+e^2+f^2 \geq \frac{1}{\cos \frac{\pi }{7}}.\]

khangnguyenthanh · Answer 1 · 10/29/2012 7:59:50 PM

Đặt: $\cos\frac{\pi}{7}=\alpha$.
Ta có:
$4\alpha^2a^2+b^2\ge4\alpha ab$
$(4\alpha^2-1)b^2+\frac{4\alpha^2}{4\alpha^2-1}c^2\ge4\alpha bc$
$\frac{16\alpha^4-8\alpha^2}{4\alpha^2-1}(c^2+d^2)\ge\frac{32\alpha^4-16\alpha^2}{4\alpha^2-1}cd$
$\frac{4\alpha^2}{4\alpha^2-1}d^2+(4\alpha^2-1)e^2\ge4\alpha de$
$e^2+4\alpha^2f^2\ge4\alpha ef$
Dễ dàng chứng minh được: $\frac{32\alpha^4-16\alpha^2}{4\alpha^2-1}=4\alpha$
Suy ra: $4\alpha^2(a^2+b^2+c^2+d^2+e^2+f^2)\ge4\alpha(ab+bc+cd+de+ef)\ge4\alpha$
$\Rightarrow a^2+b^2+c^2+d^2+e^2+f^2\ge\frac{1}{\alpha}=\frac{1}{\displaystyle \cos\frac{\pi}{7}}$

Trả lời 29-10-12 07:59 PM