Cho a,b,c>0. CMR:
$\sum \frac{a^2}{2a^2+(b+c-a)^2}\leq 1$
Cho a,b,c>0. CMR:
$\sum \frac{a^2}{2a^2+(b+c-a)^2}\leq 1$
$\sqrt[LOVE]{MATH}$
"If I feel unhappy, I do mathematics to become happy. If I am happy, I
do mathematics to keep happy" - Alfréd Rényi
Cho a,b,c>0. CMR:
$\sum \frac{a^2}{2a^2+(b+c-a)^2}\leq 1$
$2a^{2} + (b + c - a)^{2} = a^{2} + a^{2} + (b + c - a)^{2} \geq \frac{(a + a + b + c - a)^{2}}{3} = \frac{(a + b + c)^{2}}{3} => dpcm$
$\int{x^{2} + (y - \sqrt[3]{x^{2}})^{2} = 1}$
I Love CSP
$2a^{2} + (b + c - a)^{2} = a^{2} + a^{2} + (b + c - a)^{2} \geq \frac{(a + a + b + c - a)^{2}}{3} = \frac{(a + b + c)^{2}}{3} => dpcm$
Đến đây ngược dấu r ạ
$\sqrt[LOVE]{MATH}$
"If I feel unhappy, I do mathematics to become happy. If I am happy, I
do mathematics to keep happy" - Alfréd Rényi
Đặt $\left ( b+c-a;a+c-b;a+b-c \right )\rightarrow \left ( x;y;z \right )$
$\Leftrightarrow$ $\sum \frac{\left ( y+z)^{2} \right )}{\left [ 4x^{2}+2\left (y+z) ^{2}\right ) \right ]}$
$\Leftrightarrow \sum \frac{x^{2}}{2x^{2}+(y+z)^2}\geq \frac{1}{2}$
Cauchy-schwarz:
$\sum \frac{x^{4}}{2x^4+(xy+xz)^2}\geq \frac{(x^2+y^2+z^2)^2}{2\sum x^4+\sum (xy+xz)^2}\geq \frac{(x^2+y^2+z^2)^2}{2\sum x^4+4\sum x^2y^2}=\frac{1}{2}$
Cho a,b,c>0. CMR:
$\sum \frac{a^2}{2a^2+(b+c-a)^2}\leq 1$
Chuẩn hóa a+b+c=3.
Chứng minh a2$\frac{a^{2}}{2a^{2}+(b+c-a)^{2}}=\frac{a^{2}}{4a^{2}-12a+9}\leq \frac{-1}{3}+ 2/3a$
Bài viết đã được chỉnh sửa nội dung bởi Diepnguyencva: 28-02-2018 - 21:47
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