Ta có \(\left(b+c\right)^2\le2\left(b^2+c^2\right)\)

=> \(\frac{a^2}{a^2+\left(b+c\right)^2}\ge\frac{a^2}{a^2+2b^2+c^2}\)

=> \(VT\ge\Sigma\frac{a^4}{a^4+2b^2a^2+2a^2c^2}\ge\frac{\left(a^2+b^2+c^2\right)^2}{a^4+b^4+c^4+4\left(a^2b^2+b^2c^2+c^2a^2\right)}\)

=> \(VT\ge\frac{\left(a^2+b^2+c^2\right)^2}{\left(a^2+b^2+c^2\right)^2+2\left(a^2b^2+b^2c^2+a^2c^2\right)}\ge\frac{\left(a^2+b^2+c^2\right)^2}{\left(a^2+b^2+c^2\right)^2+\frac{2}{3}\left(a^2+b^2+c^2\right)^2}=\frac{3}{5}\)(ĐPCM)

Dấu bằng xảy ra khi a=b=c

\(\Leftrightarrow\frac{\left(b+c\right)^2+a^2-2a\left(b+c\right)}{\left(b+c\right)^2+a^2}+\frac{\left(a+c\right)^2+b^2-2b\left(a+c\right)}{\left(a+c\right)^2+b^2}+\frac{\left(b+a\right)^2+c^2-2c\left(a+b\right)}{\left(a+b\right)^2+c^2}\ge\frac{3}{5}\)

\(\Leftrightarrow3-2\left(\frac{a\left(b+c\right)}{\left(b+c\right)^2+a^2}+\frac{b\left(a+c\right)}{\left(a+c\right)^2+b^2}+\frac{c\left(a+b\right)}{\left(a+b\right)^2+c^2}\right)\ge\frac{3}{5}\)

\(\Leftrightarrow\frac{a\left(b+c\right)}{\left(b+c\right)^2+a^2}+\frac{b\left(a+c\right)}{\left(a+c\right)^2+b^2}+\frac{c\left(a+b\right)}{\left(a+b\right)^2+c^2}\le\frac{6}{5}\)

Chuẩn hóa \(a+b+c=3\) (hay đặt \(x=\frac{3a}{a+b+c};y=\frac{3b}{a+b+c};z=\frac{3c}{a+b+c}\))

BĐT cần chứng minh trở thành:

\(\frac{a\left(3-a\right)}{\left(3-a\right)^2+a^2}+\frac{b\left(3-b\right)}{\left(3-b\right)^2+b^2}+\frac{c\left(3-c\right)}{\left(3-c\right)^2+c^2}\le\frac{6}{5}\)

Ta có đánh giá: \(\frac{a\left(3-a\right)}{\left(3-a\right)^2+a^2}\le\frac{9a+1}{25}\) ; \(\forall a\in\left(0;3\right)\)

\(\Leftrightarrow\left(a-1\right)^2\left(2a+1\right)\ge0\) (luôn đúng)

Tương tự: \(\frac{b\left(3-b\right)}{\left(3-b\right)^2+b^2}\le\frac{9b+1}{25};\frac{c\left(3-c\right)}{\left(3-c\right)^2+c^2}\le\frac{9c+1}{25}\)

Cộng vế với vế: \(VT\le\frac{9\left(a+b+c\right)+3}{25}=\frac{30}{25}=\frac{6}{5}\) (đpcm)

Dấu "=" xảy ra khi \(a=b=c\)

a/ BĐT sai, cho \(a=b=c=2\) là thấy

b/ \(VT=\frac{a^4}{a^2+2ab}+\frac{b^4}{b^2+2bc}+\frac{c^4}{c^2+2ac}\ge\frac{\left(a^2+b^2+c^2\right)^2}{\left(a+b+c\right)^2}=\frac{\left(a^2+b^2+c^2\right)\left(a^2+b^2+c^2\right)}{\left(a+b+c\right)^2}\)

\(VT\ge\frac{\left(a^2+b^2+c^2\right)\left(a+b+c\right)^2}{3\left(a+b+c\right)^2}=\frac{1}{3}\left(a^2+b^2+c^2\right)\)

Dấu "=" xảy ra khi \(a=b=c\)

c/ Tiếp tục sai nữa, vế phải là \(\frac{3}{2}\) chứ ko phải \(2\), và hy vọng rằng a;b;c dương

\(VT=\frac{a^2}{abc.b+a}+\frac{b^2}{abc.c+b}+\frac{c^2}{abc.a+c}\ge\frac{\left(a+b+c\right)^2}{abc\left(a+b+c\right)+a+b+c}\)

\(VT\ge\frac{9}{3abc+3}\ge\frac{9}{\frac{3\left(a+b+c\right)^3}{27}+3}=\frac{9}{\frac{3.3^3}{27}+3}=\frac{9}{6}=\frac{3}{2}\)

Dấu "=" xảy ra khi \(a=b=c=1\)

Ta có:

\(a^3+b^3+b^3\ge3ab^2\) ; \(b^3+c^3+c^3\ge3bc^2\) ; \(c^3+a^3+a^3\ge3ca^2\)

Cộng vế với vế \(\Rightarrow a^3+b^3+c^3\ge ab^2+bc^2+ca^2\)

\(\frac{a^5}{b^2}+\frac{b^5}{c^2}+\frac{c^5}{a^2}=\frac{a^6}{ab^2}+\frac{b^6}{bc^2}+\frac{c^6}{ca^2}\ge\frac{\left(a^3+b^3+c^3\right)^2}{ab^2+bc^2+ca^2}\ge\frac{\left(a^3+b^3+c^3\right)^2}{a^3+b^3+c^3}=a^3+b^3+c^3\)

Dảnh àk =))

Cứ đăng đi - úng hộ ^^

https://hoc24.vn/id/2782086

@Nguyễn Việt Lâm

Câu 1 chuyên phan bội châu

câu c hà nội

câu g khoa học tự nhiên

câu b am-gm dựa vào hằng đẳng thử rồi đặt ẩn phụ

câu f đặt \(a=\frac{2m}{n+p};b=\frac{2n}{p+m};c=\frac{2p}{m+n}\)

Gà như mình mấy câu còn lại ko bt nha ! để bạn tth_pro full cho nhé !

Câu c quen thuộc, chém trước:

Ta có BĐT phụ: \(\frac{x^3}{x^3+\left(y+z\right)^3}\ge\frac{x^4}{\left(x^2+y^2+z^2\right)^2}\) \((\ast)\)

Hay là: \(\frac{1}{x^3+\left(y+z\right)^3}\ge\frac{x}{\left(x^2+y^2+z^2\right)^2}\)

Có: \(8(y^2+z^2) \Big[(x^2 +y^2 +z^2)^2 -x\left\{x^3 +(y+z)^3 \right\}\Big]\)

\(= \left( 4\,x{y}^{2}+4\,x{z}^{2}-{y}^{3}-3\,{y}^{2}z-3\,y{z}^{2}-{z}^{3 } \right) ^{2}+ \left( 7\,{y}^{4}+8\,{y}^{3}z+18\,{y}^{2}{z}^{2}+8\,{z }^{3}y+7\,{z}^{4} \right) \left( y-z \right) ^{2} \)

Từ đó BĐT \((\ast)\) là đúng. Do đó: \(\sqrt{\frac{x^3}{x^3+\left(y+z\right)^3}}\ge\frac{x^2}{x^2+y^2+z^2}\)

\(\therefore VT=\sum\sqrt{\frac{x^3}{x^3+\left(y+z\right)^3}}\ge\sum\frac{x^2}{x^2+y^2+z^2}=1\)

Done.