a) \(\left(a+b+c\right)^2=3\left(ab+bc+ac\right)\) 

  \(a^2+b^2+c^2+2ab+2ac+2bc-3ab-3ac-3bc=0\) 

 \(a^2+b^2+c^2-ab-ac-bc=0\) 

\(2\left(a^2+b^2+c^2-ab-ac-bc\right)=0\) 

 \(2a^2+2b^2+2c^2-2ab-2ac-2bc=0\) 

\(\left(a^2-2ab+b^2\right)+\left(a^2-2ac+c^2\right)+\left(b^2-2bc+c^2\right)=0\) 

\(\left(a-b\right)^2+\left(a-c\right)^2+\left(b-c\right)^2=0\) 

\(\Rightarrow a=b=c\left(đpcm\right)\)

(a-b)²+(b-c)²+(c-a)²=4(a²+b²+c²-ab-ac-bc)

<=>a²-2ab+b²+b²-2bc+c²+c²-2ac+a²=4a²+4b²+4c²-4ab-4ac-4bc

<=>2a²+2b²+2c²-2ab-2bc-2ac-4a²-4b²-4c²+4ab+4ac+4bc=0

<=>2ab+2ac+2bc-2a²-2b²-2c²=0

<=>-[(a²-2ab+b²)+(b²-2bc+c²)+(a²-2ac+c²)]=0

<=>(a²-2ab+b²)+(b²-2bc+c²)+(a²-2ac+c²)=0

<=>(a-b)²+(b-c)²+(c-a)²=0

Vì \(\hept{\begin{cases}\left(a-b\right)^2\ge0\\\left(b-c\right)^2\ge0\\\left(a-c\right)^2\ge0\end{cases}\Rightarrow\left(a-b\right)^2+\left(b-c\right)^2}+\left(a-c\right)^2\ge0\)

Dấu "=" xảy ra <=> \(\left(a-b\right)^2=\left(b-c\right)^2=\left(a-c\right)^2=0\)

<=>a-b=b-c=a-c

<=>a=b=c(đpcm)

Đặt vế trái BĐT cần chứng minh là P, ta có:

\(\dfrac{ab}{a^2+b^2}+\dfrac{bc}{b^2+c^2}+\dfrac{ca}{c^2+a^2}=\dfrac{1}{c\left(a^2+b^2\right)}+\dfrac{1}{a\left(b^2+c^2\right)}+\dfrac{1}{b\left(c^2+a^2\right)}\)

\(\ge\dfrac{9}{a\left(b^2+c^2\right)+b\left(c^2+a^2\right)+c\left(a^2+b^2\right)}\ge\dfrac{9}{2\left(a^3+b^3+c^3\right)}\)

\(\Rightarrow P\ge a^3+b^3+c^3+\dfrac{9}{2\left(a^3+b^3+c^3\right)}\ge3\sqrt[3]{\left(\dfrac{a^3+b^3+c^3}{2}\right)^2.\dfrac{9}{2\left(a^3+b^3+c^3\right)}}\)

\(=3\sqrt[3]{\dfrac{9\left(a^3+b^3+c^3\right)}{8}}\ge3\sqrt[3]{\dfrac{27abc}{8}}=\dfrac{9}{2}\)

Dấu "=" xảy ra khi \(a=b=c=1\)

Chính bài của em:

Cho \(a,b,c\ge1\). CMR: \(a\left(b+c\right)+b\left(c+a\right)+c\left(a+b\right)+2\left(\dfrac{1}{a^2+1}+\dfrac{1}{b^2+1}... - Hoc24

Áp dụng bất đẳng thức Holder ta có:

\(\left(\dfrac{a}{\sqrt{a^2+8bc}}+\dfrac{b}{\sqrt{b^2+8ca}}+\dfrac{c}{\sqrt{c^2+8ab}}\right)\left(\dfrac{a}{\sqrt{a^2+8bc}}+\dfrac{b}{\sqrt{b^2+8ca}}+\dfrac{c}{\sqrt{c^2+8ab}}\right)\left(a\left(a^2+8bc\right)+b\left(b^2+8ca\right)+c\left(c^2+8ab\right)\right)\ge\left(a+b+c\right)^3\).

Do đó ta chỉ cần chứng minh \(\left(a+b+c\right)^3\ge a\left(a^2+8bc\right)+b\left(b^2+8ca\right)+c\left(c^2+8ab\right)\Leftrightarrow3\left(a+b\right)\left(b+c\right)\left(c+a\right)\ge24abc\Leftrightarrow\left(a+b\right)\left(b+c\right)\left(c+a\right)\ge8abc\). Đây là một bđt rất quen thuộc

Không Holder thì Svacxo nha :v

Áp dụng BĐT Svacxo ta có :

\(\dfrac{a^2}{a\sqrt{a^2+8bc}}+\dfrac{b^2}{b\sqrt{b^2+8ac}}+\dfrac{c^2}{c\sqrt{c^2+8ab}}\ge\dfrac{\left(a+b+c\right)^2}{a\sqrt{a^2+8bc}+b\sqrt{b^2+8ac}+c\sqrt{c^2+8ab}}\)

Ta có sẽ đi chứng minh :

\(a\sqrt{a^2+8bc}+b\sqrt{b^2+8ac}+c\sqrt{c^2+8ab}\le\left(a+b+c\right)^2\)

Thật vậy theo Bunhiacopxki có :

\(a\sqrt{a^2+8bc}+b\sqrt{b^2+8ac}+c\sqrt{c^2+8ab}=\sqrt{a}\sqrt{a^3+8abc}+\sqrt{b}\sqrt{b^3+8abc}+\sqrt{c}\sqrt{c^3+8abc}\)

\(\le\sqrt{\left(a+b+c\right)\left(a^3+b^3+c^3+24abc\right)}\)

Ta lại đi chứng minh :

\(a^3+b^3+c^3+24abc\le\left(a+b+c\right)^3\)

\(\Leftrightarrow24abc\le3\left(a+b\right)\left(b+c\right)\left(c+a\right)\) ( Đây là BĐT đúng )

Do đó nhân vào ta có đpcm.