2: Ta có: \(\dfrac{a^2}{b+c}+\dfrac{b^2}{c+a}+\dfrac{c^2}{a+b}=\dfrac{a\left(a+b+c\right)}{b+c}+\dfrac{b\left(a+b+c\right)}{c+a}+\dfrac{c\left(a+b+c\right)}{a+b}-a-b-c=\left(a+b+c\right)\left(\dfrac{a}{b+c}+\dfrac{b}{c+a}+\dfrac{c}{a+b}\right)=a+b+c-a-b-c=0\)

1: Sửa đề: Cho \(x,y,z\ne0\) và \(\dfrac{1}{x}+\dfrac{2}{y}+\dfrac{1}{z}=\dfrac{2}{2x+y+2z}\).

CM:....

Đặt 2x = x', 2z = z'.

Ta có: \(\dfrac{2}{x'}+\dfrac{2}{y}+\dfrac{2}{z'}=\dfrac{2}{x'+y+z'}\)

\(\Leftrightarrow\dfrac{1}{x'}+\dfrac{1}{y}+\dfrac{1}{z'}=\dfrac{1}{x'+y+z'}\)

\(\Leftrightarrow\dfrac{1}{x'}-\dfrac{1}{x'+y+z'}+\dfrac{1}{y}+\dfrac{1}{z'}=0\)

\(\Leftrightarrow\dfrac{y+z'}{x'\left(x'+y+z'\right)}+\dfrac{y+z'}{yz'}=0\)

\(\Leftrightarrow\dfrac{\left(y+z'\right)\left(yz'+x'^2+x'y+x'z'\right)}{x'yz'\left(x'+y+z'\right)}=0\)

\(\Leftrightarrow\dfrac{\left(x'+y\right)\left(y+z'\right)\left(z'+x'\right)}{x'yz'\left(x'+y+z'\right)}=0\Leftrightarrow\left(2x+y\right)\left(y+2z\right)\left(2z+2x\right)=0\Leftrightarrow\left(2x+y\right)\left(y+2z\right)\left(z+x\right)=0\left(đpcm\right)\)

Ta có \(VP=\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)^2=\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}+\frac{2}{ab}+\frac{2}{bc}+\frac{2}{ac}\)\(\left(a,b,c\ne0\right)\)

\(=\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}+\frac{2a+2b+2c}{abc}\)

\(=\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}+\frac{2.\left(a+b+c\right)}{abc}\)\(=\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}+0=\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}=VT\)

Vậy đẳng thức được chứng minh

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

\(VT=-\frac{1}{2}\left(\frac{1}{ab}+\frac{1}{bc}+\frac{1}{ca}\right)=-\frac{1}{2}.\frac{a+b+c}{abc}=0\)

post ít một thôi

Ta có : \(\frac{a}{1+b^2}=a-\frac{ab^2}{1+b^2}\ge a-\frac{ab^2}{2b}=a-\frac{ab}{2}\)

Đánh giá tương tự , ta cũng có :

\(\frac{b}{1+c^2}\ge b-\frac{bc}{2},\frac{c}{1+a^2}\ge c-\frac{ab}{2}\)

Từ đó suy ra :

\(\frac{a}{1+b^2}+\frac{b}{1+c^2}+\frac{c}{1+a^2}\ge a+b+c-\frac{ab+bc+c}{2}=3-\frac{ab+bc+ca}{2}\)

Mặt khác ,ta biết rằng \(ab+bc+ca\le\frac{\left(a+b+c\right)^2}{3}=3.\)Từ đây ,kết hợp với đánh giá ở trên ,ta có kết quả cần chứng minh.

\(Ta\)\(có\) \(\frac{a}{1+b^2}\ge a-\frac{ab^2}{1+b^2}\)

Áp dụng bất đẳng thức \(a^2+b^2\ge2ab\)ta có

\(a-\frac{ab^2}{1+b^2}\ge a-\frac{ab^2}{2b}=a-\frac{ab}{2}\)

Chứng minh tương tụ với \(\frac{b}{1+c^2};\frac{c}{1+a^2}\)ta được

\(\frac{a}{1+b^2}+\frac{b}{1+c^2}+\frac{c}{1+a^2}\ge a+b+c-\frac{ab+bc+ac}{2}\) \(\left(1\right)\)

Mặt khác ta có :

\(\left(a+b+c\right)^2\ge3\left(ab+bc+ac\right)\)

\(Hay\)\(3^2\ge3\left(ab+bc+ac\right)\)

\(\Rightarrow ab+bc+ca\le3\)\(\left(2\right)\)
\(Từ\)\(\left(1\right)\)\(\left(2\right)\)\(\Rightarrow\)\(a+b+c-\frac{ab+bc+ac}{2}\)\(\ge3-\frac{3}{2}=\frac{3}{2}\)\(\left(3\right)\)

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

                                                                                                       \(\left(đpcm\right)\)

Áp dụng BĐT AM-GM ta có:\(\frac{a}{1+b^2}=\frac{a\left(1+b^2\right)-ab^2}{1+b^2}=a-\frac{ab^2}{1+b^2}\ge a-\frac{ab^2}{2b}=a-\frac{ab}{2}\)

Tương tự \(\frac{b}{1+c^2}\ge b-\frac{bc}{2};\frac{c}{1+a^2}\ge c-\frac{ca}{2}\)

Kết hợp giả thiết và \(\left(ab+bc+ca\right)\le\frac{\left(a+b+c\right)^2}{3}\) ta có:

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

Đẳng thức xảy ra khi a=b=c=1