Sửa đề: Chứng minh \(\frac{a^2+b^2}{c^2+b^2}=\frac{a}{c}\)

Ta có: \(\frac{\overline{ab}}{\overline{bc}}=\frac{b}{c}\)

=>\(\frac{10a+b}{10b+c}=\frac{b}{c}\)

=>c(10a+b)=b(10b+c)

=>10ac+bc=10b^2+bc

=>\(10ac=10b^2\)

=>\(ac=b^2\)

=>\(\frac{a}{b}=\frac{b}{c}\)

Đặt \(\frac{a}{b}=\frac{b}{c}=k\)

=>\(\begin{cases}b=ck\\ a=bk=ck\cdot k=ck^2\end{cases}\)

\(\frac{a^2+b^2}{c^2+b^2}=\frac{\left(ck^2\right)^2+\left(ck^{}\right)^2}{c^2+\left(ck\right)^2}=\frac{c^2k^4+c^2k^2}{c^2+c^2k^2}=\frac{c^2k^2\left(k^2+1\right)}{c^2\left(1+k^2\right)}=k^2\)

\(\frac{a}{c}=\frac{ck^2}{c}=k^2\)

Do đó: \(\frac{a^2+b^2}{c^2+b^2}=\frac{a}{c}\)

a.

\(\Leftrightarrow2a^2b^2+2b^2c^2+2c^2a^2\ge2abc\left(a+b+c\right)\)

\(\Leftrightarrow\left(a^2b^2-2a^2bc+c^2a^2\right)+\left(a^2b^2-2ab^2c+b^2c^2\right)+\left(b^2c^2-2abc^2+a^2c^2\right)\ge0\)

\(\Leftrightarrow\left(ab-ca\right)^2+\left(ab-bc\right)^2+\left(bc-ca\right)^2\ge0\) (luôn đúng)

Vậy BĐT đã cho đúng

b.

\(\Leftrightarrow a^2b^2+b^2c^2+c^2a^2+2abc\left(a+b+c\right)\ge3abc\left(a+b+c\right)\)

\(\Leftrightarrow a^2b^2+b^2c^2+c^2a^2\ge abc\left(a+b+c\right)\) (đúng theo câu a đã chứng minh)

Đặt \(\frac{a}{b}=\frac{c}{d}=k\Leftrightarrow a=bk;c=dk\)

Thay a = bk, c = dk vào \(\frac{a^2-b^2}{c^2-d^2}\) và \(\frac{ab}{cd}\), ta có:

\(\frac{a^2-b^2}{c^2-d^2}=\frac{\left(bk\right)^2-b^2}{\left(dk\right)^2-d^2}=\frac{b^2k^2-b^2}{d^2k^2-d^2}=\frac{b^2\left(k^2-1\right)}{d^2\left(k^2-1\right)}=\frac{b^2}{d^2}\)

\(\frac{ab}{cd}=\frac{bk.b}{dk.d}=\frac{b^2.k}{d^2.k}=\frac{b^2}{d^2}\)

\(\Rightarrow\frac{a^2-b^2}{c^2-d^2}=\frac{ab}{cd}\left(đpcm\right)\)

a.

Ta có: \(\dfrac{a^2}{b+c}+\dfrac{b+c}{4}\ge2\sqrt{\dfrac{a^2\left(b+c\right)}{4\left(b+c\right)}}=a\)

Tương tự: \(\dfrac{b^2}{c+a}+\dfrac{c+a}{4}\ge b\) ; \(\dfrac{c^2}{a+b}+\dfrac{a+b}{4}\ge c\)

Cộng vế:

\(\dfrac{a^2}{b+c}+\dfrac{b^2}{c+a}+\dfrac{c^2}{a+b}+\dfrac{a+b+c}{2}\ge a+b+c\)

\(\Leftrightarrow\dfrac{a^2}{b+c}+\dfrac{b^2}{c+a}+\dfrac{c^2}{a+b}\ge\dfrac{a+b+c}{2}\) (đpcm)

Dấu "=" xảy ra khi \(a=b=c\)

b.

Ta có:

\(a^2+bc\ge2\sqrt{a^2bc}=2\sqrt{ab.ac}\Rightarrow\dfrac{1}{a^2+bc}\le\dfrac{1}{2\sqrt{ab.ac}}\le\dfrac{1}{4}\left(\dfrac{1}{ab}+\dfrac{1}{ac}\right)\)

Tương tự: \(\dfrac{1}{b^2+ac}\le\dfrac{1}{4}\left(\dfrac{1}{ab}+\dfrac{1}{bc}\right)\) ; \(\dfrac{1}{c^2+ab}\le\dfrac{1}{4}\left(\dfrac{1}{ac}+\dfrac{1}{bc}\right)\)

Cộng vế với vế:

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

Dấu "=" xảy ra khi \(a=b=c\)