ta có: \(\frac{a^2+c^2}{b^2+a^2}\)do \(a^2=bc\)

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

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

\(\text{Ta có : }\frac{a^2+c^2}{b^2+a^2}\text{ do }a^2=bc\)

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

\(\text{Vậy }\frac{a^2+c^2}{b^2+a^2}=\frac{c}{b}\)

a: \(\dfrac{a+5}{a-5}=\dfrac{b+6}{b-6}\)

=>(a+5)(b-6)=(a-5)(b+6)

=>ab-6a+5b-30=ab+6a-5b-30

=>-6a+5b=6a-5b

=>-12a=-10b

=>6a=5b

=>\(\dfrac{a}{b}=\dfrac{5}{6}\)

b: Đặt \(\dfrac{a}{b}=\dfrac{c}{d}=k\)

=>\(a=bk;c=dk\)

\(\dfrac{a^2+b^2}{c^2+d^2}=\dfrac{b^2k^2+b^2}{d^2k^2+d^2}=\dfrac{b^2\left(k^2+1\right)}{d^2\left(k^2+1\right)}=\dfrac{b^2}{d^2}\)

\(\dfrac{ab}{cd}=\dfrac{bk\cdot b}{dk\cdot d}=\dfrac{b^2k}{d^2k}=\dfrac{b^2}{d^2}\)

Do đó: \(\dfrac{a^2+b^2}{c^2+d^2}=\dfrac{ab}{cd}\)

Lời giải:

\(a^3+b^3+c^3=3abc\)

\(\Leftrightarrow (a+b+c)^3-3(a+b)(b+c)(c+a)=3abc\)

\(\Leftrightarrow (a+b+c)^3-3[(a+b+c)(ab+bc+ac)-abc]=3abc\)

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

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

Vì \(a,b,c>0\Rightarrow a+b+c>0\)

Do đó \(a^2+b^2+c^2-ab-bc-ac=0\)

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

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

Ta thấy \((a-b^2;(b-c)^2;(c-a)^2\geq 0\), do đó điều trên xảy ra khi mà:
\(\left\{\begin{matrix} (a-b)^2=0\\ (b-c)^2=0\\ (c-a)^2=0\end{matrix}\right.\Leftrightarrow a=b=c\)

Ta có đpcm.

\(\text{Ta có }:a^3+b^3+c^3=3abc\\ \Leftrightarrow a^3+b^3+c^3-3abc=0\\ \Leftrightarrow\left(a+b+c\right)\left(a^2+b^2+c^2-ab-bc-ac\right)=0\)

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

\(\Leftrightarrow2\left(a^2+b^2+c^2-ab-ac-bc\right)\)

\(\Leftrightarrow2a^2+2b^2+2c^2-2ab-2ac-2bc\)

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

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

\(Do\left(a-b\right)^2\ge0\forall x\\ \left(a-c\right)^2\ge0\forall x\\ \left(b-c\right)^2\ge0\forall x\\ \Leftrightarrow\left(a-b\right)^2+\left(a-c\right)^2+\left(b+c\right)^2\ge0\forall x\)

\(\text{Dấu "=" xảy ra khi: }\left\{{}\begin{matrix}\left(a-b\right)^2=0\\\left(a-c\right)^2=0\\\left(b-c\right)^2=0\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}a-b=0\\a-c=0\\b-c=0\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}a=b\\a=c\\b=c\end{matrix}\right.\Leftrightarrow a=b=c\)

Vậy \(a=b=c\text{ }khi\text{ }a^3+b^3+c^3=3abc\)

b, Ta có \(m=a+b+c\)

          \(\Rightarrow am+bc=a\left(a+b+c\right)+bc=a\left(a+b\right)+ac+bc=\left(a+c\right)\left(a+b\right)\)

CMTT \(bm+ac=\left(b+c\right)\left(b+a\right)\);\(cm+ab=\left(c+a\right)\left(c+b\right)\)

Suy ra \(\left(am+bc\right)\left(bm+ac\right)\left(cm+ab\right)=\left(a+b\right)^2\left(a+c\right)^2\left(b+c\right)^2\)

\(4,VT=-a+b+c-a+b-c+a-b-c=-a+b-c=-\left(a-b+c\right)=VP\\ 5,M=-a+b-b-c+a+c-a=-a\\ M>0\Rightarrow-a>0\Rightarrow a< 0\)

Chỉ có thể đưa ra ví dụ thôi chứ đây đã là kiến thức cơ bản r nhé bn.

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