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

Bài 1:

Đặt \(\frac{a}{b}=\frac{c}{d}=k\)

=>a=bk; c=dk

\(\frac{5a+4b}{5a-4b}=\frac{5\cdot bk+4b}{5\cdot bk-4b}=\frac{b\left(5k+4\right)}{b\left(5k-4\right)}=\frac{5k+4}{5k-4}\)

\(\frac{5c+4d}{5c-4d}=\frac{5\cdot dk+4d}{5\cdot dk-4d}=\frac{d\left(5k+4\right)}{d\left(5k-4\right)}=\frac{5k+4}{5k-4}\)

Do đó: \(\frac{5a+4b}{5a-4b}=\frac{5c+4d}{5c-4d}\)

Bài 2:

a: Đặt \(\frac{a}{b}=\frac{b}{c}=k\)

=>b=ck; a=bk=ck^2

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

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

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

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

=>\(\begin{cases}c=dk\\ b=ck=dk\cdot k=dk^2\\ a=bk=dk^2\cdot k=dk^3\end{cases}\)

\(\left(\frac{a+b-c}{b+c-d}\right)^3=\left(\frac{dk^3+dk^2-dk}{dk^2+dk-d}\right)^3\)

\(=\left\lbrack\frac{dk\left(k^2+k-1\right)}{d\left(k^2+k-1\right)}\right\rbrack^3=k^3\)

\(\frac{a}{d}=\frac{dk^3}{d}=k^3\)

Do đó: \(\left(\frac{a+b-c}{b+c-d}\right)^3=\frac{a}{d}\)

ta có: \(\frac{a}{b}=\frac{b}{d}\Rightarrow\frac{ab}{bd}=\frac{a^2}{b^2}=\frac{b^2}{d^2}\) (*)

mà \(\frac{a^2}{b^2}=\frac{b^2}{d^2}=\frac{a^2+b^2}{b^2+d^2}\)

Từ (*) \(\Rightarrow\frac{ab}{bd}=\frac{a^2+b^2}{b^2+d^2}\)

          \(\Rightarrow\frac{a}{d}=\frac{a^2+b^2}{b^2+d^2}\left(đpcm\right)\) ( do \(\frac{ab}{bd}=\frac{a}{d}\)) 

toi bạn rùi cmr là chết mày rùi

a² + b² + c² + d² + e² ≥ a(b + c + d + e) 

Ta có: a² + b² + c² + d² + e² 

= (a²/4 + b²) + (a²/4 + c²) + (a²/4 + d²) + (a²/4 + e²) 

Lại có: (a/2 - b)² ≥ 0 <=> a²/4 - ab + b² ≥ 0 <=> a²/4 + b² ≥ ab 

Tương tự ta có: 

. a²/4 + c² ≥ ac 
. a²/4 + d² ≥ ad 
. a²/4 + e² ≥ ae 

--> (a²/4 + b²) + (a²/4 + c²) + (a²/4 + d²) + (a²/4 + e²) ≥ ab + ac + ad + ae 

<=> a² + b² + c² + d² + e² ≥ a(b + c + d + e) --> đ.p.c.m 

Dấu " = " xảy ra <=> a/2 = b = c = d = e 

P/s: Hơi hơi dễ nhỉ