A B C H D E F

Gọi D, E, F lần lượt là chân đường cao hạ từ A, B, C của tam giác ABC.

+) \(\Delta AHE~\Delta ACD\)( vì ^HAE =^CAD, ^HEA=^CDA )

=> \(\frac{HA}{CA}=\frac{EA}{AD}\)=> \(\frac{HA}{CA}.\frac{HB}{BC}=\frac{EA}{CA}.\frac{HB}{BC}=\frac{2.EA.HB}{2.CA.BC}=\frac{S_{\Delta AHB}}{S_{ABC}}\)(1)

+) \(\Delta CHD~\Delta CBF\)( vì ^DCH=^FCB, ^CDH=^CFB )

=> \(\frac{CH}{CB}=\frac{CD}{CF}\)=> \(\frac{CH}{CB}.\frac{AH}{AB}=\frac{CD.AH}{CF.AB}=\frac{S_{AHC}}{S_{ABC}}\)(2)

+) \(\Delta ABE~\Delta HBF\)

=> \(\frac{HB}{AB}=\frac{BF}{BE}\Rightarrow\frac{HB}{AB}.\frac{HC}{AC}=\frac{BF.HC}{BE.AC}=\frac{S_{BHC}}{S_{ABC}}\)(3)

Từ (1) ; (2) ; (3) => \(\frac{HA}{CA}.\frac{HB}{BC}+\frac{CH}{CB}.\frac{AH}{AB}+\frac{HB}{AB}.\frac{HC}{AC}=\frac{S_{ABE}}{S_{ABC}}+\frac{S_{ABE}}{S_{ABC}}+\frac{S_{ABE}}{S_{ABC}}=1\)

=> \(\frac{HA}{BC}.\frac{HB}{AC}+\frac{HB}{AC}.\frac{HC}{AB}+\frac{HC}{AB}.\frac{HA}{BC}=1\)

Đặt: \(\frac{HA}{BC}=x;\frac{HB}{AC}=y;\frac{HC}{AB}=z\); x, y, z>0

Ta có: \(xy+yz+zx=1\)

=> \(\left(x+y+z\right)^2\ge3\left(xy+yz+zx\right)=3\)

=> \(x+y+z\ge\sqrt{3}\)

"=" xảy ra khi và chỉ khi x=y=z

Vậy : \(\frac{HA}{BC}+\frac{HB}{AC}+\frac{HC}{AB}\ge\sqrt{3}\)

"=" xảy ra <=> \(\frac{HA}{BC}=\frac{HB}{AC}=\frac{HC}{AB}\)

-1 có căn k bạn?

39 < 103

Áp dụng cô si

\(\hept{\begin{cases}\frac{1}{a}+\frac{1}{b}\ge2\sqrt{\frac{1}{ab}}\\\frac{1}{c}+\frac{1}{b}\ge2\sqrt{\frac{1}{cb}}\\\frac{1}{a}+\frac{1}{c}\ge2\sqrt{\frac{1}{ac}}\end{cases}}\)\(\Rightarrow\frac{1}{c}+\frac{1}{b}+\frac{1}{a}\ge\frac{1}{\sqrt{ab}}+\frac{1}{\sqrt{bc}}+\frac{1}{\sqrt{ac}}\)

\("="\Leftrightarrow a=b=c=0\)

\(\hept{\begin{cases}\sqrt{x}\le\frac{x+1}{2}\\\sqrt{y-1}\le\frac{y-1+1}{2}\\\sqrt{z-2}\le\frac{z-2+1}{2}\end{cases}}\)\(\Rightarrow\sqrt{x}+\sqrt{y-1}+\sqrt{z-2}\le\frac{x+1+y-1+1+z-2+1}{2}\)

\(\Leftrightarrow\sqrt{x}+\sqrt{y-1}+\sqrt{z-2}\le\frac{x+y+z}{2}\)

\("="\Leftrightarrow\hept{\begin{cases}x=1\\y=2\\z=3\end{cases}}\)

Sửa ĐK của c) : a, b, c > 0

Áp dụng bất đẳng thức Cauchy ta có :

\(\frac{1}{a}+\frac{1}{b}\ge2\sqrt{\frac{1}{ab}}=\frac{2}{\sqrt{ab}}\)

\(\frac{1}{b}+\frac{1}{c}\ge2\sqrt{\frac{1}{bc}}=\frac{2}{\sqrt{bc}}\)

\(\frac{1}{c}+\frac{1}{a}\ge2\sqrt{\frac{1}{ca}}=\frac{2}{\sqrt{ca}}\)

Cộng các vế tương ứng

=> \(\frac{1}{a}+\frac{1}{b}+\frac{1}{b}+\frac{1}{c}+\frac{1}{c}+\frac{1}{a}\ge\frac{2}{\sqrt{ab}}+\frac{2}{\sqrt{bc}}+\frac{2}{\sqrt{ca}}\)

=> \(2\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)\ge2\left(\frac{1}{\sqrt{ab}}+\frac{1}{\sqrt{bc}}+\frac{1}{\sqrt{ca}}\right)\)

=> \(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\ge\frac{1}{\sqrt{ab}}+\frac{1}{\sqrt{bc}}+\frac{1}{\sqrt{ca}}\)

=> đpcm

Đẳng thức xảy ra khi a = b = c