đây đau phải la lớp1

khong phai toan lop 1

dung day la toan lop 2

\(\left(a-b\right)^2+\left(b-c\right)^2+\left(c-a\right)^2=4\left(a^2+b^2+c^2-ab-ac-bc\right)\\ \Leftrightarrow a^2-2ab+b^2+b^2-2bc-c^2+c^2-2ac+a^2\\ =4a^2+4b^2+4c^2-4ab-4ac-4bc\\ \Leftrightarrow0=2a^2+2b^2+2c^2-2ab-2ac-2bc\\ \Leftrightarrow\left(a^2-2ab+b^2\right)+\left(a^2-2ac+c^2\right)+\left(b^2-2bc+c^2\right)=0\\ \Leftrightarrow\left(a-b\right)^2+\left(a-c\right)^2+\left(b-c\right)^2=0\Leftrightarrow\left\{\begin{matrix}\left(a-b\right)^2=0\Leftrightarrow a-b=0\Leftrightarrow a=b\\\left(a-c\right)^2=0\Leftrightarrow a-c=0\Leftrightarrow a=c\\\left(b-c\right)^2=0\Leftrightarrow b-c=0\Leftrightarrow b=c\end{matrix}\right.\)

Vậy a=b=c

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

Ta có: \(a^2+b^2+c^2+2ab+2bc+2ac\ge3ab+3bc+3ac\)

\(\Leftrightarrow a^2+b^2+c^2-ab-bc-ac\ge0\)

\(\Leftrightarrow2a^2+2b^2+2c^2-2ab-2bc-2ac\ge0\) (nhân cả hai vế cho 2)

\(\Leftrightarrow a^2-2ab+b^2+a^2-2ac+c^2+b^2-2bc+c^2\ge0\)

\(\Leftrightarrow\left(a+b\right)^2+\left(a+c\right)^2+\left(b+c\right)^2\ge0\) ( đúng )

Lời giải:

Ba số thực $a,b,c$ cần có thêm điều kiện không âm mới đúng.

BĐT cần chứng minh tương đương với:

$ab^3+bc^3+ca^3+2abc(a+b+c)\leq a^3b+b^3c+c^3a+ab^3+bc^3+ca^3+abc(a+b+c)$

$\Leftrightarrow abc(a+b+c)\leq a^3b+b^3c+c^3a(*)$

Áp dụng BĐT Bunhiacopxky:

$(a^3b+b^3c+c^3a)(abc^2+bca^2+cab^2)\geq (a^2bc+b^2ca+c^2ab)^2$

$\Rightarrow a^3b+b^3c+c^3a\geq abc(a+b+c)$

BĐT $(*)$ đúng nên ta có đpcm.

Dấu "=" xảy ra khi $a=b=c$

SOS là ra, khá đơn giản. Ta có:

$$\text{VP}-\text{VT}=ab \left( -c+a \right) ^{2}+ca \left( b-c \right) ^{2}+cb \left( a-b
\right) ^{2}\geqq 0.$$

Đẳng thức xảy ra khi $a=b=c.$

a,b,c>0 

\(VP-VT=a^3b+b^3c+c^3a-abc\left(a+b+c\right)=abc\Sigma\frac{\left(a-b\right)^2}{a}\ge0\)

\(\dfrac{a^2+\left(a-c\right)^2}{b^2+\left(b-c\right)^2}\)

\(=\dfrac{a^2+\left(a-c\right)^2+c^2+2\left(ab-ac-bc\right)}{b^2+\left(b-c\right)^2+c^2+2\left(ab-ac-bc\right)}\)

\(=\dfrac{a^2+a^2-2ac+c^2+c^2+2ab-2ac-2bc}{b^2+b^2-2bc+c^2+c^2+2ab-2ac-2bc}\)

\(=\dfrac{2a^2+2c^2-4ac+2ab-2bc}{2b^2+2c^2-4bc+2ab-2ac}\)

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

\(=\dfrac{\left(a-c\right)\left(a-c+b\right)}{\left(b-c\right)\left(a-c+b\right)}=\dfrac{a-c}{b-c}\left(đpcm\right)\)

Ta có:

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

\(\Leftrightarrow abc^2+ab^2c+a^2bc-ab-bc-ca=0\left(1\right)\)

Ta cần chứng minh

\(b\left(a^2-bc\right)\left(1-ac\right)=a\left(1-bc\right)\left(b^2-ac\right)\)

\(\Leftrightarrow ab^2c^2-a^2bc^2+ab^3c-b^2c-a^3bc+a^2c-ab^2+a^2b=0\)

\(\Leftrightarrow b\left(abc^2+ab^2c-bc-ab\right)-a^2bc^2-a^3bc+a^2c+a^2b=0\)

\(\Leftrightarrow b\left(ac-a^2bc\right)-a^2bc^2-a^3bc+a^2c+a^2b=0\)

\(\Leftrightarrow-a\left(ab^2c+abc^2+a^2bc-bc-ac-ab\right)=0\)(theo (1) thì đúng)

\(\RightarrowĐPCM\)