a/b = b/c = c/a và a + b + c khác 0
chứng minh a = b = c.
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H
1
7 tháng 3 2022
\(\dfrac{a}{b}=\dfrac{c}{d}\\ \Rightarrow\dfrac{b}{a}=\dfrac{d}{c}\\ \Rightarrow\dfrac{b}{a}-1=\dfrac{d}{c}-1\\ \Rightarrow\dfrac{b-a}{a}=\dfrac{d-c}{c}\\ \Rightarrow\dfrac{-\left(a-b\right)}{a}=\dfrac{-\left(c-d\right)}{c}\\ \Rightarrow\dfrac{a-b}{a}=\dfrac{c-d}{c}\)
18 tháng 4 2023
đb bị thiếu nhá bn, mik bổ sung ns sẽ thành: thỏa mãn a\(\le b\le c\)
LL
3
30 tháng 6 2021
$a+b+c \ge \sqrt{ab}+\sqrt{bc}+\sqrt{ca}$
$\Leftrightarrow 2a+2b+2c \ge 2\sqrt{ab}+2\sqrt{bc}+2\sqrt{ca}$
$\Leftrightarrow a-2\sqrt{ab}+b+b-2\sqrt{bc}+c+c-2\sqrt{ca}+a \ge 0$
$\Leftrightarrow (\sqrt{a}-\sqrt{b})^2+(\sqrt{c}-\sqrt{b})^2+(\sqrt{a}-\sqrt{c})^2 \ge 0$ luôn đúng với $a,b,c \ge 0$
Dấu "=" xảy ra khi a=b=c
30 tháng 6 2021
Ta có: \(a+b+c\ge\sqrt{ab}+\sqrt{bc}+\sqrt{ca}\)
\(\Leftrightarrow2a+2b+2c-2\sqrt{ab}-2\sqrt{bc}-2\sqrt{ca}\ge0\)
\(\Leftrightarrow\left(a-2\sqrt{ab}+b\right)+\left(b-2\sqrt{bc}+c\right)+\left(c-2\sqrt{ca}+a\right)\ge0\)
\(\Leftrightarrow\left(\sqrt{a}-\sqrt{b}\right)^2+\left(\sqrt{b}-\sqrt{c}\right)^2+\left(\sqrt{c}-\sqrt{a}\right)^2\ge0\)(luôn đúng với mọi a,b,c không âm)
NV
Nguyễn Việt Lâm
Giáo viên
23 tháng 1 2022
Đặt \(\left(\sqrt{b^2+c^2};\sqrt{c^2+a^2};\sqrt{a^2+b^2}\right)=\left(x;y;z\right)\)
\(\Rightarrow\left\{{}\begin{matrix}a^2=\dfrac{y^2+z^2-x^2}{2}\\b^2=\dfrac{x^2+z^2-y^2}{2}\\c^2=\dfrac{x^2+y^2-z^2}{2}\end{matrix}\right.\)
\(\Rightarrow VT=\dfrac{y^2+z^2-x^2}{2x}+\dfrac{x^2+z^2-y^2}{2y}+\dfrac{x^2+y^2-z^2}{2z}\)
\(VT\ge\dfrac{\left(y+z\right)^2}{4x}+\dfrac{\left(x+z\right)^2}{4y}+\dfrac{\left(x+y\right)^2}{4z}-\dfrac{1}{2}\left(x+y+z\right)\)
\(VT\ge\dfrac{\left(2x+2y+2z\right)^2}{4\left(x+y+z\right)}-\dfrac{1}{2}\left(x+y+z\right)=\dfrac{1}{2}\left(x+y+z\right)\)
\(VT\ge\dfrac{1}{2}\left(\sqrt{a^2+b^2}+\sqrt{b^2+c^2}+\sqrt{c^2+a^2}\right)\)
\(VT\ge\dfrac{1}{2}\left(\sqrt{\dfrac{1}{2}\left(a+b\right)^2}+\sqrt{\dfrac{1}{2}\left(b+c\right)^2}+\sqrt{\dfrac{1}{2}\left(c+a\right)^2}\right)\)
\(VT\ge\dfrac{a+b+c}{\sqrt{2}}\) (đpcm)
NM
1
T
20 tháng 8 2023
Ta có: \(a+b+c=0\)
\(\Rightarrow\left(a+b+c\right)^2=0\)
\(\Rightarrow a^2+b^2+c^2+2ab+2bc+2ac=0\)
Mặt khác: \(a^2\ge0\forall a;b^2\ge0\forall b;c^2\ge0\forall c\)
\(\Rightarrow a^2+b^2+c^2\ge0\)
Suy ra: \(2ab+2bc+2ac=0\)
\(\Rightarrow2\left(ab+bc+ac\right)=0\)
\(\Rightarrow ab+bc+ac=0\Leftrightarrow2\left(ab+bc+ac\right)^2=0\) (1)
Lại có: \(a^4+b^4+c^4\)
\(=\left(a^2+b^2+c^2\right)^2-2\left[\left(ab\right)^2+\left(bc\right)^2+\left(ac\right)^2\right]\)
\(=0-2\left[\left(ab\right)^2+\left(bc\right)^2+\left(ac\right)^2+2\left(ab+bc+ac\right)-2\left(ab+bc+ac\right)\right]\)
\(=-2\left(ab+bc+ac\right)^2-4\left(ab+bc+ac\right)\)
\(=0\) (2)
Từ (1) và (2) \(\Rightarrow a^4+b^4+c^4=2\left(ab+bc+ac\right)^2=0\)
hay \(a^4+b^4+c^4=2\left(ab+ac+bc\right)^2\)
Kiểm tra hộ mình xem có đúng không ạ!
11 tháng 7 2021
Ta có: \(\sqrt{\dfrac{1}{a^2}+\dfrac{1}{b^2}+\dfrac{1}{c^2}}\)
\(=\sqrt{\dfrac{1}{a^2}+\dfrac{1}{b^2}+\dfrac{1}{c^2}-2\left(\dfrac{c}{abc}+\dfrac{b}{abc}+\dfrac{a}{abc}\right)}\)
\(=\sqrt{\dfrac{1}{a^2}+\dfrac{1}{b^2}+\dfrac{1}{c^2}-2\cdot\dfrac{a+b+c}{abc}}\)
\(=\sqrt{\left(\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}\right)^2}=\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}\)
T
2 tháng 10 2023
\(a^2+b^2+c^2-ab-ac-bc=0\\\Leftrightarrow 2a^2+2b^2+2c^2-2ab-2ac-2bc=0\\\Leftrightarrow (a^2-2ab+b^2)+(b^2-2bc+c^2)+(a^2-2ac+c^2)=0\\\Leftrightarrow (a-b)^2+(b-c)^2+(a-c)^2=0\)
Ta thấy: \(\left(a-b\right)^2\ge0\forall a;b\)
\(\left(b-c\right)^2\ge0\forall b;c\)
\(\left(a-c\right)^2\ge0\forall a;c\)
\(\Rightarrow\left(a-b\right)^2+\left(b-c\right)^2+\left(a-c\right)^2\ge0\forall a;b;c\)
Mặt khác: \(\left(a-b\right)^2+\left(b-c\right)^2+\left(a-c\right)^2=0\)
nên: \(\left\{{}\begin{matrix}a-b=0\\b-c=0\\a-c=0\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}a=b\\b=c\\a=c\end{matrix}\right.\)
\(\Leftrightarrow a=b=c\left(dpcm\right)\)
#\(Toru\)
MY
1
10 tháng 5 2021
\(\dfrac{a}{x}+\dfrac{b}{y}+\dfrac{c}{z}=0\) \(\Rightarrow\dfrac{ayz+bxz+cxy}{xyz}=0\) \(\Rightarrow ayz+bxz+cxy=0\) \(\dfrac{x}{a}+\dfrac{y}{b}+\dfrac{z}{c}=1\) \(\Rightarrow\left(\dfrac{x}{a}+\dfrac{y}{b}+\dfrac{z}{c}\right)^2=1\) \(\Rightarrow\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}+\dfrac{z^2}{c^2}+2\left(\dfrac{xy}{ab}+\dfrac{xz}{ac}+\dfrac{yz}{bc}\right)=1\) \(\Rightarrow\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}+\dfrac{z^2}{c^2}+2\left(\dfrac{cxy+bxz+ayz}{abc}\right)=1\) \(\Rightarrow\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}+\dfrac{z^2}{c^2}+2\left(\dfrac{0}{abc}\right)=1\) \(\Rightarrow\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}+\dfrac{z^2}{c^2}+0=1\) \(\Rightarrow\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}+\dfrac{z^2}{c^2}=1\)
\(\frac{a}{b}=\frac{b}{c}=\frac{c}{a}\Rightarrow\frac{a+b+c}{b+c+a}\)mà \(a+b+c\)khác 0 nên \(\frac{a+b+c}{b+c+a}=1\)hay \(a=b=c\)