cho x/a+y/b+z/c=1 va a/x+b/y+c/z=0 .CMR:x^2/a+y^2/b+z^2/c=1
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cho a,b, c, x, y, z :{a/x+b/y+c/z=0;x/a+y/b+z/c=1
CMR:x^2/a^2+y^2/b^2+z^2/c^2=1
cho a, b, c, x, y, z:{a/x+b/y+c/z=0;x/a+y/b+z/c=1
CMR:x^2/a^2+y^2/b^2+z^2/c^2=1
Ta có :\(\frac{a}{x}+\frac{b}{y}+\frac{c}{z}=0\Rightarrow\frac{ayz+bxz+cxy}{xyz}=0\Rightarrow ayz+bxz+cxy=0\)
Lại có \(\frac{x}{a}+\frac{y}{b}+\frac{z}{c}=1\Rightarrow\left(\frac{x}{a}+\frac{y}{b}+\frac{z}{c}\right)^2=1\)
=> \(\left(\frac{x}{a}\right)^2+\left(\frac{y}{b}\right)^2+\left(\frac{z}{c}\right)^2+\frac{2xy}{ab}+\frac{2yz}{bc}+\frac{2xz}{ac}=1\)
=> \(\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}+\frac{2xyc}{abc}+\frac{2ayz}{abc}+\frac{2bxz}{abc}=1\)
=> \(\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}+\frac{2}{abc}\left(xyc+ayz+bxz\right)=1\)
=> \(\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1\left(\text{vì }xyc+ayz+bxz=0\right)\)(đpcm)
1, x/y = 9/7;y/z = 7/9 va x-y+z=-15
b.6/11 x= 9/2 y=18/5z va -x+y+z=3
c,x/5=y/7=z/3 va x^2+y^2-z^2=585io
d,cho x/y/z =5/4/3 tinh P=x+2y-3z/x-2y+3z
e,cho 2a+b+c/a = a+2b+c/b = a+b+2c/c tinh S=a+b/c + b+c/a + c+a/b
1. Cho a,b,c biết: a.b.c khác 0
và ab+bc+ca=0. Tính: P=(a+b)(b+c)(c+a)/abc
2.CMR: Nếu a,b,c>0 và a,b,c khác nhau thì
A=a^3+b^3+c^3-3abc > 0
3.Cho(x+y+z)(xy+yz+zx)=xyz
Cmr:x^2017+y^2017+z^2017=(x+y+z)^2017
bài 1
ab+bc+ca=0
=>ab+bc=-ca
ta có (a+b)(b+c)(c+a)/abc
=> (ab+ac+bc+b2)(c+a)/abc
=> (0+b2)(c+a)/abc
=>b2c+b2a/abc
=>b(ab+bc)/abc
=>b(-ac)/abc
=>-abc/abc=-1
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Cmr \(\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1\)
mả x/a + y/b + z/c = 1 va a/x + b/y + c/z = 0
`Cho x/a + y/b + z/c = 1 va a/x + b/y+ c/z = 0 cmr x2/ a2 + y2/ b2+ z2/ c2 = 1
Ta có: \(\frac{a}{x}+\frac{b}{y}+\frac{c}{z}=0\)
\(\Rightarrow\frac{ayz+bxz+cxy}{xyz}=0\)
\(\Rightarrow ayz+bxz+cxy=0\)
Lại có: \(\frac{x}{a}+\frac{y}{b}+\frac{z}{c}=1\)
\(\Rightarrow\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}+2.\frac{ayz+bxz+cxy}{abc}=1\)(bình phương hai vế)
\(\Rightarrow\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1\)(Vì \(ayz+bxz+cxy=0\))
1) cho a/b=c/d chung minh:
a) a^2+c^2/b^2+d^2=a^2-c^2/b^2+d^2
b) (a+c)^2/(b+d)^2=(a-c)^2/b^2+d^2
2) a) cho x/y=y/z=z/x va x+y+z khac 0
tinhx^333.z^666/y^999
b) cho a.c=b^2 ; a.b=c^2 va a+b khac 0 ; a ; b ; c kha 0 ,tinh b^333/a^111.c^222
1, Cho x; y; z ≠0 và dfrac{1}{x} + dfrac{1}{y}+ dfrac{1}{z}dfrac{2}{2x+y+2z}. Cmr: (2x+y)(y+2z)(z+x) 0 2, Cho dfrac{a}{b+c}+dfrac{b}{c+a}+dfrac{c}{a+b}1. Cmr: dfrac{a^2}{b+c}+dfrac{b^2}{c+a}+dfrac{c^2}{a+b}0Gấp ạ, ai giúp mình với!!!!
Đọc tiếp
1, Cho x; y; z ≠0 và \(\dfrac{1}{x}\) + \(\dfrac{1}{y}\)+ \(\dfrac{1}{z}\)=\(\dfrac{2}{2x+y+2z}\). Cmr: (2x+y)(y+2z)(z+x)= 0
2, Cho \(\dfrac{a}{b+c}+\dfrac{b}{c+a}+\dfrac{c}{a+b}=1\). Cmr: \(\dfrac{a^2}{b+c}+\dfrac{b^2}{c+a}+\dfrac{c^2}{a+b}=0\)
Gấp ạ, ai giúp mình với!!!!
2: Ta có: \(\dfrac{a^2}{b+c}+\dfrac{b^2}{c+a}+\dfrac{c^2}{a+b}=\dfrac{a\left(a+b+c\right)}{b+c}+\dfrac{b\left(a+b+c\right)}{c+a}+\dfrac{c\left(a+b+c\right)}{a+b}-a-b-c=\left(a+b+c\right)\left(\dfrac{a}{b+c}+\dfrac{b}{c+a}+\dfrac{c}{a+b}\right)=a+b+c-a-b-c=0\)
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1: Sửa đề: Cho \(x,y,z\ne0\) và \(\dfrac{1}{x}+\dfrac{2}{y}+\dfrac{1}{z}=\dfrac{2}{2x+y+2z}\).
CM:....
Đặt 2x = x', 2z = z'.
Ta có: \(\dfrac{2}{x'}+\dfrac{2}{y}+\dfrac{2}{z'}=\dfrac{2}{x'+y+z'}\)
\(\Leftrightarrow\dfrac{1}{x'}+\dfrac{1}{y}+\dfrac{1}{z'}=\dfrac{1}{x'+y+z'}\)
\(\Leftrightarrow\dfrac{1}{x'}-\dfrac{1}{x'+y+z'}+\dfrac{1}{y}+\dfrac{1}{z'}=0\)
\(\Leftrightarrow\dfrac{y+z'}{x'\left(x'+y+z'\right)}+\dfrac{y+z'}{yz'}=0\)
\(\Leftrightarrow\dfrac{\left(y+z'\right)\left(yz'+x'^2+x'y+x'z'\right)}{x'yz'\left(x'+y+z'\right)}=0\)
\(\Leftrightarrow\dfrac{\left(x'+y\right)\left(y+z'\right)\left(z'+x'\right)}{x'yz'\left(x'+y+z'\right)}=0\Leftrightarrow\left(2x+y\right)\left(y+2z\right)\left(2z+2x\right)=0\Leftrightarrow\left(2x+y\right)\left(y+2z\right)\left(z+x\right)=0\left(đpcm\right)\)
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Giup mink nhanh nha:
1. Cho: x+y+z=3
va x^3+y^3+z^3+6=3(x^2+y^2+z^2)
Tinh P= (x^2015-1)(y^2015-1)(z^2015-1)
2.Cho a,b,c khac nhau va a^2-b=b^2-c=c^2-a. Tinh Q=(a+b+1)(b+c+1)(c+a+1)