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\(\dfrac{x}{xyz+xy+x+1}+\dfrac{y}{yzt+yz+y+1}+\dfrac{z}{xzt+zt+z+1}+\dfrac{t}{xyt+tx+t+1}\)
= \(\dfrac{x}{xyz+xy+x+1}+\dfrac{xy}{xyzt+xyz+xy+x}+\dfrac{xyz}{x^2yzt+xyzt+xyz+xy}+\dfrac{xyzt}{x^{2^{ }}y^2zt+x^2yzt+xyzt+xyz}\)
= \(\dfrac{x}{xyz+xy+x+1}+\dfrac{xy}{1+xyz+xy+x}+\dfrac{xyz}{x+1+xyz+xy}+\dfrac{1}{xy+x+1+xyz}\)
= \(\dfrac{x+xy+xyz+1}{x+xy+xyz+1}\)
= 1
Thay xyzt = 1 vào P, có:
P= \(\frac{x}{xyz+xy+x+xyzt\ }\) + \(\frac{y}{yzt+yz+y+1}+\frac{z}{xzt+zt+z+xyzt}+\frac{t}{xyt+tx+t+1}\)
\(P=\frac{x}{x.\left(yz+y+1+yzt\right)}+\frac{y}{yzt+yz+y+1}+\frac{z}{z.\left(xt+t+1+xyt\right)}+\frac{t}{xyt+tx+t+1}\)
\(P=\frac{1\ +y}{yz+y+yzt+1}\) \(+\frac{1+t}{xyt+tx+t+1}\)
\(P=\frac{1+y}{yz+y+yzt+xyzt\ }+\frac{1+t}{xyt+tx+t+1}\)
\(P=\frac{1+y}{y.z.\left(xyt+tx+t+1\right)}+\frac{yz+tyz}{yz.\left(xyt+tx+t+1\right)}\)
\(P=\frac{1+y+yz+tyz}{yz.\left(xyt+tx+t+1\right)}=\frac{1+y+yz+tyz}{xyzt.\left(1+y+yz+tyz\right)}=\frac{1}{xyzt}=1\)
KL: P = 1 tại xyzt=1
Theo bài ra, ta có:
\(P=\frac{x}{xy+x+1}+\frac{y}{yz+y+1}+\frac{z}{xz+z+1}\)
\(=\frac{x}{xy+x+1}+\frac{xy}{x\left(yz+y+1\right)}+\frac{z}{xz+z+xyz}\)
\(=\frac{x}{xy+x+1}+\frac{xy}{xyz+xy+x}+\frac{z}{z\left(x+1+xy\right)}\)
\(=\frac{x}{xy+x+1}+\frac{xy}{xy+x+1}+\frac{1}{xy+x+1}\)
\(=\frac{x+xy+1}{xy+x+1}\)
\(=1\)
Vậy P = 1
Ta có: P = \(\dfrac{x}{xy+x+1}\)+\(\dfrac{y}{yz+y+1}\)+\(\dfrac{z}{xz+z+1}\)
=\(\dfrac{x}{xy+x+1}\)+\(\dfrac{xy}{xyz+xy+x}\)+\(\dfrac{xyz}{x^2yz+xyz+xy}\)
=\(\dfrac{x}{xy+x+1}\)+\(\dfrac{xy}{xy+x+1}\)+\(\dfrac{1}{xy+x+1}\)(vì xyz=1)
=\(\dfrac{x+xy+1}{xy+x+1}\)
=1
Vậy P = 1
T/c:xyz=1
=>x=1;y=1;z=1
=>T=1/1+1+1 +1/1+1+1 +1/1+1+1
=>T=1/3 +1/3 +1/3
=>T=1
Ta co : x.y.z=1
Hay : x=1 ; y=1 va z=1
\(\Rightarrow T=\frac{x}{xy+x+1}+\frac{y}{yz+y+1}+\frac{z}{xz+z+1}\)
\(=\frac{1}{1.1+1+1}+\frac{1}{1.1+1+1}+\frac{1}{1.1+1+1}\)
\(=\frac{1}{3}+\frac{1}{3}+\frac{1}{3}\)
\(\Rightarrow\)T=1
abcd=1 đâu ra zậy
\(S=\left(xy+yz+zx\right)\cdot\frac{xy+yz+zx}{xyz}-\frac{xyz\left(x^2y^2+y^2z^2+z^2x^2\right)}{x^2y^2z^2}\)
\(=\frac{\left(xy+yz+zx\right)^2}{xyz}-\frac{x^2y^2+y^2z^2+z^2x^2}{xyz}\)
\(=\frac{x^2y^2+y^2z^2+z^2x^2+2xyz\left(x+y+z\right)-x^2y^2-y^2z^2-z^2x^2}{xyz}\)
\(=\frac{2xyz\left(x+y+z\right)}{xyz}=2\left(x+y+z\right)\)
\(M=\frac{1}{1+x+xy}+\frac{1}{1+y+yz}+\frac{1}{1+z+xz}\)
\(M=\frac{xyz}{x\left(1+y+yz\right)}+\frac{1}{1+y+yz}+\frac{y}{y+yz+xyz}\)
\(M=\frac{yz}{1+y+yz}+\frac{1}{1+y+yz}+\frac{y}{y+yz+1}\)
\(M=\frac{yz+y+1}{1+y+yz}\)
Tham khảo nhé~