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If \[xyz=1\] prove that:

\[\frac { x }{ xy+x+1 } +\frac { y }{ yz+y+1 } +\frac { z }{ zx+z+1 } =1\]

Note by Mobin Moradi 2 years, 6 months ago

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Substitute \( x=\dfrac{1}{yz} \) into the first term of the expression:

\( \dfrac{\dfrac{1}{yz}}{\dfrac{y}{yz} + \dfrac{1}{yz}+ 1} = \dfrac{1}{yz+y + 1} .\)

Again, substitute \(x=\dfrac{1}{yz} \) into the third term of the expression:

\( \dfrac{z}{\dfrac{z}{yz} + z+ 1} = \dfrac{yz}{yz+y + 1} .\)

Put the terms together we have:

\(\dfrac{1 + y + yz}{yz+y+1} = 1. \blacksquare \)

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TopNewestSubstitute \( x=\dfrac{1}{yz} \) into the first term of the expression:

\( \dfrac{\dfrac{1}{yz}}{\dfrac{y}{yz} + \dfrac{1}{yz}+ 1} = \dfrac{1}{yz+y + 1} .\)

Again, substitute \(x=\dfrac{1}{yz} \) into the third term of the expression:

\( \dfrac{z}{\dfrac{z}{yz} + z+ 1} = \dfrac{yz}{yz+y + 1} .\)

Put the terms together we have:

\(\dfrac{1 + y + yz}{yz+y+1} = 1. \blacksquare \)

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