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If \(x,y\) and \(z\) are positive proper fractions satisfying \(x+y+z=2\), prove that \[ \dfrac x{1-x} \cdot \dfrac y{1-y} \cdot \dfrac z{1-z} \ge 8 . \]

Note by Aniket Sen 1 year, 9 months ago

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Substitute \(a=1-x,b=1-y,c=1-z\). Simplify the inequality and you find that it is true by AM-HM inequality.

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Easy Math Editor

`*italics*`

or`_italics_`

italics`**bold**`

or`__bold__`

boldNote: you must add a full line of space before and after lists for them to show up correctlyparagraph 1

paragraph 2

`[example link](https://brilliant.org)`

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Remember to wrap math in \( ... \) or \[ ... \] to ensure proper formatting.`2 \times 3`

`2^{34}`

`a_{i-1}`

`\frac{2}{3}`

`\sqrt{2}`

`\sum_{i=1}^3`

`\sin \theta`

`\boxed{123}`

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TopNewestSubstitute \(a=1-x,b=1-y,c=1-z\). Simplify the inequality and you find that it is true by AM-HM inequality.

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