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Let \(x, y, z\) be three positive integers such that \(x+y+z=1\). Show that \[x^5\sqrt{x}+y^5\sqrt{y}+z^5\sqrt{z}\leq\frac{1}{10}\sum\limits_{i=2}^{10}\left(x^i+y^i+z^i+\frac{1}{9}\right )\]

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x,y ,z positive reals?

It follows from AM-GM: \(\sum_{i=1}^{10} x^i \ge 10 x^5\sqrt{x}\). Then just add three together.

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Comment deleted Dec 12, 2013

\(\sum_{i=2}^{10}(x^i+y^i+z^i + \frac{1}{9}) = \sum^{10}_{i=2}(x^i+y^i+z^i ) + 1 = \sum^{10}_{i=1}(x^i+y^i+z^i)\)

@George G – sorry, brainfart, failed to realize that the \(x + y + z\) is also multiplied by \(\frac{1}{10}\), silly me.

yeah.. precisely!

if the base of the isosceles triangle joints the points (2,-4), (1,-3); the area is 9/2 . find the third vertex of the triangle.

please help me, show your solution

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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)`

`> This is a quote`

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}`

## Comments

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TopNewestx,y ,z positive reals?

It follows from AM-GM: \(\sum_{i=1}^{10} x^i \ge 10 x^5\sqrt{x}\). Then just add three together.

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Comment deleted Dec 12, 2013

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\(\sum_{i=2}^{10}(x^i+y^i+z^i + \frac{1}{9}) = \sum^{10}_{i=2}(x^i+y^i+z^i ) + 1 = \sum^{10}_{i=1}(x^i+y^i+z^i)\)

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yeah.. precisely!

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if the base of the isosceles triangle joints the points (2,-4), (1,-3); the area is 9/2 . find the third vertex of the triangle.

please help me, show your solution

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