[Brilliant Blog] Hard Made Easy II

Main post link -> http://blog.brilliant.org/2013/02/06/hard-made-easy-ii/

Please see the Blog for the complete story. Feel free to discuss the post, or the Test Yourself questions. If you show your work, there is more to discuss.

Test Yourself

Complete the proof of Question 4.
Prove that if $x$ , $y$ and $z$ are rational numbers such that $\sqrt{x} + \sqrt{y} + \sqrt{z}$ is rational, then $\sqrt{x}$ , $\sqrt{y}$ and $\sqrt{z}$ are all rational. How many terms can you show this for?
How many ordered triples of integers $(x, y, z)$ are there such that $\sqrt{x} + \sqrt{y} + \sqrt{z} = \sqrt{2000}$ ?
(**) Prove that if $a, b, c, x, y, z$ are rational numbers such that $\sqrt{a} + \sqrt{b} + \sqrt{c} + \sqrt{x} + \sqrt{y} + \sqrt{z}$ is rational, then $\sqrt{a}, \sqrt{b}, \sqrt{c}, \sqrt{x}, \sqrt{y}$ and $\sqrt{z}$ are all rational. Note: This is extremely hard, and not approachable by the methods discussed in this post.

#BrilliantBlog #Math

Easy Math Editor

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Markdown	Appears as
`italics` or `_italics_`	italics
`bold` or `__bold__`	bold
- bulleted - list	bulleted list
1. numbered 2. list	numbered list
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Math	Appears as
Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Comments

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Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$