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.

No vote yet

4 votes

×

Problem Loading...

Note Loading...

Set Loading...

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

There are no comments in this discussion.