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# [Brilliant Blog] 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

1. Complete the proof of Question 4.

2. 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?

3. How many ordered triples of integers $$(x, y, z)$$ are there such that $$\sqrt{x} + \sqrt{y} + \sqrt{z} = \sqrt{2000}$$?

4. (**) 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.

Note by Peter Taylor
4 years, 6 months ago