Cool problems (Part 2)

This is the continue of Cool nt problems. These are some problems I've found on internet and I've really liked them.

Find all real solutions of this sistem:

$\sqrt{x^{2}+y^{2}+6x+9}+ \sqrt{ x^{2}+y^{2}-8y+16} = 5$

$9y^{2}-4x^{2}= 60$

Find all real solutions of $x^{6}+y^{6}+z^{6}= 6xyz - 3$
Let n be a positive integer. Show that $N = 11111(2n times)2222(n times)$ is a perfect square.

#NumberTheory #DiophantineEquations #Integers

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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

I believe the last problem has an issue.

Bogdan Simeonov - 7 years ago

yes it must be 1111(2n times)2222(n times)

Jordi Bosch - 7 years ago

The lat digit of the number given in the third problem is 2 right? So how can it be a perfect square.......

Eddie The Head - 7 years ago

*last

Eddie The Head - 7 years ago

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}`	$\boxed{123}$