# Squares and Square roots | HANOI 2018

Try this beautiful problem from American Invitational Mathematics Examination, HANOI, 2018 based on Squares and square roots.

## Squares and square roots - HANOI 2018

Let a=$(\sqrt2+\sqrt3+\sqrt6)(\sqrt2+\sqrt3-\sqrt6)(\sqrt3+\sqrt6-\sqrt2)(\sqrt6+\sqrt2-\sqrt3)$

b=$(\sqrt2+\sqrt3+\sqrt5)(\sqrt2+\sqrt3-\sqrt5)(\sqrt3+\sqrt5-\sqrt2)(\sqrt5+\sqrt2-\sqrt3)$. The difference a-b belongs to the set

• is [-4,0)
• is {6}
• is [-8,-6]
• cannot be determined from the given information

### Key Concepts

Algebra

Squares and square roots

Number Theory

HANOI, 2018

Elementary Number Theory by David Burton

## Try with Hints

First hint

(x+y+z)(x+y-z)(x-y+z)(-x+y+z)=2$(x^{2}y{2}+y^{2}z^{2}+z^{2}x^{2})-x^{4}-y^{4}-z^{4}$.

Second Hint

We get a-b=2(2+3)(6-5)-$6^{2}+5^{2}$=-1.

Final Step

Then a-b belongs to [-4,0).

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