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March 20, 2020

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

Check the Answer


Answer: is [-4,0).

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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