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

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

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

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

Squares and Inequality | HANOI 2018

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

Squares and inequality - HANOI 2018

Write down all real numbers (x,y) satisfying two conditions \(x^{2018}+y^{2}=2\) and \(x^{2}+y^{2018}=2\).

is [-1,0)
is (0,1),(-1,0)
is (-1,-1),(-1,1),(1,-1),(1,1)
cannot be determined from the given information

Key Concepts

Algebra

Squares and square roots

Inequality

Check the Answer

Answer: is (-1,-1),(-1,1),(1,-1),(1,1).

HANOI, 2018

Inequalities (Little Mathematical Library) by Korovkin

Try with Hints

If \(x^{2}>1\) then\(x^{2018}>x^{2}>1\) and \(y^{2}<1\) implies that \(y^{2} \gt y^{2018}\) Then \(x^{2018}+y^{2} \gt x^{2}+y^{2018}\) (contradiction) .

Analogically, if \(x^{2} \lt 1\) implies that \(x^{2018}+y^{2} \lt x^{2}+y^{2018}\)(contradiction).

Then \(x^{2}=y^{2}=1\).

Squares and Square roots | HANOI 2018