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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

But try the problem first…

Source

HANOI, 2018

Inequalities (Little Mathematical Library) by Korovkin

Try with Hints

First hint

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) .

Second Hint

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

Final Step

Then $x^{2}=y^{2}=1$.