Try this beautiful problem from American Invitational Mathematics Examination, HANOI, 2018 based on Squares and inequality.
Write down all real numbers (x,y) satisfying two conditions \(x^{2018}+y^{2}=2\) and \(x^{2}+y^{2018}=2\).
Algebra
Squares and square roots
Inequality
Answer: is (-1,-1),(-1,1),(1,-1),(1,1).
HANOI, 2018
Inequalities (Little Mathematical Library) by Korovkin
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 \(x^{2}=y^{2}=1\).
