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March 20, 2020

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


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

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