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Squares and Inequality | HANOI 2018

Try this beautiful problem from American Invitational Mathematics Examination, AIME, 2009 based on geometric sequence. Use hints to solve the problem.

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


But try the problem first…

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

Source
Suggested Reading

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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