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    OlympiadHardNumber Theory7–9

    If x+1x=3x + \frac{1}{x} = 3, what is x2+1x2x^{2} + \frac{1}{x^{2}}?

    Se x+1x=3x + \frac{1}{x} = 3, qual é o valor de x2+1x2x^{2} + \frac{1}{x^{2}}?

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    Solution

    (x+1x)2=x2+2+1x2=9\left(x + \frac{1}{x}\right)^{2} = x^{2} + 2 + \frac{1}{x^{2}} = 9, so x2+1x2=7x^{2} + \frac{1}{x^{2}} = 7.