This problem was suggested by Yatir Halevi.
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<br />
Solve the equation:
<br />
<table width="100%"><tr><td align="center">
    <math xmlns="http://www.w3.org/1998/Math/MathML">
    <mstyle displaystyle="true"><mrow>
<msup><mrow><mi>x</mi></mrow><mrow><mo stretchy="false">(</mo>
<msup><mrow><mi>x</mi></mrow><mrow><mn>3</mn></mrow>
</msup>
<mo stretchy="false">)</mo></mrow>
</msup>
<mo>=</mo><mn>3</mn><mo>.</mo></mrow>
    </mstyle></math>
</td></tr></table>
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<br />
<br />
Solve the equation:
<br />
<table width="100%"><tr><td align="center">
    <math xmlns="http://www.w3.org/1998/Math/MathML">
    <mstyle displaystyle="true"><mrow>
<msup><mrow><mi>x</mi></mrow><mrow><mo stretchy="false">(</mo>
<msup><mrow><mi>x</mi></mrow><mrow><mo stretchy="false">(</mo>
<msup><mrow><mi>x</mi></mrow><mrow><mn>3</mn></mrow>
</msup>
<mo stretchy="false">)</mo></mrow>
</msup>
<mo stretchy="false">)</mo></mrow>
</msup>
<mo>=</mo><mn>3</mn><mo>.</mo></mrow>
    </mstyle></math>
</td></tr></table>
<br />

<br />
<br />
Can you find solutions to all equations of the
form:
<br />
<table width="100%"><tr><td align="center">
    <math xmlns="http://www.w3.org/1998/Math/MathML">
    <mstyle displaystyle="true"><mrow>
<msup><mrow><mi>x</mi></mrow><mrow>
<msup><mrow><mi>x</mi></mrow><mrow>
<msup><mrow><mi>x</mi></mrow><mrow>
<msup><mrow><mi>x</mi></mrow><mrow>
<msup><mrow><mi>x</mi></mrow><mrow>
<msup><mrow><mi>x</mi></mrow><mrow>
<msup><mrow><mo>...</mo></mrow><mrow><mi>a</mi></mrow>
</msup>
</mrow>
</msup>
</mrow>
</msup>
</mrow>
</msup>
</mrow>
</msup>
</mrow>
</msup>
</mrow>
</msup>
<mo>=</mo><mi>a</mi></mrow>
    </mstyle></math>
</td></tr></table>
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where the sequence of powers is defined in the
same way and 
<math xmlns="http://www.w3.org/1998/Math/MathML">
<mrow><mi>a</mi></mrow></math> is a positive integer?
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<br />
Explain what you have done and prove that you
have found all possible solutions.

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