This problem was suggested by Yatir Halevi.
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<br />
Solve the equation:



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<table align="center"><tr><td nowrap="nowrap" align="center">
<i>x</i><sup>(<i>x</i><sup>3</sup>)</sup>=3.</td></tr></table>
</td></tr></table>


<br />
<br />
Solve the equation:



<br clear="all" /><table border="0" width="100%"><tr><td>
<table align="center"><tr><td nowrap="nowrap" align="center">
<i>x</i><sup>(<i>x</i><sup>(<i>x</i><sup>3</sup>)</sup>)</sup>=3.</td></tr></table>
</td></tr></table>


<br />
<br />
Can you find solutions to all equations of the
form:



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<table align="center"><tr><td nowrap="nowrap" align="center">
<i>x</i><sup><i>x</i><sup><i>x</i><sup><i>x</i><sup><i>x</i><sup><i>x</i><sup>...<sup><i>a</i></sup></sup></sup></sup></sup></sup></sup>=<i>a</i></td></tr></table>
</td></tr></table>


where the sequence of powers is defined in the
same way and <i>a</i> is a positive integer?
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<br />
Explain what you have done and prove that you
have found all possible solutions.