The 5772 Ulpaniada included the following question:
Consider a four digit square number (a number which is the square of a whole number).
Its digit notation is $aabb$ (the thousands digit is $a$, the hundreds digit is $a$, the tens digit is $b$ and the units digit is $b$).
What is the value of $a^2+b^2$?
It offered five choices: $51$, $85$, $73$, $65$, and $60$.
$85$ and $65$ are the only choices equal to $a^2+b^2$ for $b\in\{1,4,9,5,6\}$ and $1\le a\le9$. That analysis also shows $aabb$ to be one of $2299$, $7766$, $8811$, and $7744$. Since $11$ goes into $aabb$, so does $11^2$, which eliminates $7766$ and $8811$ (as $11\nmid706$, $11\nmid801$). But I can't seem to narrow it down further, to decide between $7744$ and $2299$ (without, of course, reference to or generation of a list of squares), or by any other means to decide between $85$ and $65$.
Any suggestions?