Putnam 1989 
Topic: Complex Numbers Algebra Analysis

Problem: Given \[ 11z^{10}+10iz^9+10iz-11=0, \] find the value of $|z|$. (Here $z$ is a complex number and $i^2=-1$.)

Solution: We have $z^9 = \frac{11 - 10iz}{11z + 10i}$. If $z = a + bi$, then 
\[ 
|z^9| = \left| \frac{11 - 10iz}{11z + 10i} \right| = \sqrt{\frac{11^2 + 220b + 10^2(a^2 + b^2)}{11^2(a^2 + b^2) + 220b + 10^2}} \equiv \frac{f(a,b)}{g(a,b)}. 
\]

If $a^2 + b^2 > 1$, then $g(a,b) > f(a,b)$, making $|z^9| < 1$, a contradiction. If $a^2 + b^2 < 1$, then $f(a,b) > g(a,b)$, making $|z^9| > 1$, a contradiction. Thus, $|z| = \boxed{1}$.