Sum of all the solutions of $z^2 + |z| = |z(z)|^2$ is ____.
Step-by-Step Solution
Key Concept: Count valid 3-digit numbers by fixing the middle digit and determining the ranges for the first and last digits independently.
For 3-digit numbers $xyz$ with $x ≥ 1$ and $y ≥ 2$, if $y = n$ then $x$ ranges from 1 to $n-1$ (giving $n-1$ values) and $z$ ranges from 0 to $n-1$ (giving $n$ values). Thus for each $y$ value from 2 to 9, there are $n(n-1)$ valid 3-digit numbers. The total is $\sum_{n=2}^{9} n(n-1) = \sum_{n=1}^{9} n^2 - \sum_{n=1}^{9} n = \frac{9(10)(19)}{6} - \frac{9(10)}{2} = 285 - 45 = 240$.
Correct Answer: 0