Matt's means

The arithmetic mean, geometric mean, and harmonic mean of $a,b,c$ are 8, 5, 3, respectively. What is the value of $a^2 + b^2 + c^2$?

This problem is posed by Matt.

Details and Assumptions:

The arithmetic mean, geometric mean, and harmonic mean of $n$ numbers $a_1, a_2, \ldots, a_n$ are (respectively)

$\frac {\sum_{i=1}^{n} a_n}{n},\quad \sqrt[n]{\prod_{i=1}^{n} a_n},\quad \left( \frac{n} { \sum_{i=1}^{n} \frac{1}{a_n} } \right) .$