# Matt's means

**Algebra**Level 3

The arithmetic mean, geometric mean, and harmonic mean of \(a,b,c\) are 8, 5, and 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, \(a_1, a_2, \ldots, a_n\), is (respectively)

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