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)

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

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