# 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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