jmc

We start by trying to get as much information as possible from what is given to us about $a$, $b$, and $c$. Since their arithmetic mean is $8$, we know that $\frac{a+b+c}{3}=8$ which when we multiply both sides by $3$ gives us $a+b+c=24$. Since their geometric mean is $5$ we have $\sqrt[3]{abc}=5$, which when we cube both sides gives us $abc=125$. Now since the harmonic mean is $3$, we have $$\frac{3}{\frac{1}{a}+\frac{1}{b}+\frac{1}{c}}=3.$$We can simplify to get $$\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=1.$$Converting to a common denominator gives $$\frac{ab+bc+ca}{abc}=1$$which tells us that $ab+bc+ca=abc=125$.

Now, we try and use this information to find $a^2+b^2+c^2$. Since we already know $a+b+c$, we can start by squaring that whole expression and expanding. This gives us, $$(a+b+c{)}^2=a^2+b^2+c^2+2(ab+bc+ca).$$We can rewrite the equation above as $$a^2+b^2+c^2=(a+b+c{)}^2-2(ab+bc+ca).$$Since we know both expressions on the right, we can substitute them in and solve to get $$a^2+b^2+c^2=(24{)}^2-2(125)=576-250=\boxed{326}.$$