Maximum Value of Eh?

The first thing we can do is to express $b$ and $c$ in terms of $a$ .

$b^{2} + c^{2}$ = $2014 - a^{2}$ $(1)$

$2b + 3c$ = $86 - a$ $(2)$

Then applying the Cauchy- Schwarz Inequality (As in the tag), We will have:

( $2^{2} + 3^{2})$ ( $b^{2} + c^{2})$ $\geq$ $(2b+3c)^{2}$

Substituting $(1)$ and $(2)$ ,

$(13)$ $(2014 - a^{2})$ $\geq$ $(86 - a)^{2}$

Expanding and Manipulating, we will get:

$0 \geq 7 a^{2} - 86 a - 9393$

Which gives the range of possible values for a which is:

$-31 \leq a \leq \frac{303}{7}$

The maximum value of a is $\frac{303}{7}$ and gives $m = 303, n = 7$ and the required answer which is $\boxed{m+n = 310}$