đk$\Leftrightarrow \begin{cases}a+2b+3c=6 \\ 6a+3b+2c=11 \end{cases}\Leftrightarrow \begin{cases}6a+12b+18c=36\\ 6a+3b+2c=11 \end{cases}\Leftrightarrow \begin{cases}9b +16c=25\\ 18a+9b+6c=33 \end{cases}$$\Leftrightarrow \begin{cases}18a-10c=8 \\ 9b+16c=25 \end{cases}\Leftrightarrow \begin{cases}c=\frac{9a-4}{5} \\ b= \frac{21-16a}{5} \end{cases}$
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Khi đó $F-2015=81a^2-26\left(\dfrac{9a-4}5 \right)^2+\frac{9(21-16a)}{5}-\frac{22(9a-4)}{5}$
$=\frac{-81a^2+162a+969}{25} \le \frac{1050}{25}$
$\Leftrightarrow F\le 2057$
Vậy $\max F=2057\Leftrightarrow a=b=c=1$