cubic equation has $3$ distinct roots $\alpha,\beta,\gamma$ equation $x^3-9x^2+24x+c=0$ has $3$ distinct roots $\alpha,\beta,\gamma$, then $\lfloor \alpha \rfloor+\lfloor \beta \rfloor +\lfloor \gamma \rfloor =$
and $\lfloor \alpha \rfloor = \alpha-\{\alpha\},\lfloor \beta \rfloor = \beta-\{\beta\},\lfloor \gamma \rfloor = \gamma-\{\gamma\},0\leq \{\alpha\},\{\beta\},\{\gamma\}<1$
we assume $f(x) = x^3-9x^2+24x+c$, using $f'(x) = 3x^2-18x+24$
put maximum , minimum  $f'(x)=0$
$3x^2-18x+24=0$
$x^2-6x+8=(x-4)(x-2)=0$
$x=2,x=4$
put $x=2,3$ in $f''(x)=6x-18=6(x-3)$ 
$f''(2) = -6<0$ means $x=2$ is a point of local maximum
and $f''(4)=6>0$ means $x=4$ is a point of local minimum
i can not go further

Direct from vieta's relation, we get $\alpha^{2}+\beta^{2}+\gamma^{2}=-b/a=9$
