24. Let $f(x)=x^{3}+3 x+1$, where $x$ is a real number. Given that the inverse function of $f$ exists and is given by
$$
f^{-1}(x)=\left(\frac{x-a+\sqrt{x^{2}-b x+c}}{2}\right)^{1 / 3}+\left(\frac{x-a-\sqrt{x^{2}-b x+c}}{2}\right)^{1 / 3}
$$
where $a, b$ and $c$ are positive constants, find the value of $a+10 b+100 c$.
24. Answer: 521 .
Let $y=x^{3}+3 x+1$. Then $x^{3}+3 x+(1-y)=0$. Let $A, B$ be constants such that
$$
\left(x^{3}+A^{3}+B^{3}\right)-3 A B x=x^{3}+3 x+(1-y) \text {. }
$$

Hence
$$
\begin{array}{c}
A^{3}+B^{3}=1-y \\
-3 A B=3
\end{array}
$$

By (2), we have $B=-1 / A$. Substitute this into (1) and simplify.
$$
A^{6}+(y-1) A^{3}-1=0 \text {. }
$$

Hence
$$
A^{3}=\frac{1-y \pm \sqrt{(y-1)^{2}+4}}{2}=\frac{1-y \pm \sqrt{y^{2}-2 y+5}}{2} .
$$

Therefore $B=\left(\frac{1-y \mp \sqrt{y^{2}-2 y+5}}{2}\right)^{1 / 3}$. Hence
$$
\begin{aligned}
x & =-A-B=\left(\frac{y-1+\sqrt{y^{2}-2 y+5}}{2}\right)^{1 / 3}+\left(\frac{y-1-\sqrt{y^{2}-2 y+5}}{2}\right)^{1 / 3} \\
& \equiv\left(\frac{y-a+\sqrt{y^{2}-b y+c}}{2}\right)^{1 / 3}+\left(\frac{y-a-\sqrt{y^{2}-b y+c}}{2}\right)^{1 / 3}
\end{aligned}
$$

Thus $a=1, b=2, c=5$ and $a+10 b+100 c=521$.