5$\cdot$8 If $y=-|x-a|+b$ and $y=|x-c|+d$ intersect at the points $(2,5)$ and $(8,3)$, then $a+c$ equals
(A) 7.
(B) 8.
(C) 10.
(D) 13.
(E) 18.
(50th American High School Mathematics Examination, 1999)
［Solution］Given the problem:
$$
\left\{\begin{array} { l } 
{ 5 = - | 2 - a | + b , } \\
{ 8 = - | 8 - a | + b }
\end{array} \quad \text { and } \quad \left\{\begin{array}{l}
5=|2-c|+d, \\
8=|8-c|+d .
\end{array}\right.\right.
$$

Subtracting the two equations in each set, we get:
$$
\left\{\begin{array}{l}
|2-a|-|8-a|=3 \\
|8-c|-|2-c|=3
\end{array}\right.
$$

Discuss equation (1) in the interval $a \leqslant 2,28$;
Discuss equation (2) in the interval $c \leqslant 2,28$ to solve for $a=6 \frac{1}{2}$, $c=3 \frac{1}{2}$.
Thus, $a+c=10$.
Therefore, the answer is $(C)$.