Question

Suppose the equation $||x-a|-b|=2008$ has $3$ distinct real roots and $a\ne 0.$ Find the value of $\frac{b}{502}.$
(correct answer + 5, wrong answer 0)

Answer 1

$||x-a|-b|=2008$
$\Rightarrow |x-a|=b\pm 2008$

Case $1:$ If $b<2008$,
then $|x-a|=b-2008$ has no real root since $b-2008<0$ and $|x-a|=b+2008$ has atmost $2$ real roots.

Case $2:$ If $b>2008$,
then both $|x-a|=b-2008\text{and}|x-a|=b+2008$ has $2$ real roots, which gives $4$ distinct real roots $a\pm (b-2008)\text{and}a\pm (b+2008)$ since $a+b+2008>a+b-2008>a-b+2008>a-b-2008$

Case $3:$ If $b=2008$,
then $|x-a|=b-2008=0$ has only $1$ real root $x=a$,
and $|x-a|=b+2008=4016$ has $2$ real roots $x=a\pm 4016$
Hence, $b=2008$
$\therefore \frac{b}{502}=\frac{2008}{502}=4$