Answer
The paint bomb explodes.
Work Step by Step
We can find $\gamma$:
$\gamma = \frac{1}{\sqrt{1-\beta^2}}$
$\gamma = \frac{1}{\sqrt{1-0.900^2}}$
$\gamma = 2.294$
We can find the length of the tunnel in the train view:
$L = \frac{L_0}{\gamma}$
$L = \frac{200~m}{2.294}$
$L = 87.2~m$
In the train view, the length of the train is $200~m$
When the front of the train passes the far end of the tunnel (FF), the rear of the train has still not reached the near end of the tunnel (RN)
The paint bomb explodes when event FF occurs, which is before event RN occurs.
Therefore, the paint bomb explodes.
