Let the sum of scores of students be $x.$
$\Rightarrow \frac{x}{50} = 25$
$\Rightarrow x= 25 \times 50$
$\Rightarrow \boxed{x= 1250}$
For the scores of the top $5$ students to be as high as possible, the score of the bottom $20$ students should be as low as possible.
The minimum score is $30,$ and the scores of the bottom $20$ students are distinct integers.
So, the score of bottom $20$ students must be $30,31,32, \dots, 49.$
Let the score of the topper be $y.$
So, $5y+(30+31+32+ \dots +49) = x$
$\Rightarrow 5y + \frac{20}{2}(30+49) = 1250$
$\Rightarrow 5y+790 = 1250$
$\Rightarrow 5y = 1250-790$
$\Rightarrow 5y = 460$
$\Rightarrow \boxed{y=92}$
$\therefore$ The maximum possible score of the topper is $92.$
Correct Answer $:92$