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When I started solving this problem, I realized that there were several, unique solutions to the first part of the problem. For example, one solution I got was:

a = 2
b = 1
c = 8
d = 4
In this case, the averages and the mediant are both 25%.
Another example could be:
a = 4
b = 2
c = 8
d = 4
In this case, the averages and the mediant are both 50%.

The second part of the problem can be solved using simple algebra.

Pass Mark = 50%
25% of n questions = 0.25n

0.25n/n +b/d >= 0.5
0.25n/n = 0.5 - b/d
0.25 = 0.5 = -b/d
-0.25 = -b/d
b/d = 0.25

The only way that he will pass is if he scores greater than or equal to 25% in his remaining questions (b/d).