The most common and most likely answer to this crossword clue is the 5 plus 6 letter word AGONY COLUMN
I've seen this crossword clue in The Daily Telegraph
A
G
O
N
Y
C
O
L
U
M
N
Buddy explains!
The straight part
We know that "Where problems for examination are solved by specialist on paper" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case where problems for examination are solved by specialist on paper = agony column.The cryptic/wordplay part
In cryptic crossword there's often an wordplay/cryptic part as well.This part involves a more complex hint, such as an anagram, homophone, hidden word, etc.
Sadly, We don't have an explanation for this specific crossword clue yet.