Since it's been years since the question was asked, I'm not using spoilers.
There are 4 possible answers, and we cannot deduce which is correct.
Let the numbers be $a, b,$ and $c$. Each player knows that their number is either the sum or the positive difference between the other two numbers, a total of two options. To conclude what the other two numbers are, they must eliminate one of the options.
Let $a = ix$, $b = jx$, $c = kx$, where $x$ is the greatest common factor of $a, b,$ and $c$. It must be that $i, j,$ and $k$ are all relatively prime to one another, since any number that is a factor of two numbers is also a factor of their sum and their difference.
The players all know what $x$ is; it's the GCF of the two numbers they see. So they're all trying to deduce the triad $(i,j,k)$.
Which patterns (triads) can we rule out?
Rule 4 (all numbers are positive integers) rules out the patterns
(1,1,0) &(1,0,1) &(0,1,1)\
and all patterns containing negative numbers.
Rule 5 (no two numbers are the same) rules out the patterns
(1,1,2) &(1,2,1) &(2,1,1)\
What do we deduce from each person's answer?
Person A, first round
If person A saw the partial pattern $(-,1,2)$, they would know the only possible patterns are $(1,1,2)$ and $(3,1,2)$. They know the first one is ruled out, so if they saw that, they would know the pattern had to be $(3,1,2)$, and they would know their number.
To determine which patterns a "Don't know" answer eliminates, take each already-eliminated pattern and find the complement pattern for the person who answered.
Person A said "Don't know", so the new patterns that are eliminated by that answer are:
Person B, first round
Likewise, person B's "I don't know" answer will eliminate
(1,3,2) & (2,3,1)\\
(3,5,2) & (3,4,1)
The first two were eliminated based on the initial ruled out patterns; the last two were eliminated based on the patterns ruled out by A's answer.
Person C, first round
Likewise, the answers eliminated by C's "Don't know" answer are:
(1,2,3) & (2,1,3) \\
(3,1,4) & (3,2,5) \\
(1,3,4) & (2,3,5) & (3,5,8) & (3,4,7)
Person A, second round
Person A says "Don't know" again, eliminating more patterns based on the patterns eliminated in the first round by B and C. The newly eliminated patterns are:
(5,3,2) & (4,3,1) & (7,5,2) & (5,4,1) \\
(5,2,3) & (4,1,3) & (5,1,4) & (7,2,5) \\
(7,3,4) & (8,3,5) & (13,5,8) & (11,4,7)
Person B, second round
This time, B said "Yes, I do know the answer." So we proceed as above, identifying complementary patterns to the recently eliminated ones, except one of these must be the correct pattern.
The complementary patterns for B at this round are:
(1,4,3) & (2,5,3) & (3,7,4) & (3,8,5) \\
(1,5,4) & (2,7,5) & (3,11,8) & (3,10,7) \\
(5,7,2) & (4,5,1) & (7,9,2) & (5,6,1) \\
(5,8,3) & (4,7,3) & (5,9,4) & (7,12,5) \\
(7,11,4) & (8,13,5) & (13,21,8) & (11,18,7)
Since B declared their answer is $50,$ the pattern must have a factor of 50 as the second number. The patterns satisfying that are:
(2,5,3) & (1,5,4) & (3,10,7) & (4,5,1)
Thus, the three numbers must be one of:
(20,50,30) & (10,50,40) & (15,50,35) & (40,50,10)
Side note: of these four possible solutions, only for $(40,50,10)$ does B need to rely on A's second-round answer. For the other three, B concluded their own number after C's first-round answer.