Question 770173
Pretty tricky isn't it?
I name the 5 boxes a,b,c,d,e and assume they are in order from least to most heavy. We can write at least four equations for sure. The two lightest boxes, a and b, must be
(1) a + b = 110 the smallest sum. The next heavier pair must be 
(2) a + c = 112
The two heaviest boxes, d and e, must be
(3) d + e = 121 and the second heaviest pair is
(4) c + e = 120
If we subtract (4) from (3) we get
(5) d = c + 1 which when we apply to (2) we get
(6) a + c + 1 = 112 + 1 or
(7) a + d = 113
This equation (7) is not independent of the others because we used (3) and (4) to generate it. So we still only have 4 independent equations, but 5 unknowns. We need another independent equation to solve this problem.
The sum of the next pair
(8) a + e = {114,115,116,117,118} and likewise
(9) c + d = {114,115,116,117,118}
To be honest I don't have a good mathematical reason for choosing either pair. I sort of argue that (8) should be the median of the set and (9) could be {117,118}. The only method I can prove is trial and error!
I used 
(10) c + d = 117
This gives us the following 5 independent equations:
(11) a + b = 110
(12) a + c = 112
(13) c + d = 117
(14) c + e = 120
(15) d + e = 121
To solve add (13) and (14) to get
(16) 2c + (d + e) = 117 + 120 or
(17) 2c + (d+e) = 237
Use (15) in (17) to get
(18) 2c + 121 = 237 or
(19) 2c = 116 or
(20) c = 58
Now we can use (12) to get
(21) a = 54 and (13) to get
(22) d = 59 and (14) to get
(23) e = 62 and (11) to find
(24) b = 56
Let's check the answer.
Is (a+b=110)?
Is (54+56=110)?
Is (110=110)? Yes
Is (a+c=112)?
Is (54+58=112)?
Is (112=112)? Yes
Is (a+d=113)?
Is (54+59=113)?
Is (113=113)? Yes
Is (a+e=116)?
Is (54+62=116)?
Is (116=116)? Yes
Is (b+c=114)?
Is (56+58=114)?
Is (114=114)? Yes
Is (b+d=115)?
Is (56+59=115)?
Is (115=115)? Yes
Is (b+e=118)?
Is (56+62=118)?
Is (118=118)? Yes
Is (c+d=117)?
Is (58+59=117)?
Is (117=117)? Yes
Is (c+e=120)?
Is (58+62=120)?
Is (120=120)? Yes
Is (d+e=121)?
Is (59+62=121)?
Is (121=121)? Yes
Are all of the weightings used?
Is {110,112,113,116,114,115,118,117,120,121} = {110,112,113,114,115,116,117,118,120,121) Yes
Are the weights in order?
Is (a,b,c,d,e) = (54,56,58,59,62) Yes