Every year, there is a contest to see who has the heaviest pumpkins for that year.
Last year, a farmer brought 5 pumpkins to the contest. Instead of weighing them one at a time, he informed the judges,
"When I weighed two at a time, I got the following weights: 108, 112, 113, 114, 115, 116, 117, 118, 120, and 122."
How much did each pumpkin weigh?
I tried solving this problem by summing all the weights (1,155) and dividing by 10 to get the average weight of two pumpkins (115.5). I then divided by 2 to get the average weight of each pumpkin (57.75). To determine the middle pumpkin, I multiplied by 5 to find the total weight of the five pumpkins (288.75) and then subtracted the lightest and heaviest weight ('C' = 58.75). Fast forward, I determined 'A' = 53.25, 'B' = 54.75, 'C' = 58.75, 'D' = 60.75, and 'E' = 61.25.
However upon review, I determined that these values could not produce the 10 weights. In addition, I concluded the values must end in .5 to increase the combinations.
The question is: is there a correct answer and what is it, or is this unsolvable?