Number the pumpkins from 1 (heaviest) to 5 (lightest)

The two heaviest weigh 121 and the two lightest weigh 110.

Find the average weight of the 5 pumpkins.

=(110 + 112 + 113 + 114 + 115 + 116 + 117 + 118 + 120 + 121)/20 = 57.8

Knowing the weights of the two heaviest and two lightest work out the weight of the pumpkin in the middle.

121 + 110 + Pumpkin3 = 5 * (57.8)

231 + Pumpkin3 = 289

ie Pumpkin3 weighs 58 pounds.

The second heaviest weight comes from Pumpkin3 and Pumpkin1.

ie Pumpkin1=120 - 58 = 62 pounds

Pumpkin1 + Pumpkin2 = 121

ie Pumpkin2 = 59 Pounds

The second lightest weight comes from Pumpkin3 and Pumpkin5.

ie Pumpkin5=112 - 58 = 54 pounds

Pumpkin4 + pumpkin5 = 110

ie Pumpkin4 = 56 pounds

So the pumpkin wights are 62, 59, 58, 56 and 54 pounds.