A distinct integer from 1 to 10 is placed in each red circle on the figure below.

Is it possible, that the sum of the four numbers in all of the blue lines is the same?

No, it's not possible.
Yes, it's possible.

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It is not possible because the sum of the numbers 1-10 = 55 and all 10 numbers are used twice which adds up to 110. All the sums are in lines of 4 and 110 is not divisible by 4