Let's name the integers around the circle as
a_1, a_2, ..., a_10 (they are the integers 1 through 10 in some order).

Now let's go around the circle calculating partial sums
s_1 = a_1 + a_2 + a_3
s_2 = a_2 + a_3 + a_4
....
s_8 = a_8 + a_9 + a_10
s_9 = a_9 + a_10 + a_1
s_10= a_10 + a_1 + a_2
All possible triplets of integers in consecutive locations around the circle are represented here, as well as their sums.
We can rephrase the question now:
prove that there is at least one of those sums greater than or equal to 17.