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You were on the right track but you didn't listen to the advice of the question. Try to use the previous lemma to prove this question.
First, lets think about all the possible groups of these consequtive integers that are arranged around this circle. We want to show that there necessarily exists a group whose elements sum to a number greater than or equal to 17. So, because we're trying to prove a property of this set of groups (that there exists one with our special property), lets take a look at the nature of these groups, in general.
So, lets ask the question "what is the average of each group's average?" This may sound like a funny question but it's easy enough to answer because each group is the same size. Therefore, the average of their averages is simply the average of all the different group's elements. If you are not convinced of this I'll leave this fact as an exercise for the reader. This is your homework, after all.
You've established, yourself, that the sum of all the elements of all the groups is 165 and that there are 30 of them. So, that makes the average 165 / 30 = 5.5. This is not surprising since it is also the average of the consecutive integers 1 through 10.
Now, I don't want to do your homework for you (you are supposed to learn this stuff) but you should be wondering how to apply your lemma (at least one element must be greater than the average) to this situation. Then the proof should become obvious to you.
If you totally give up, come back and I will show you the answer.
Good luck!
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