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The numbers 1-12 are to be placed around a circle, as on a clock, but in any order.
Show that there are three consecutive numbers in the arrangement with a sum of at least 19.

Proof #1:

The sum of the 12 numbers is 78.
If we add up all the sums of three adjacent numbers,
we use each number three times, for a total of 234.
If all the sums are to be < 19, then their maximum
total would be 12 * 18 = 216, which is too small.

In fact, there must be a sum of at least 20,
since 12 * 19 = 228, which is still too small.

Proof #2:

To avoid a sum of at least 19, the numbers 12, 11, 10, and 9
must all be separated by at least two other numbers.
Otherwise, two of them, plus any of the others totals at least 20.
Let the 12 places be as follows:
_ X _ _ X _ _ 9 _ _ X _
where X indicates 10, 11, and 12 in some order.
Since each of the dashes is part of some 3-element sum
with one of the X's, none of them can accommodate 8,
without creating a sum that is at least 8 + 10 + 1 = 19.