The integers 1, 2, ..., 10 are written on a circle, in any order. Show that there are 3 adjacent numbers whose sum is 17 or greater.

Solution The integers 1, 2, ..., 10 are written on a circle, in any order. Show that there are 3 adjacent numbers whose sum is 17 or greater.

There are 10 triples of adjacent numbers with sums S1, S2, ..., S10. If each is less than 17, they all add up to 16×10 = 160, at most. However, in the latter sum each of the numbers 1, 2, ..., 10 appears 3 times, so that the sum must be 3×55 = 165, at least. (55 = 1 + 2 + ... + 10.) It follows that some of Si are bound to be greater than 16. • 200 points have been cosen on a circle, all with integer number of degrees. Prove that the points there are at least one pair of antipodes, i.e., the points 180° apart.
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