Show that if the first 10 positive integers 1,2,3,···,10 are placed around a circle, in any order, there exists three integers in consecutive locations around the circle that have a sum greater thanor equal to 17. Is your proof constructive or non-constructive?

Question

Genesis Ellison

11 January, 06:37

+5

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Espinoza · Answer 1

Let A1=a1+a2+a3, A2=a2+a3+a4, and so on, A10=a10+a1+a2. Then A1+A2+⋯+A10=3 (a1+a2+⋯+a10) = (3) (55) = 165, so some Ai≥165/10=16.5, so some Ai≥17.