Question

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 ?

Answer 1

Answer:

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.

Step-by-step explanation: