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Let $n$ be a positive integer and that $2n$ numbers are arranged at different points around a circle, half of these numbers being $+1$ and half of being $-1$. Moving clockwise around the circle from a given starting position, let $T_i$ be the total of the first $i$ numbers passed.

$(i)$ Prove that there is a starting position on the circle for which no $T_i$ is negative.

$(ii)$ For any starting position prove that $$n+\sum_{i=1}^{2n}T_i$$ is even.

I have solved $(ii)$. How can I approach the first one? Any help?

Christoph
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Kousik Sett
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  • "...being +1..." stands for "the $x$ component of the point is positive"? "...being -1..." stands for "the $x$ component of the point is negative"? – the_candyman Sep 28 '20 at 21:17
  • I think the author means that those values are assigned to those points, rather than having any direct connection to their locations. – Lieutenant Zipp Sep 28 '20 at 23:10

1 Answers1

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HINT: Start anywhere and get $2n$ partial sums $T_1,\ldots,T_{2n}(=0)$. Say that $T_k$ is minimal among these $2n$ numbers. What happens if you start at $k+1$ instead?

Brian M. Scott
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