Problem 1

Let

O

lie on a line

l

. The vectors

\overrightarrow{OP_1},\ldots,\overrightarrow{OP_n}

are unit vectors, and the points

P_i

and the line

l

lie in one plane, with every

P_i

in the same half-plane bounded by

l

. Prove that, when

n

is odd,

\left|\sum_{i=1}^n\overrightarrow{OP_i}\right|\ge 1

Step 2 of 4: Order the directions

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In plain words

Remove the two outside directions and keep an odd middle block.

n=2m+1

Detailed analysis

Order the vectors by their angles in the common half-plane. Let $u=\overrightarrow{OP_1}+\overrightarrow{OP_n}$ and let $v=\sum_{i=2}^{n-1}\overrightarrow{OP_i}$ . The middle block has $n-2=2m-1$ vectors.