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 3 of 4: The bisector estimate

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In plain words

The bisector points toward every middle vector rather than away from it.

u\cdot v\ge 0

Detailed analysis

The direction of $u$ is the bisector of the two extreme directions. Every middle vector makes an angle at most $90^\circ$ with $u$ , because its angle lies between the two extreme angles and the angular spread is at most $180^\circ$ . Thus each dot product with $u$ is nonnegative and therefore $u\cdot v\ge0$ . If $u=0$ , the same inequality is immediate.