MathLabs

Problem 1

Let OO lie on a line ll. The vectors OP1→,…,OPn→\overrightarrow{OP_1},\ldots,\overrightarrow{OP_n} are unit vectors, and the points PiP_i and the line ll lie in one plane, with every PiP_i in the same half-plane bounded by ll. Prove that, when nn is odd, ∣∑i=1nOPi→∣≥1\left|\sum_{i=1}^n\overrightarrow{OP_i}\right|\ge 1.
Step 1 of 4: Base case
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In plain words

One unit arrow already has the required length.

n=1n=1
A representative unit vector at angle θ\theta from the boundary line.
The unit circle shows a radius vector and its horizontal and vertical components.
Detailed analysis

For one unit vector the norm is exactly 11.