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 4 of 4: Finish the induction
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In plain words

Adding the extreme pair cannot shorten the already large middle sum.

∣u+v∣2=∣u∣2+2u⋅v+∣v∣2≥∣v∣2≥1|u+v|^2=|u|^2+2u\cdot v+|v|^2\ge|v|^2\ge1
Detailed analysis

By induction, ∣v∣≥1|v|\ge1 because the middle block has odd size. The displayed identity gives ∣u+v∣≥∣v∣≥1|u+v|\ge|v|\ge1, which is the desired bound.