MathDB

length of sum of 2n vectors from center of a semicircle is an odd integer

Source: Danube 2008 p3

July 22, 2019

geometryvectorSumsemicircleprojectiongeometric inequality

Problem Statement

On a semicircle centred at

O

and with radius

1

choose the respective points

A_1,A_2,...,A_{2n}

, for

n \in N^*

. The lenght of the projection of the vector

\overrightarrow {u}=\overrightarrow{OA_1} +\overrightarrow{OA_2}+...+\overrightarrow{OA_{2n}}

on the diameter is an odd integer. Show that the projection of that vector on the diameter is at least

1