Prove that in any polygon, there exist two sides whose radio is less than $2$.(Essentialy if $a_1\geq a_2\geq...\geq a_n$ are the sides of a polygon prove that there exist $i,j\in\{1,2,..,n\}$ so that $i<j$ and $\frac {a_i}{a_j}<2$).
Source: Kosovo MO 2010 Grade 9, Problem 3
Tags: algebra
Prove that in any polygon, there exist two sides whose radio is less than $2$.(Essentialy if $a_1\geq a_2\geq...\geq a_n$ are the sides of a polygon prove that there exist $i,j\in\{1,2,..,n\}$ so that $i<j$ and $\frac {a_i}{a_j}<2$).