Let n geq 4 be a positive integer and let l_1 | vedclass

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(D) Given $\frac{l_1}{l_2} + \frac{l_2}{l_3} + \ldots + \frac{l_n}{l_1} = n$.By the Arithmetic Mean-Geometric Mean $(AM \geq GM)$ inequality,we have $

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$I$ and $III$ are true

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(D) Given $\frac{l_1}{l_2} + \frac{l_2}{l_3} + \ldots + \frac{l_n}{l_1} = n$.By the Arithmetic Mean-Geometric Mean $(AM \geq GM)$ inequality,we have $\frac{1}{n} \sum_{i=1}^{n} \frac{l_i}{l_{i+1}} \geq \sqrt[n]{\prod_{i=1}^{n} \frac{l_i}{l_{i+1}}}$,where $l_{n+1} = l_1$.Since the product of the terms is $\frac{l_1}{l_2} \cdot \frac{l_2}{l_3} \cdots \frac{l_n}{l_1} = 1$,the $GM$ is $1$.Thus,$\frac{1}{n} \sum \frac{l_i}{l_{i+1}} \geq 1$,which implies $\sum \frac{l_i}{l_{i+1}} \geq n$.The equality holds if and only if all terms are equal,i.e.,$\frac{l_1}{l_2} = \frac{l_2}{l_3} = \ldots = \frac{l_n}{l_1} = 1$.This implies $l_1 = l_2 = \ldots = l_n$,so statement $I$ is true.If $P$ is cyclic and all sides are equal,then the arcs subtended by these sides are equal,which implies the angles are equal. Thus,$P$ is a regular polygon,making statement $III$ true.Statement $II$ is not necessarily true because equal sides do not imply equal angles in a general polygon (e.g.,a rhombus is not necessarily a square).

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