In this paper we consider one well known cyclic algebraic inequality \[ \frac{a_1}{a_2+a_3}+\frac{a_2}{a_3+a_4}+\ldots+\frac{a_{n-1}}{a_n+a_1}+\frac{a_n}{a_1+a_2}\geq\frac n2 \] where is $a_i>0$, $a_i+a_{i+1}>0$ $(a_{n+1}=a_1, \ i=1,2,\ldots,n)$. We give the proofs of this inequality for $n=3,4,5,6$ by using the inequality \[ \frac{a^2_1}{x_1}+\frac{a^2_2}{x_2}+\ldots\frac{a^2_n}{x_n}\geq\frac{(a_1+a_2+\ldots+a_n)}{x_1+x_2+\ldots+x_n}, \] where $x_i>0$; $a_i\in$\textexclamdown $\quad (i=1,2,\ldots,n)$.