RU-ALLRUSSIAN 1969 Problem 14

Prove that for any positive numbers $\mathbf{a}_{1}, \mathbf{a}_{2}, \ldots , \mathbf{a}_{\mathrm{n}}$ we have:

$$\mathrm{a}_{1} / (\mathrm{a}_{2} + \mathrm{a}_{3}) + \mathrm{a}_{2} / (\mathrm{a}_{3} + \mathrm{a}_{4}) + \ldots +\mathrm{a}_{\mathrm{n} - 1} / (\mathrm{a}_{\mathrm{n}} + \mathrm{a}_{1}) + \mathrm{a}_{\mathrm{n}} / (\mathrm{a}_{1} + \mathrm{a}_{2}) > \mathrm{n} / 4.$$