Question 6
Let $n\ge4$ be an integer. Find all positive real solutions of the cyclic system of $2n$ equations below, where indices are taken modulo $2n$:
$$\begin{aligned}a_1&=\frac1{a_{2n}}+\frac1{a_2},&a_2&=a_1+a_3,\\ a_3&=\frac1{a_2}+\frac1{a_4},&a_4&=a_3+a_5,\\ a_5&=\frac1{a_4}+\frac1{a_6},&a_6&=a_5+a_7,\\ &\vdots&&\vdots\\ a_{2n-1}&=\frac1{a_{2n-2}}+\frac1{a_{2n}},&a_{2n}&=a_{2n-1}+a_1.\end{aligned}$$
Proposed by Mohsen Jamaali. Evan Chen’s third-party solution notes: https://aops.com/community/p21498967; source video https://youtu.be/9WNgDETHOlI?t=1700. One point is nonofficial editorial normalization; manual essay grading only.
$$\begin{aligned}a_1&=\frac1{a_{2n}}+\frac1{a_2},&a_2&=a_1+a_3,\\ a_3&=\frac1{a_2}+\frac1{a_4},&a_4&=a_3+a_5,\\ a_5&=\frac1{a_4}+\frac1{a_6},&a_6&=a_5+a_7,\\ &\vdots&&\vdots\\ a_{2n-1}&=\frac1{a_{2n-2}}+\frac1{a_{2n}},&a_{2n}&=a_{2n-1}+a_1.\end{aligned}$$
Proposed by Mohsen Jamaali. Evan Chen’s third-party solution notes: https://aops.com/community/p21498967; source video https://youtu.be/9WNgDETHOlI?t=1700. One point is nonofficial editorial normalization; manual essay grading only.