Let $x_1,x_2,\dots,x_5>0$. Prove that the only solution to the following system of equations is $(1,1,1,1,1)$ (it looks complicated, but has a simple cyclic structure):
I have pasted a picture instead of writing it in LaTeX because of some parsing issues.
I am looking for nice proofs that I can generalize to the other such systems of equations that I have. Note that:
One particularly painful way of going about this is to substitute $x_1=a^2,x_2=b^2,\dots,x_5=e^2$. However, I don't think this is the best way, and there might be more clever ways about this. At least the arguments that I could think of were long.
One can directly check that $(1,1,1,1,1)$ is a solution. Hence, a way of proving this would be to show that this system of equations can have only one solution.