Problem 4

Find all real solutions

x_1,\ldots,x_5

x_i+x_{i+2}=yx_{i+1}

for

i=1,\ldots,5

, where subscripts are reduced modulo

5

Step 4 of 4: Classify the golden-ratio cases

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In plain words

The exceptional quadratic values give a two-dimensional cyclic eigenspace; the uniform value gives a one-dimensional one.

y^2+y-1=0\Longrightarrow (x_1,\ldots,x_5)=(s,t,yt-s,-y(s+t),ys-t)

Detailed analysis

For $y=(-1\pm\sqrt5)/2$ , any $s,t\in\mathbb R$ produce a solution through the displayed two-parameter family. For $y=2$ , the solutions are precisely $(s,s,s,s,s)$ . For every other real $y$ , only $(0,0,0,0,0)$ remains.