Problem 2

Find all integers

n \ge 3

for which there exist real numbers

a_1, a_2, \ldots, a_n

satisfying

a_{n+1}=a_1

a_{n+2}=a_2

, and

a_ia_{i+1}+1=a_{i+2}

for

i=1,2,\ldots,n

Step 6 of 7: A constant sequence is impossible

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a_1=a_2=\cdots=a_n=c\ \Longrightarrow\ c^2-c+1=0,\ \Delta=-3<0

Detailed analysis

If $a_1=a_2=\cdots=a_n=c$ , the relation $a_1a_2+1=a_3$ becomes $c^2+1=c$ , i.e. $c^2-c+1=0$ . This quadratic has discriminant $\Delta=1-4=-3<0$ , so it has no real root $c$ , a contradiction. Hence $3\nmid n$ is impossible.