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 1 of 7: The answer and the extremal construction

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3\mid n,\qquad a_{3j+1}=a_{3j+2}=-1,\ a_{3j+3}=2

Detailed analysis

The claim is that such real numbers exist exactly when $3\mid n$ . For $n=3k$ , repeating the block $(-1,-1,2)$ a total of $k$ times — i.e. $a_{3j+1}=a_{3j+2}=-1$ and $a_{3j+3}=2$ for $j=0,1,\ldots,k-1$ — gives a valid sequence: every three cyclically consecutive entries are a rotation of $(-1,-1,2)$ , and $(-1)(-1)+1=2$ , $(-1)(2)+1=-1$ , $(2)(-1)+1=-1$ show that $a_ia_{i+1}+1=a_{i+2}$ holds for every $i=1,\ldots,n$ . It remains to show that no other $n\ge 3$ works.