MathLabs

Problem 2

Find all integers n≥3n \ge 3 for which there exist real numbers a1,a2,…,ana_1, a_2, \ldots, a_n satisfying an+1=a1a_{n+1}=a_1, an+2=a2a_{n+2}=a_2, and aiai+1+1=ai+2a_ia_{i+1}+1=a_{i+2} for i=1,2,…,ni=1,2,\ldots,n.
Step 7 of 7: Conclusion
Previous stepNext step
n≥3 works  ⟺  3∣nn\ge 3 \text{ works} \iff 3\mid n
Detailed analysis

Combining Steps 5–6: if 3∤n3\nmid n then Step 5 forces the sequence to be constant, and Step 6 shows a constant sequence is impossible, so no such real numbers exist. Hence the required real numbers exist only if 3∣n3\mid n, and Step 1 exhibits an explicit sequence whenever 3∣n3\mid n. So the integers n≥3n\ge 3 for which the required real numbers exist are exactly the multiples of 33.