BdMO Online Forum

Find the maximum value of the positive integer $n$ that satisfies the following inequality,
$a_{1}^{2}+a_{2}^{2}+a_{3}^{2}+a_{4}^{2}+...+a_{n+1}^{2} \geq a_{n+1}(a_{1}+a_{2}+....+a_{n})$.
where $a_{i}$ is an arbitrary real number.$i \in (1,2,3,....,n+1)$.

sakibtanvir wrote:Find the maximum value of the positive integer $n$ that satisfies the following inequality,
$a_{1}^{2}+a_{2}^{2}+a_{3}^{2}+a_{4}^{2}+...+a_{n+1}^{2} \geq a_{n+1}(a_{1}+a_{2}+....+a_{n})$.
where $a_{i}$ is an arbitrary real number.$i \in (1,2,3,....,n+1)$.

The problem is from brilliant.org , for moral reasons, I request the other members of this forum to not post the solution/answer before next Monday.