Find all integers $n \geq 3$ such that among any $n$ positive real numbers $a_{1}, a_{2}, \ldots$, $a_{n}$ with 
$$ \max \left(a_{1}, a_{2}, \ldots, a_{n}\right) \leq n \cdot \min \left(a_{1}, a_{2}, \ldots, a_{n}\right), $$
there exist three that are the side lengths of an acute triangle.
The answer is all $n \geq 13$. Define $\left(F_{n}\right)$ as the sequence of Fibonacci numbers, by $F_{1}=F_{2}=1$ and $F_{n+1}=$ $F_{n}+F_{n-1}$. We will find that Fibonacci numbers show up naturally when we work through the main proof, so we will isolate the following calculation now to make the subsequent solution easier to read.  Claim - For positive integers $m$, we have $F_{m} \leq m^{2}$ if and only if $m \leq 12$.  $$ \begin{array}{rrrrrrrrrrrrrr} F_{1} & F_{2} & F_{3} & F_{4} & F_{5} & F_{6} & F_{7} & F_{8} & F_{9} & F_{10} & F_{11} & F_{12} & F_{13} & F_{14} \\ \hline 1 & 1 & 2 & 3 & 5 & 8 & 13 & 21 & 34 & 55 & 89 & 144 & 233 & 377 \end{array} $$  By examining the table, we see that $F_{m} \leq m^{2}$ is true for $m=1,2, \ldots 12$, and in fact $F_{12}=12^{2}=144$. However, $F_{m}>m^{2}$ for $m=13$ and $m=14$.  Now it remains to prove that $F_{m}>m^{2}$ for $m \geq 15$. The proof is by induction with base cases $m=13$ and $m=14$ being checked already. For the inductive step, if $m \geq 15$ then we have  $$ \begin{aligned} F_{m} & =F_{m-1}+F_{m-2}>(m-1)^{2}+(m-2)^{2} \\ & =2 m^{2}-6 m+5=m^{2}+(m-1)(m-5)>m^{2} \end{aligned} $$  as desired. We now proceed to the main problem. The hypothesis $\max \left(a_{1}, a_{2}, \ldots, a_{n}\right) \leq n$. $\min \left(a_{1}, a_{2}, \ldots, a_{n}\right)$ will be denoted by $(\dagger)$.   $$ \begin{aligned} a_{3}^{2} & \geq a_{2}^{2}+a_{1}^{2} \geq 2 a_{1}^{2} \\ a_{4}^{2} & \geq a_{3}^{2}+a_{2}^{2} \geq 2 a_{1}^{2}+a_{1}^{2}=3 a_{1}^{2} \\ a_{5}^{2} & \geq a_{4}^{2}+a_{3}^{2} \geq 3 a_{1}^{2}+2 a_{1}^{2}=5 a_{1}^{2} \\ a_{6}^{2} & \geq a_{5}^{2}+a_{4}^{2} \geq 5 a_{1}^{2}+3 a_{1}^{2}=8 a_{1}^{2} \end{aligned} $$  and so on. The Fibonacci numbers appear naturally and by induction, we conclude that $a_{i}^{2} \geq F_{i} a_{1}^{2}$. In particular, $a_{n}^{2} \geq F_{n} a_{1}^{2}$.  However, we know $\max \left(a_{1}, \ldots, a_{n}\right)=a_{n}$ and $\min \left(a_{1}, \ldots, a_{n}\right)=a_{1}$, so $(\dagger)$ reads $a_{n} \leq n \cdot a_{1}$. Therefore we have $F_{n} \leq n^{2}$, and so $n \leq 12$, contradiction!