1
$\begingroup$

While learning about inequalities, I encountered this problem

12 positive real numbers $s_1 \leq s_2 \leq \ldots \leq s_{11} \leq s_{12}$ have the property that no three of them are the side lengths of an acute triangle. Compute the minimum value of $\frac{s_{12}}{s_1}$.

Since the triangle cannot be acute, $s_i^2+s_j^2 \leq s_k^2 \quad (i, j, k \in \{1, 2, \ldots, 12\}, i \neq j \neq k$.

However, I don't know how to proceed from here. I'm not sure if listing all the possible inequalities out would help.

Also, the answer is $12$, which leads me to wonder if there were $n$ numbers, would the minimum be $n$?

If anyone could provide any insight, it would be greatly appreciated!

$\endgroup$
0
$\begingroup$

If $a<b<c$ and $s_a,s_b,s_c$ are not the sides of an acute-angled triangle, then $s_c^2\geq s_a^2+s_b^2$.
We may assume WLOG $s_1=\sqrt{1}$ and $s_2=\sqrt{x}\geq 1$. It follows that $s_3\geq\sqrt{x+1}$, then $s_4\geq \sqrt{2x+1}$, then $s_5\geq \sqrt{3x+2}$, then $\ldots$, then $s_{12}\geq \sqrt{F_{11}x+F_{10}} = \sqrt{89x+55}$.
In particular the ratio between $s_{12}$ and $s_1$ is at least $\sqrt{F_{12}}=\color{red}{12}$, and the ratio between $s_n$ and $s_1$ is at least $\color{red}{\sqrt{F_n}}$ in the general case.

| cite | improve this answer | |
$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy