Prove that the vectors follow the inequality

Question

(Unicamp Mathematics Olympiad) Let $S$ be a set with 7 distinct points on $\Bbb R^2$. For every $v \in S$, it's valid that $1 \leq ||v|| \leq 8$. Prove that it's always possible to find $a,b,c \in S$ such that $\frac{3}{2} < \frac{||a||}{||b||} + \frac{||b||}{||c||} + \frac{||c||}{||a||} < 6$.

Answer 1

By AM-GM it is easy to establish the minimum of $3$ for any choice of vectors. For the upper bound, partitioning the norms into $[1,2), [2,4),[4,8]$ as suggested ensures by Pigeonhole at least $3$ vectors are in one of the partitions. So the maximum ratios possible in that partition is $2$, and not all three ratios can be simultaneously $2$, so the upper bound is also established.

In fact the maximum for such choice of vectors is restricted to $3\frac12$, which can be also shown if needed.