Given arbitrary $5$ vectors $v_1,\cdots,v_5$ in $\mathbb{R}^3$, there always exists some vector $v$ such that there are three vectors among $v_1,\cdots,v_5$ that the three angles between $v$ and them are at most $\pi/2$.
We want to figure out the minimum possible substitution for the number $\pi/2$, and further conjecture that the minimum is $\arccos(1/\sqrt{6+2\sqrt{3}})$, which will be attained when $v_1,\cdots,v_5$ form a trigonal bipyramid, namely these vectors are $(1,0,0),(-\sqrt{3},1,0),(-\sqrt{3},-1,0),(0,0,1),(0,0,-1)$.
All kinds of discussion are welcomed.