Given a regular triangular pyramid $S A B C$. Point $S$ is the apex of the pyramid, $S A=2 \sqrt{3}, B C=3, B M$ is the median of the base of the pyramid, $A R$ is the height of triangle $A S B$. Find the length of segment $M R$.

Answer

## $\frac{9}{4}$

Given a unit cube $A B C D A 1 B 1 C 1 D 1$. Find the radius of the sphere passing through points $A, B, C 1$ and the midpoint of edge $B 1 C 1$.