2. In the cube $A B C D A^{\prime} B^{\prime} C^{\prime} D^{\prime}$, consider the points $M$ and $P$ as the midpoints of the edges $\left[D^{\prime} C^{\prime}\right]$ and $[B C]$. If $A C \cap B D=\left\{O_{1}\right\}, A D^{\prime} \cap A^{\prime} D=\left\{O_{2}\right\}$ and the points $S, T$ are the midpoints of the segments $\left[O_{1} M\right],\left[P O_{2}\right]$, then determine the measure of the angle between the lines $S T$ and $B^{\prime} C^{\prime}$.

Nicolae Stănică, Brăila
## Solution.

$D^{\prime} M P O_{1}$ parallelogram $\Rightarrow D^{\prime}, S, P$ collinear and $D^{\prime} S=S P$. 3 p

From $D^{\prime} S=S P$ and $O_{2} T=T P \Rightarrow(S T)$ midline in $\Delta D^{\prime} O_{2} P \Rightarrow$ 2 p

$\Rightarrow S T \| D^{\prime} O_{2} \Rightarrow m\left(\angle S T, B^{\prime} C^{\prime}\right)=m\left(\angle D^{\prime} O_{2}, A^{\prime} D^{\prime}\right)=45^{\circ}$. 2 p