RO-NMO 2004 Problem 83

Let $ABCD A^{\prime} B^{\prime} C^{\prime} D^{\prime}$ be a truncated regular pyramid in which $BC^{\prime}$ and $DA^{\prime}$ are perpendicular.

(a) Prove that $\measuredangle \left( AB^{\prime},DA^{\prime} \right) = 60^{\circ}$ ;

(b) If the projection of $B^{\prime}$ on $(ABC)$ is the center of the incircle of $ABC$ , then prove that $d \left( CB^{\prime},AD^{\prime} \right) = \frac12 BC^{\prime}$ .

Mircea Fianu