4. Let $ABC$ be a triangle, $M \in (AB)$ such that $\overrightarrow{AM} = \frac{1}{2} \overrightarrow{AB}$ and $N \in (AC)$ such that $\overrightarrow{AN} = \frac{3}{4} \overrightarrow{AC}$. Denote by $P$ the midpoint of $(BC)$ and by $R$ the intersection point of the lines $AP$ and $MN$. If $\overrightarrow{AR} = k \cdot \overrightarrow{AP}$, determine the value of $k$.

B.M.
Solution: From MP||AN we deduce $\frac{M P}{A N}=\frac{P R}{A R} \Leftrightarrow \frac{A C: 2}{3 A C: 4}=\frac{P R}{A R} \Leftrightarrow \frac{2}{3}=\frac{P R}{A R} \Rightarrow$ $\frac{5}{3}=\frac{A P}{A R} \Rightarrow \overrightarrow{A R}=\frac{3}{5} \overrightarrow{A P}$. (7p)

Note: Working time 3 hours.

Each subject is graded with $7 p$.