PROBLEM 3. Let triangle $ABC$ and $M$ be the midpoint of side [ $BC$ ]. If $N$ is the midpoint of segment $[AM]$ and $BN \cap AC=\{P\}$, determine the ratio between the area of quadrilateral $MCPN$ and the area of triangle $ABC$.
PROBLEM 3. Let triangle $ABC$ and $M$ be the midpoint of side $[BC]$. If $N$ is the midpoint of segment $[AM]$ and $BN \cap AC=\{P\}$, determine the ratio between the area of quadrilateral $MCPN$ and the area of triangle $ABC$.

Grading criteria. Let $MR \| BP (R \in PC)$ and $S=A_{ANP}$.

(1p) Since $MR=\frac{BP}{2}$,

(2p) we obtain that $NP=\frac{MR}{2}=\frac{BP}{4}$, hence $A_{ABN}=3 A_{ANP}=3S$.

(1p) But since $A_{ABN}=A_{BMN}=3S$

(1p) and $A_{ABM}=A_{AMC}=6S$, it follows that $A_{ABC}=12S$

(1p) and $A_{MCPN}=A_{AMC}-A_{ANP}=5S$.

(1p) Therefore, $\frac{A_{MCPN}}{A_{ABC}}=\frac{5}{12}$.