4. ( $7 \mathrm{p}$ ) In the square $A B C D$, consider $M$ and $N$ as the midpoints of the sides $[A D]$ and $[D C]$. Let $P \in(M B$ such that $M B=B P$. If $B D=8 \mathrm{~cm}$, find the distance from $B$ to $N P$.

Note: All questions are mandatory,

Time allowed: 3 hours

Each question is scored from 0 to 7 p.

## GRADING AND SCORING GUIDE  MATHEMATICS OLYMPIAD  LOCAL PHASE
$$
\Delta M A B \equiv \triangle N C B \Rightarrow \left\lvert\, \begin{aligned}
& B M=B N \\
& B N=B P
\end{aligned} \Rightarrow N B=\frac{M P}{2}\right.
$$

$\triangle M N P$ right-angled $\left(\widehat{N}=90^{\circ}\right) \Rightarrow M N \perp N P, B R \perp N P \Rightarrow M N \| B R$

B is the midpoint of MP, so BR is the midline in $\triangle M N P \Rightarrow B R=\frac{M N}{2}=\frac{A C}{4}=2 \mathrm{~cm}$