4. On the plane of rectangle $\mathrm{ABCD}$, $\mathrm{with} \mathrm{AB}=4 \mathrm{~cm}$ and $\mathrm{BC}=8 \mathrm{~cm}$, perpendicular EA is raised. Let $B M \perp E C$ and $D N \perp E C, M, N \in(E C)$. If $M N=3 \mathrm{~cm}$, calculate the length of segment $E C$.

7 points

## NATIONAL MATHEMATICS OLYMPIAD

Local stage - 20.02.2016

## 8th Grade

## Grading and marking rubric
4. $\mathrm{EA} \perp(\mathrm{ABC}), \mathrm{AB} \perp \mathrm{BC} \Rightarrow(\mathrm{T} 3 \perp) \Rightarrow \mathrm{EB} \perp \mathrm{BC}$....................................................... 1 point

Analog ED $\perp \mathrm{DC}$................................................................................................. 1 point

$\triangle E B C$ ( theorem of the leg $) \Rightarrow B C^{2}=C M \cdot E C$.................................................... 1 point

$\triangle E D C$ ( theorem of the leg ) $\Rightarrow C D^{2}=C N \cdot E C$...................................................... 1 point

$B C^{2}-C D^{2}=E C \cdot(C M-C N)=E C \cdot M N$......................................................... 2 points

$48=3 \cdot E C \Rightarrow E C=16 \mathrm{~cm}$................................................................................ 1 point

NOTE. For any correct solution, even if it is different from the one in the grading rubric, the corresponding score will be awarded.