## Problem 4

In the cube $\mathrm{ABCDA}^{\prime} \mathrm{B}^{\prime} \mathrm{C}^{\prime} \mathrm{D}^{\prime}$, points $\mathrm{M}, \mathrm{N}$, and $\mathrm{P}$ are the midpoints of the edges $\mathrm{CC}$ ', $\mathrm{A}^{\prime} \mathrm{D}^{\prime}$, and $\mathrm{C}^{\prime} \mathrm{D}^{\prime}$, respectively. Find the sine of the angle formed by the lines BM and NP.

Problems selected by Lung Ioan and Nicoara Florin

Note: a) The actual working time is 3 hours.

b) All problems are mandatory.

c) Each problem is scored from 0 to 7.

## NATIONAL MATHEMATICS OLYMPIAD

Local stage - 15.02.2014

## Grade 8

## Grading Rubric
## Problem 4

$B M \cap B^{\prime} C^{\prime}=\{Q\} ; D^{\prime} Q \| \mathrm{NP}$ (2p). The angle determined by the lines NP and BM is $\square D^{\prime} Q M$ (1p)

Let $l$ be the side length of the cube $D^{\prime} M=\frac{l}{2} \sqrt{5}, M Q=\frac{l}{2} \sqrt{5}, D^{\prime} Q=l \sqrt{2} \quad$ (2p). Let $R \in\left(D^{\prime} Q\right)$ such that

$M R \perp D^{\prime} Q \Rightarrow M R=\frac{l}{2} \sqrt{3} \quad$ (1p) Therefore, $\sin \left(\square D^{\prime} Q M\right)=\frac{M R}{M Q}=\frac{\sqrt{3}}{\sqrt{5}} \quad$ (1p)

Note: a) Each corrector awards an integer number of points.

b) Any other correct solution is graded accordingly.