2006 Romanian NMO Problems/Grade 7/Problem 1

Problem

Let $ABC$ be a triangle and the points $M$ and $N$ on the sides $AB$ respectively $BC$ , such that $2 \cdot \frac{CN}{BC} = \frac{AM}{AB}$ . Let $P$ be a point on the line $AC$ . Prove that the lines $MN$ and $NP$ are perpendicular if and only if $PN$ is the interior angle bisector of $\angle MPC$ .

Solution

Let $L$ be a point on $BC$ such that $N$ is the midpoint of LC, then $2CN$ = $LC$ , the given information is the same as $\frac{LN}{BC} = \frac{AM}{AB}$ , applicating Thales theorem it follows that $ML$ is parallel to $AC$ .

Let $R$ be the point on $MN$ such that $MN$ = $NR$ , in view of $MN$ = $NR$ and $LN$ = $NC$ it follows that $RLMC$ is a parallelogram, implying that $CR$ is parallel to $ML$ , but we know that $ML$ is parallel to $AC$ , then $A$ , $C$ , $R$ are collineal.

$MN$ is perpendicular to $PN$ if and only if $NP$ is the perpendicular bisector of $MC$ if and only if $PN$ is the angle bisector of $\angle MPR$ if and only if $PN$ is the angle bisector of $\angle MPC$ , as requiered.

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2006 Romanian NMO Problems/Grade 7/Problem 1

Problem

Solution

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