### China Northern 2017-18 4p

geometry problems from China Northern Mathematical Olympiads
with aops links in the names
China Sijiazhuang
2017 - 2018

Let $D$ be the midpoint of side $BC$ of triangle $ABC$. Let $E, F$ be points on sides $AB, AC$ respectively such that $DE = DF$. Prove that $AE + AF = BE + CF \iff \angle EDF = \angle BAC$.

Triangle $ABC$ has $AB > AC$ and $\angle A = 60^\circ$. Let $M$ be the midpoint of $BC$, $N$ be the point on segment $AB$ such that $\angle BNM = 30^\circ$. Let $D,E$ be points on $AB, AC$ respectively. Let $F, G, H$ be the midpoints of $BE, CD, DE$ respectively. Let $O$ be the circumcenter of triangle $FGH$. Prove that $O$ lies on line $MN$.

China Northern 2018 grade 10 P1
In triangle $ABC$, let the circumcenter, incenter, and orthocenter be $O$, $I$, and $H$ respectively. Segments $AO$, $AI$, and $AH$ intersect the circumcircle of triangle $ABC$ at $D$, $E$, and $F$. $CD$ intersects $AE$ at $M$ and $CE$ intersects $AF$ at $N$. Prove that $MN$ is parallel to $BC$.

China Northern 2018 grade 10 P6
Let $H$ be the orthocenter of triangle $ABC$. Let $D$ and $E$ be points on $AB$ and $AC$ such that $DE$ is parallel to $CH$. If the circumcircle of triangle $BDH$ passes through $M$, the midpoint of $DE$, then prove that $\angle ABM=\angle ACM$