## Σελίδες

### China Northern 2016-20 (CNMO) 13p

geometry problems from China Northern Mathematical Olympiads
with aops links in the names
China Sijiazhuang
2016 - 2020

China Northern 2016 grade 10 p2
In isosceles triangle $ABC$, $\angle CAB=\angle CBA=\alpha$, points $P,Q$ are on different sides of line $AB$, and $\angle CAP=\angle ABQ=\beta,\angle CBP=\angle BAQ=\gamma$. Prove that $P,C,Q$ are collinear.

China Northern 2016 grade 10 p6 , 11 p6
Four points $B,E,A,F$ lie on line $AB$ in order, four points $C,G,D,H$ lie on line $CD$ in order, satisfying: $\frac{AE}{EB}=\frac{AF}{FB}=\frac{DG}{GC}=\frac{DH}{HC}=\frac{AD}{BC}.$ Prove that $FH\perp EG$.

China Northern 2016 grade 11 p2
Inscribed Triangle $ABC$ on circle $\odot O$. Bisector of $\angle ABC$ intersects $\odot O$ at $D$. Two lines $PB$ and $PC$ that are tangent to $\odot O$ intersect at $P$. $PD$ intersects $AC$ at $E$, $\odot O$ at $F$. Prove that $M,F,C,E$ are concyclic.

China Northern 2017 grade 10 p3 , 11 p2
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 $MN$.

Length of sides of regular hexagon $ABCDEF$ is $a$. Two moving points $M,N$ moves on sides $BC,DE$, satisfy that $\angle MAN=\frac{\pi}{3}$. Prove that $AM\cdot AN-BM\cdot DN$ is a definite value.

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 , 11 p5
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 $\angle ABM=\angle ACM$

China Northern 2018 grade 11 p3
$A,B,C,D,E$ lie on $\odot O$ in that order,and$$BD \cap CE=F,CE \cap AD=G,AD \cap BE=H,BE \cap AC=I,AC \cap BD=J.$$Prove that $\frac{FG}{CE}=\frac{GH}{DA}=\frac{HI}{BE}=\frac{IJ}{AC}=\frac{JF}{BD}$ when and only when $F,G,H,I,J$ are concyclic.

Two circles $O_1$ and $O_2$ intersect at $A,B$. Diameter $AC$ of $\odot O_1$ intersects $\odot O_2$ at $E$, Diameter $AD$ of $\odot O_2$ intersects $\odot O_1$ at $F$. $CF$ intersects $O_2$ at $H$, $DE$ intersects $O_1$ at $G,H$. $GH\cap O_1=P$. Prove that $PH=PK$.
Two circles $O_1$ and $O_2$ intersect at $A,B$. Bisector of outer angle $\angle O_1AO_2$ intersects $O_1$ at $C$, $O_2$ at $D$. $P$ is a point on $\odot(BCD)$, $CP\cap O_1=E,DP\cap O_2=F$. Prove that $PE=PF$.

In $\triangle ABC$, $\angle BAC = 60^{\circ}$, point $D$ lies on side $BC$, $O_1$ and $O_2$ are the centers of the circumcircles of $\triangle ABD$ and $\triangle ACD$, respectively. Lines $BO_1$ and $CO_2$ intersect at point $P$. If $I$ is the incenter of $\triangle ABC$ and $H$ is the orthocenter of $\triangle PBC$, then prove that the four points $B,C,I,H$ are on the same circle.

In $\triangle ABC$, $AB>AC$. Let $O$ and $I$ be the circumcenter and incenter respectively. Prove that if $\angle AIO = 30^{\circ}$, then $\angle ABC = 60^{\circ}$.