### China Hong Kong 1999-2018 (CHKMO) 18p (UC)

geometry problems from China Hong Kong Mathematical Olympiads (CHKMO)
which serves as a TST for Hong Kong
(only those not in IMO Shortlist)
with aops links in the names

1999 -
under construction, missing 2012

Note 2005 stands for school year 2004-2005, and for IMO 2005

1999 CHKMO P1
In a concyclic quadrilateral $PQRS$,$\angle PSR=\frac{\pi}{2}$ , $H,K$ are perpendicular foot from $Q$ to sides $PR,RS$ , prove that $HK$ bisect segment$SQ$.

2000 CHKMO P2
Let $ABC$ be a non-equilateral triangle. Denote by $I$ the incenter and by $O$ the circumcenter of the triangle $ABC$. Prove that $\angle AIO\leq\frac{\pi}{2}$ holds if and only if $2\cdot BC\leq AB +AC$.

2001 CHKMO P1
Let $O$ be the circumcentre of a triangle $ABC$ with $AB > AC > BC$. Let $D$ be a point on the minor arc $BC$ of the circumcircle and let $E$ and $F$ be points on $AD$ such that $AB \perp OE$ and $AC \perp OF$ . The lines $BE$ and $CF$ meet at $P$. Prove that if $PB=PC+PO$, then $\angle BAC = 30^{\circ}$.

2002 CHKMO P1
A triangle $ABC$ is given. A circle $\Gamma$, passing through $A$, is tangent to side $BC$ at point $P$ and intersects sides $AB$ and $AC$ at $M$ and $N$ respectively. Prove that the smaller arcs $MP$ and $NP$ of $\Gamma$ are equal iff $\Gamma$ is tangent to the circumcircle of $\Delta ABC$ at $A$.

2003 CHKMO P1
Two circles meet at points $A$ and $B$. A line through $B$ intersects the first circle again at $K$ and the second circle at $M$. A line parallel to $AM$ is tangent to the first circle at $Q$. The line $AQ$ intersects the second circle again at $R$.
(a) Prove that the tangent to the second circle at $R$ is parallel to $AK$.
(b) Prove that these two tangents meet on $KM$.

2004 CHKMO P3
Let $K, L, M, N$ be the midpoints of sides $AB, BC, CD, DA$ of a cyclic quadrilateral $ABCD$. Prove that the orthocentres of triangles $ANK, BKL, CLM, DMN$ are the vertices of a parallelogram.

2005 CHKMO P3
Points $P$ and $Q$ are taken sides $AB$ and $AC$ of a triangle $ABC$ respectively such that $\hat{APC}=\hat{AQB}=45^{0}$. The line through $P$ perpendicular to $AB$ intersects $BQ$ at $S$, and the line through $Q$ perpendicular to $AC$ intersects $CP$ at $R$. Let $D$ be the foot of the altitude of triangle $ABC$ from $A$. Prove that $SR\parallel BC$ and $PS,AD,QR$ are concurrent.

2007 CHKMO P3
A convex quadrilateral $ABCD$ with $AC \ne BD$ is inscribed in a circle with center $O$. Let $E$ be the intersection of diagonals $AC$ and BD.  If $P$ is a point inside $ABCD$ such that $\angle PCB+ \angle PAB = \angle PDC + \angle PBC=90^{o}$. Prove that $O, P$ and $E$ are collinear

2008 CHKMO P2
Let $D$ be a point on the side $BC$ of triangle $ABC$ such that $AB+BD= AC+CD$. The line segment $AD$ cut the incircle of triangle $ABC$ at $X$ and $Y$ with $X$ closer to $A$. Let $E$ be the point of contact of the incircle of triangle $ABC$ on the side $BC$. Show that
(i) $EY$ is perpendicular to $AD$,
(ii) $XD$ is $2IA'$, where $I$ is the incentre of the triangle $ABC$ and $A'$ is the midpoint of $BC$.

2009 CHKMO P3
$\Delta ABC$ is a triangle such that $AB \neq AC$. The incircle of $\Delta ABC$ touches $BC, CA, AB$ at $D, E, F$ respectively. $H$ is a point on the segment $EF$ such that $DH \bot EF$. Suppose $AH \bot BC$, prove that $H$ is the orthocentre of $\Delta ABC$.

2010  CHKMO P3
Let $\triangle ABC$ be a right-angled triangle with $\angle C=90^\circ$. $CD$ is the altitude from $C$ to $AB$, with $D$ on $AB$. $\omega$ is the circumcircle of $\triangle BCD$. $\omega_1$ is a circle situated in $\triangle ACD$, it is tangent to the segments $AD$ and $AC$ at $M$ and $N$ respectively, and is also tangent to circle $\omega$.
(i) Show that $BD\cdot CN+BC\cdot DM=CD\cdot BM$.
(ii) Show that $BM=BC$.

2011 CHKMO P1
Let $ABC$ be an arbitrary triangle. A regular $n$-gon is constructed outward on the three sides of $\triangle ABC$. Find all $n$ such that the triangle formed by the three centres of the $n$-gons is equilateral.

missing 2012

In $\triangle ABC$, $AB>AC$. In the circumcircle $(O)$ of $\triangle ABC$, $M$ is the midpoint of arc $BAC$. The incircle $(I)$ of $\triangle ABC$ touches $BC$ at $D$, the line through $D$ parallel to $AI$ intersects $(I)$ again at $P$. Prove that $AP$ and $IM$ intersect at a point on $(O)$.

Let $ABC$ be a triangle with $CA>BC>AB$. Let $O$ and $H$ be the circumcentre and orthocentre of triangle $ABC$ respectively. Denote by $D$ and $E$ the midpoints of the arcs $AB$ and $AC$ of the circumcircle of triangle $ABC$ not containing the opposite vertices. Let $D'$ be the reflection of $D$ about $AB$ and $E'$ the reflection of $E$ about $AC$. Prove that $O,H,D',E'$ are concylic if and only if $A,D',E'$ are collinear.

2015 CHKMO P4
Let $\triangle ABC$ be a scalene triangle, and let $D$ and $E$ be points on sides $AB$ and $AC$ respectively such that the circumcircles of triangles $\triangle ACD$ and $\triangle ABE$ are tangent to $BC$. Let $F$ be the intersection point of $BC$ and $DE$. Prove that $AF$ is perpendicular to the Euler line of $\triangle ABC$.

2016 CHKMO P3
Let $ABC$ be a triangle. Let $D$ and $E$ be respectively points on the segments $AB$ and $AC$, and such that $DE||BC$. Let $M$ be the midpoint of $BC$. Let $P$ be a point such that $DB=DP$, $EC=EP$ and such that the open segments (segments excluding the endpoints) $AP$ and $BC$ intersect. Suppose $\angle BPD=\angle CME$. Show that $\angle CPE=\angle BMD$

2017 CHKMO P3
Let $ABC$ be an acute-angled triangle. Let $D$ be a point on the segment $BC, I$ the incentre of $ABC$. The circumcircle of $ABD$ meets $BI$ at $P$ and the circumcircle of $ACD$ meets $CI$ at $Q$. If the area of $PID$ and the area of $QID$ are equal, prove that $PI \cdot QD=QI \cdot PD$.

2018 CHKMO P2
Suppose $ABCD$ is a cyclic quadrilateral. Extend $DA$ and $DC$ to $P$ and $Q$ respectively such that $AP=BC$ and $CQ=AB$. Let $M$ be the midpoint of $PQ$. Show that $MA\perp MC$.