geometry problems from China Hong Kong Mathematical Olympiads (CHKMO)
which serves as a TST for Hong Kong
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
2003 CHKMO P1
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 $MA\perp MC$.
2019 CHKMO P3
The incircle of $\triangle{ABC}$, with incentre $I$, meets $BC, CA$, and $AB$ at $D,E$, and $F$, respectively. The line $EF$ cuts the lines $BI$, $CI, BC$, and $DI$ at $K,L,M$, and $Q$, respectively. The line through the midpoint of $CL$ and $M$ meets $CK$ at $P$.
(a) Determine $\angle{BKC}$.
(b) Show that the lines $PQ$ and $CL$ are parallel.
2020 CHKMO P3
Let $\Delta ABC$ be an isosceles triangle with $AB=AC$. The incircle $\Gamma$ of $\Delta ABC$ has centre $I$, and it is tangent to the sides $AB$ and $AC$ at $F$ and $E$ respectively. Let $\Omega$ be the circumcircle of $\Delta AFE$. The two external common tangents of $\Gamma$ and $\Omega$ intersect at a point $P$. If one of these external common tangents is parallel to $AC$, prove that $\angle PBI=90^{\circ}$.
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
under construction, missing 2012
Note : 2005 stands for school year 2004-2005, and for IMO 2005
numbering of the years may change in this page
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 $A$.
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
2007 CHKMO P3
(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.
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.
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 $D$ 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
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 $\triangle ABC$.
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 $\angle CPE=\angle BMD$
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 $MA\perp MC$.
2019 CHKMO P3
The incircle of $\triangle{ABC}$, with incentre $I$, meets $BC, CA$, and $AB$ at $D,E$, and $F$, respectively. The line $EF$ cuts the lines $BI$, $CI, BC$, and $DI$ at $K,L,M$, and $Q$, respectively. The line through the midpoint of $CL$ and $M$ meets $CK$ at $P$.
(a) Determine $\angle{BKC}$.
(b) Show that the lines $PQ$ and $CL$ are parallel.
2020 CHKMO P3
Let $\Delta ABC$ be an isosceles triangle with $AB=AC$. The incircle $\Gamma$ of $\Delta ABC$ has centre $I$, and it is tangent to the sides $AB$ and $AC$ at $F$ and $E$ respectively. Let $\Omega$ be the circumcircle of $\Delta AFE$. The two external common tangents of $\Gamma$ and $\Omega$ intersect at a point $P$. If one of these external common tangents is parallel to $AC$, prove that $\angle PBI=90^{\circ}$.
Let $ABCD$ be a cyclic quadrilateral inscribed in a circle $\Gamma$ such that $AB=AD$.
Let $E$ be a point on the segment $CD$ such that $BC=DE$. The line $AE$ intersect $\Gamma$ again
at $F$. The chords $AC$ and $BF$ meet at $M$. Let $P$ be the symmetric point of $C$ about $M$.
Prove that $PE$ and $BF$ are parallel.
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