Singapore 2nd Round (SMO) 2005-19 42p (UC)

geometry problems from Singapore Mathematical Olympiads Round 2
with aops links in the names

2005 - 2013 (complete)

2014-2019 (under construction)

Junior Round 2 

2006 Singapore Junior Round 2 P4
In $\vartriangle ABC$, the bisector of $\angle B$ meets $AC$ at $D$ and the bisector of $\angle C$ meets $AB$ at $E$. These bisectors intersect at $O$ and $OD = OE$. If $AD \ne AE$, prove that $\angle = 60^o$.

2007 Singapore Junior Round 2 P1
Let $ABCD$ be a trapezium with $AB// DC, AB = b, AD = a ,a<b$ and $O$ the intersection point of the diagonals. Let $S$ be the area of the trapezium $ABCD$. Suppose the area of $\vartriangle DOC$ is $2S/9$. Find the value of $a/b$.

2007 Singapore Junior Round 2 P2
Equilateral triangles $ABE$ and $BCF$ are erected externally onthe sidess $AB$ and $BC$ of a parallelogram $ABCD$. Prove that $\vartriangle DEF$ is equilateral.

2008 Singapore Junior Round 2 P1
In $\vartriangle ABC, \angle ACB = 90^o, D$ is the foot of the altitude from $C$ to $AB$ and $E$ is the point on the side $BC$ such that $CE = BD/2$. Prove that $AD + CE = AE$.

2008 Singapore Junior Round 2 P3
In the quadrilateral $PQRS, A, B, C$ and $D$ are the midpoints of the sides $PQ, QR, RS$ and $SP$ respectively, and $M$ is the midpoint of $CD$. Suppose $H$ is the point on the line $AM$ such that $HC = BC$. Prove that $\angle BHM = 90^o$.

2009 Singapore Junior Round 2 P1
In $\vartriangle ABC, \angle A= 2 \angle B$. Let $a,b,c$ be the lengths of its sides $BC,CA,AB$, respectively. Prove that$a^2 = b(b + c)$.

2010 Singapore Junior Round 2 P1
Let the diagonals of the square $ABCD$ intersect at $S$ and let $P$ be the midpoint of $AB$. Let $M$ be the intersection of $AC$ and $PD$ and $N$ the intersection of $BD$ and $PC$. A circle is incribed in the quadrilateral $PMSN$. Prove that the radius of the circle is $MP- MS$.

2011 Singapore Junior Round 2 P2
Two circles $\Gamma_1,  \Gamma_2$ with radii $r_i, r_2$, respectively, touch internally at the point $P$. A tangent parallel to the diameter through $P$ touches $ \Gamma_1$ at $R$ and intersects  $\Gamma_2$ at $M$ and $N$. Prove that $PR$ bisects $\angle MPN$.

2012 Singapore Junior Round 2 P1
Let $O$ be the centre of a parallelogram $ABCD$ and $P$ be any point in the plane. Let $M, N$ be the midpoints of $AP, BP$, respectively and $Q$ be the intersection of $MC$ and $ND$. Prove that $O, P$ and $Q$ are collinear.

2012 Singapore Junior Round 2 P3
In $\vartriangle ABC$, the external bisectors of $\angle A$ and $\angle B$ meet at a point $D$. Prove that the circumcentre of  $\vartriangle ABD$ and the points $C, D$ lie on the same straight line.

2013 Singapore Junior Round 2 P2
In the triangle $ABC$, points $D, E, F$ are on the sides $BC, CA$ and $AB$ respectively such that $FE$ is parallel to $BC$ and $DF$ is parallel to $CA$, Let P be the intersection of $BE$ and $DF$, and $Q$ the intersection of $FE$ and $AD$. Prove that $PQ$ is parallel to $AB$.

2014 Singapore Junior Round 2 P3
In the triangle $ABC$, the bisector of $\angle A$ intersects the bisection of $\angle B$ at the point $I, D$ is the foot of the perpendicular from $I$ onto $BC$. Prove that the bisector of $\angle BIC$ is perpendicular to the bisector of $\angle AID$.

2015 Singapore Junior Round 2 P
2016 Singapore Junior Round 2 P

2017 Singapore Junior Round 2 P3
 In $\triangle ABC, AB=AC, D$ is a point on the side $BC$ and $E$ is a point on the segment $AD$. Given $\angle{BED}=\angle{BAC}=2\angle{CED}$, prove that $BD=2CD$.

2018 Singapore Junior Round 2 P2
In $\vartriangle ABC, AB=AC=14 \sqrt2 , D$ is the midpoint of $CA$ and $E$ is the midpoint of $BD$. Suppose $\vartriangle CDE$ is similar to $\vartriangle ABC$. Find the length of $BD$.

2019 Singapore Junior Round 2 P1
In the triangle $ABC, AC=BC, \angle C=90^o, D$ is the midpoint of $BC, E$ is the point on $AB$ such that $AD$ is perpendicular to $CE$. Prove that $AE=2EB$.

Senior Round 2

2005 Singapore Senior Round 2 P2 
Consider the nonconvex quadrilateral $ABCD$ with $\angle C>180$ degrees. Let the side $DC$ extended to meet $AB$ at $F$ and the side $BC$ extended to meet $AD$ at $E$. A line intersects the interiors of the sides $AB,AD,BC,CD$ at points $K,L,J,I$ respectively. Prove that if $DI=CF$ and $BJ=CE$, then $KJ=IL$

2006 Singapore Senior Round 2 P2
Let $ABCD$ be a cyclic quadrilateral, let the angle bisectors at $A$ and $B$ meet at $E$, and let the line through $E$ parallel to side $CD$ intersect $AD$ at $L$ and $BC$ at $M$. Prove that $LA + MB = LM$.

2006 Singapore Senior Round 2 P3
Two circles are tangent to each other internally at a point $T$. Let the chord $AB$ of the larger circle be tangent to the smaller circle at a point $P$. Prove that the line TP bisects $\angle ATB$.

2007 Singapore Senior Round 2 P3
In the equilateral triangle $ABC, M, N$ are the midpoints of the sides $AB, AC$, respectively. The line $MN$ intersects the circumcircle of $\vartriangle ABC$ at $K$ and $L$ and the lines $CK$ and $CL$ meet the line $AB$ at $P$ and $Q$, respectively. Prove that $PA^2 \cdot QB = QA^2 \cdot PB$.

2008 Singapore Senior Round 2 P1
Let $ABCD$ be a trapezium with $AD // BC$. Suppose $K$ and $L$ are, respectively, points on the sides $AB$ and $CD$ such that $\angle  BAL = \angle  CDK$. Prove that $\angle BLA = \angle  CKD$.

2009 Singapore Senior Round 2 P1
Given triangle $ ABC $ with points $ M $ and $ N $ are in the sides $ AB $ and $ AC $ respectively.
If  $ \frac{BM}{MA} +\frac{CN}{NA} = 1 $ , then prove that the centroid of  $ ABC $ lies on $ MN $ .

2010 Singapore Senior Round 2 P1
In the $\triangle ABC$ with $AC>BC$ and $\angle B<90^{\circ}$, $D$ is the foot of the perpendicular from $A$ onto $BC$ and $E$ is the foot of perpendicular from $D$ onto $AC$. Let $F$ be the point on the segment $DE$ such that $EF \cdot DC=BD \cdot DE$. Prove that $AF$ is perpendicular to $BF$.

2011 Singapore Senior Round 2 P1
In the triangle $ABC$, the altitude at $A$, the bisector of $\angle B$ and the median at $C$ meet at a common point. Prove  that the triangle $ABC$ is equilateral.

2012 Singapore Senior Round 2 P1
A circle $\omega$ through the incentre$ I$ of a triangle $ABC$ and tangent to $AB$ at $A$, intersects the segment $BC$ at $D$ and the extension of$ BC$ at $E$. Prove that the line $IC$ intersects $\omega$ at a point $M$ such that $MD=ME$.

2013 Singapore Senior Round 2 P1
In triangle∆$ABC$,$AB>AC$,the extension of the altitude $AD$ with $D$ lying inside $BC$ intersects the circumcircle of $ABC$ at $P$. The circle through $P$ and tangent to BC to BC at D intersects the circumcircle of ∆$ABC$ at $Q$ distinct from P with PQ=DQ.Prove that $AD=BD-DC$

2014 Singapore Senior Round 2 P1
In the triangle $ABC$, the excircle opposite to the vertex $A$ with centre $I$ touches the side BC at D. (The circle also touches the sides of $AB$, $AC$ extended.) Let  $M$ be the midpoint of $BC$ and $N$ the midpoint of $AD$. Prove that $I,M,N$ are collinear.

2015 Singapore Senior Round 2 P
2016 Singapore Senior Round 2 P
2017 Singapore Senior Round 2 P

2018 Singapore Senior Round 2 P2
Let $O$ be a point inside a triangle such that $\angle BOC=90^o$ and $\angle BAO = \angle BCO$. Prove that $\angle OMN =90^o$, where $M,N$ are the midpoints of $AC$ and $BC$ respectively.

In a parallelogram $ABCD$, the bisector of $\angle A$ intersects $BC$ at $M$ and the extension of $DC$ at $N$. Let $O$ be the circumcircle of the triangle $MCN$. Prove that $\angle OBC = \angle ODC$

Open Round 2

2005 Singapore Open Round 2 P2
Let $G$ be the centroid of triangle $ABC$. Through $G$ draw a line parallel to $BC$ and intersecting the sides $AB$ and $AC$ at $P$ and $Q$ respectively. Let $BQ$ intersect $GC$ at $E$ and $CP$ intersect $GB$ at $F$. If $D$ is the midpoint of $BC$, prove that triangles $ABC$ and $DEF$ are similar.

2006 Singapore Open Round 2 P1 (also here)
In the triangle $ABC,\angle A=\frac{\pi}{3},D,M$ are points on the line $AC$ and $E,N$ are points on the line $AB$ such that $DN$ and $EM$ are the perpendicular bisectors of $AC$ and $AB$ respectively. Let $L$ be the midpoint of $MN$. Prove that $\angle EDL=\angle ELD$

2007 Singapore Open Round 2 P3
Let $A_1$, $B_1$ be two points on the base $AB$ of an isosceles triangle $ABC$, with $\angle C>60^{\circ}$, such that $\angle A_1CB_1=\angle ABC$. A circle externally tangent to the circumcircle of $\triangle A_1B_1C$ is tangent to the rays $CA$ and $CB$ at points $A_2$ and $B_2$, respectively. Prove that $A_2B_2=2AB$.

2008 Singapore Open Round 2 P2
In the acute triangle $\triangle ABC$ M is a point in the interior of the segment AC and N is a point on the extension of segment AC such that MN=AC.Let D,E be the feet of perpendiculars from M,N onto lines BC,AB respectively. Prove that the orthocentre of $\triangle ABC$ lies on circumcircle of $\triangle BED$

2009 Singapore Open Round 2 P1
Let $O$ be the center of the circle inscribed in a rhombus $ABCD$. points $E,F,G,H$ are chosen on sides $AB, BC, CD, DA$ respectively so that $EF$ and $GH$ are tangent to inscribed circle. show that $EH$ and $FG$ are parallel.

2010 Singapore Open Round 2 P1
Let $CD$ be a chord of a circle $\Gamma_1$ and $AB$ a diameter of $\Gamma_1$ perpendicular to $CD$ at $N$ with $AN > NB$. A circle $\Gamma_2$ centered at $C$ with radius $CN$ intersects $\Gamma_1$ at points $P$ and $Q$. The line $PQ$ intersects $CD$ at $M$ and $AC$ at $K$; and the extension of $NK$ meets $\Gamma_2$ at $L$. Prove that $PQ$ is perpendicular to $AL$

2011 Singapore Open Round 2 P1 (also here)
In the acute-angled non-isosceles triangle $ABC$, $O$ is its circumcenter, $H$ is its orthocenter and $AB>AC$. Let $Q$ be a point on $AC$ such that the extension of $HQ$ meets the extension of $BC$ at the point $P$. Suppose $BD=DP$, where $D$ is the foot of the perpendicular from $A$ onto $BC$. Prove that $\angle ODQ=90^{\circ}$.

2012 Singapore Open Round 2 P1
The incircle with centre $I$ of the triangle $ABC$ touches the sides $BC, CA$ and $AB$ at $D, E, F$ respectively. The line $ID$ intersects the segment $EF$ at $K$. Proof that $A, K, M$ collinear, where $M$ is the midpoint of $BC$.

2013 Singapore Open Round 2 P2
Let $ABC$ be an acute-angled triangle and let $D$, $E$, and $F$ be the midpoints of $BC$, $CA$, and $AB$ respectively. Construct a circle, centered at the orthocenter of triangle $ABC$, such that triangle $ABC$ lies in the interior of the circle. Extend $EF$ to intersect the circle at $P$, $FD$ to intersect the circle at $Q$ and $DE$ to intersect the circle at $R$. Show that $AP=BQ=CR$.

2013 Singapore Open Round 2 P5
Let $ABC$ be a triangle with integral side lengths such that $\angle A=3\angle B$. Find the minimum value of its perimeter.

2014 Singapore Open Round 2 P1 (also here) (and here)
The quadrilateral $ABCD$ is inscribed in a circle which has diameter $BD$. Points $A'$ and $B'$ are symmetric to $A$ and $B$ with respect to the lines $BD$ and $AC$ respectively. If the lines $A'C$ and $BD$ intersect at $P$, and the lines $AC$ and $B'D$ intersect at $Q$, prove that $PQ$ is perpendicular to $AC$.

2015 Singapore Open Round 2 P1
In an acute-angled triangle $ABC, D$ is the point on $BC$ such that $AD$ bisects $\angle BAC$,
$E$ and $F$ are the feet of the perpendiculars from $D$ onto $AB$ and $AC$ respectively. The
segments $BF$ and $CE$ intersect at $K$. Prove that $AK$ is perpendicular to $BC$.

2016 Singapore Open Round 2 P1
Let $D$ be a point in the interior of $\triangle{ABC}$ such that $AB=ab$, $AC=ac$, $BC=bc$, $AD=ad$, $BD=bd$, $CD=cd$. Show that $\angle{ABD}+\angle{ACD}=60^{\circ}$.

2017 Singapore Open Round 2 P

2018 Singapore Open Round 2 P1
Consider a regular cube with side length $2$. Let $A$ and $B$ be $2$ vertices that are furthest apart. construct a sequence of points on the surface of the cube $A_1,A_2,...,A_k$ so that $A_1=A$, $A_k=B$ and for any $i = 1,..,k-1$, the distance from $A_i$ to $A_{i+1}$ is $3$. find the minimum value of $k$.

2018 Singapore Open Round 2 P2
Let $O$ be a point inside triangle $ABC$ such that BOC is $90^\circ$ and $\angle BAO = \angle BCO$.Prove that $\angle OMN$ is $90$ degrees, where $M,N$ are the midpoints of $AC$ and $BC$ respectively.

2019 Singapore Open Round 2 P1
In the acute-angled triangle $ABC$ with circumcircle $\omega$ and orthocenter $H$, points $D$ and $E$ are the feet of the perpendiculars from $A$ onto $BC$ and from $B$ onto $AC$ respecively. Let $P$ be a point on the minor arc $BC$ of $\omega$ . Points $M$ and $N$ are the  feet of the perpendiculars from $P$ onto lines $BC$ and $AC$ respectively. Let $PH$ and $MN$ intersect at $R$. Prove that $\angle DMR=\angle MDR$.

missing data

unknown date, probably 2016
In the triangle $ABC$, $\angle A=90^\circ$, the bisector of $\angle B$ meets the altitude $AD$ at the point $E$, and the bisector of $\angle CAD$ meets the side $CD$ at $F$. The line through $F$ perpendicular to $BC$ intersects $AC$ at $G$. Prove that $B,E,G$ are collinear.

unknown date, at most 2016
Let $ABCD$ be a cyclic quadrilateral with $AB=AD$. Let $M$ and $N$ be on the line segments $BC$ and $CD$ respectively such that $BM+ND=MN$. Let $O$ be the circumcentre of $\triangle{AMN}$. Show that the points $A, O, C$ are collinear.


source: wwwdontmesswith6a.blogspot.com/

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