Korea S. 2nd Round 1994-97, 2004-22 (KMO) 46p

geometry problems from South Korean Mathematical Olympiads (KMO) - Second Round

with aops links in the names

[2 days, 4p per day]

collected inside aops here

1994 - 1997, 2004 - 2022

1994 Korean MO p3

In a triangle $ABC$, $I$ and $O$ are the incenter and circumcenter respectively, $A',B',C'$ the excenters, and $O'$ the circumcenter of $\triangle A'B'C'$. If $R$ and $R'$ are the circumradii of triangles $ABC$ and $A'B'C'$, respectively, prove that:

(i) $R'= 2R$

(ii) $IO' = 2IO$

1995 Korean MO p3

Let $ABC$ be an equilateral triangle of side $1$, $D$ be a point on $BC$, and $r_1, r_2$ be the inradii of triangles $ABD$ and $ADC$. Express $r_1r_2$ in terms of $p = BD$ and find the maximum of $r_1r_2$.

1995 Korean MO p4

Let $O$ and $R$ be the circumcenter and circumradius of a triangle $ABC$, and let $P$ be any point in the plane of the triangle. The perpendiculars $PA_1,PB_1,PC_1$ are drawn from $P$ on $BC,CA,AB$. Express $S_{A_1B_1C_1}/S_{ABC}$ in terms of $R$ and $d = OP$, where $S_{XYZ}$ is the area of $\triangle XYZ$.

1996 Korean MO p4

Circle $C$(the center is $C$.) is inside the $\angle XOY$ and it is tangent to the two sides of the angle. Let $C_1$ be the circle that passes through the center of $C$ and tangent to two sides of angle and let $A$ be one of the endpoint of diameter of $C_1$ that passes through $C$ and $B$ be the intersection of this diameter and circle $C.$ Prove that the cirlce that $A$ is the center and $AB$ is the radius is also tangent to the two sides of $\angle XOY.$

1996 Korean MO p8

Let $\triangle ABC$ be the acute triangle such that $AB\ne AC.$ Let $V$ be the intersection of $BC$ and angle bisector of $\angle A.$ Let $D$ be the foot of altitude from $A$ to $BC.$ Let $E,F$ be the intersection of circumcircle of $\triangle AVD$ and $CA,AB$ respectively. Prove that the lines $AD, BE,CF$ is concurrent.

1997 Korean MO p3

Let $ABCDEF$ be a convex hexagon such that $AB=BC,CD=DE, EF=FA.$

Prove that $\frac{BC}{BE}+\frac{DE}{DA}+\frac{FA}{FC}\ge\frac{3}{2}$ and find when equality holds.

1997 Korean MO p5

Let $a,b,c$ be the side lengths of any triangle $\triangle ABC$ opposite to $A,B$ and $C,$ respectively. Let $x,y,z$ be the length of medians from $A,B$ and $C,$ respectively.

If $T$ is the area of $\triangle ABC$, prove that $\frac{a^2}{x}+\frac{b^2}{y}+\frac{c^2}{z}\ge\sqrt{\sqrt{3}T}$

1997 Korean MO p7

Let $X,Y,Z$ be the points outside the $\triangle ABC$ such that $\angle BAZ=\angle CAY,\angle CBX=\angle ABZ,\angle ACY=\angle BCX.$ Prove that the lines $AX, BY, CZ$ are concurrent.

2004 Korean MO p5
$A, B, C$, and $D$ are the four different points on the circle $O$ in the order. Let the centre of the scribed circle of triangle $ABC$, which is tangent to $BC$, be $O_1$. Let the centre of the scribed circle of triangle $ACD$, which is tangent to $CD$, be $O_2$.
i) Show that the circumcentre of triangle $ABO_1$ is on the circle $O$.
ii) Show that the circumcircle of triangle $CO_1O_2$ always pass through a fixed point on the circle $O$, when $C$ is moving along arc $BD$.

2005 Korean MO p2
For triangle $ABC$, $P$ and $Q$ satisfy $\angle BPA + \angle AQC=90^{\circ}$. It is provided that the vertices of the triangle $BAP$ and $ACQ$ are ordered counterclockwise(or clockwise). Let the intersection of the circumcircles of the two triangles be $N$ ($A \neq N$, however if $A$ is the only intersection $A=N$), and the midpoint of segment $BC$ be $M$. Show that the length of $MN$ does not depend on $P$ and $Q$.

2005 Korean MO p5
Let $P$ be a point that lies outside of circle $O$. A line passes through $P$ and meets the circle at $A$ and $B$, and another line passes through $P$ and meets the circle at $C$ and $D$. The point $A$ is between $P$ and $B$, $C$ is between $P$ and $D$. Let the intersection of segment $AD$ and $BC$ be $L$ and construct $E$ on ray $(PA$ so that $BL \cdot PE = DL \cdot PD$. Show that $M$ is the midpoint of the segment $DE$, where $M$ is the intersection of lines $PL$ and $DE$.

2006 Korean MO p4
On the circle $O,$ six points $A,B,C,D,E,F$ are  on the circle counterclockwise. $BD$ is the diameter of the circle and it is perpendicular to $CF.$ Also, lines $CF, BE, AD$ is concurrent. Let $M$ be the foot of altitude from $B$ to $AC$ and let $N$ be the foot of altitude from $D$ to $CE.$ Prove that the area of $\triangle MNC$ is less than half the area of $\square ACEF.$

2006 Korean MO p7
Points $A,B,C,D,E,F$ is on the circle $O.$ A line $\ell$ is tangent to $O$ at $E$ is parallel to $AC$ and $DE>EF.$ Let $P,Q$ be the intersection of $\ell$ and $BC,CD$ ,respectively and let $R,S$ be the intersection of $\ell$ and $CF,DF$ ,respectively. Show that $PQ=RS$ if and only if $QE=ER.$

2007 Korean MO p2
$A_{1}B_{1}B_{2}A_{2}$ is a convex quadrilateral, and $A_{1}B_{1}\neq A_{2}B_{2}$. Show that there exists a point $M$ such that
$\frac{A_{1}B_{1}}{A_{2}B_{2}}=\frac{MA_{1}}{MA_{2}}=\frac{MB_{1}}{MB_{2}}$

2007 Korean MO p6
$ABC$ is a triangle which is not isosceles. Let the circumcenter and orthocenter of $ABC$ be $O$, $H$, respectively, and the altitudes of $ABC$ be $AD$, $BC$, $CF$. Let $K\neq A$ be the intersection of $AD$ and circumcircle of triangle $ABC$, $L$ be the intersection of $OK$ and $BC$, $M$ be the midpoint of $BC$, $P$ be the intersection of $AM$ and the line that passes $L$ and perpendicular to $BC$, $Q$ be the intersection of $AD$ and the line that passes $P$ and parallel to $MH$, $R$ be the intersection of line $EQ$ and $AB$, $S$ be the intersection of $FD$ and $BE$.  If $OL = KL$, then prove that two lines $OH$ and $RS$ are orthogonal.

2008 Korean MO p3
Points $A,B,C,D,E$ lie in a counterclockwise order on a circle $O$, and $AC = CE$
$P=BD \cap AC$, $Q=BD \cap CE$ . Let $O_1$ be the circle which is tangent to $\overline {AP}, \overline {BP}$ and arc $AB$ (which doesn't contain $C$) . Let $O_2$ be the circle which is tangent $\overline {DQ}, \overline {EQ}$ and arc $DE$ (which doesn't contain $C$) . Let $O_1 \cap O = R, O_2 \cap O = S, RP \cap QS = X$ . Prove that $XC$ bisects $\angle ACE$

2008 Korean MO p6
Let $ABCD$ be inscribed in a circle $\omega$. Let the line parallel to the tangent to $\omega$ at $A$ and passing $D$ meet $\omega$ at $E$. $F$ is a point on $\omega$ such that lies on the different side of $E$ wrt $CD$.  If $AE \cdot AD \cdot CF = BE \cdot BC \cdot DF$ and $\angle CFD = 2\angle AFB$, Show that the tangent to $\omega$ at $A, B$ and line $EF$ concur at one point. ($A$ and $E$ lies on the same side of $CD$)

2009 Korean MO p1
Let $I, O$ be the incenter and the circumcenter of triangle $ABC$, and $D,E,F$ be the circumcenters of triangle $BIC, CIA, AIB$. Let $P, Q, R$ be the midpoints of segments $DI, EI, FI$. Prove that the circumcenter of triangle $PQR$, $M$, is the midpoint of segment $IO$.

2009 Korean MO p6
Let $ABC$ be a triangle and $P, Q ( \ne A, B, C )$ are the points lying on segments $BC , CA$. Let $I, J, K$ be the incenters of triangle $ABP, APQ, CPQ$. Prove that $PIJK$ is a convex quadrilateral.

2010 Korean MO p3
Let $I$ be the incenter of triangle $ABC$. The incircle touches $BC, CA, AB$ at points $P, Q, R$. A circle passing through $B , C$ is tangent to the circle $I$ at point $X$, a circle passing through $C , A$ is tangent to the circle $I$ at point $Y$, and a circle passing through $A , B$ is tangent to the circle $I$ at point $Z$, respectively. Prove that three lines $PX, QY, RZ$ are concurrent.

2010 Korean MO p6
Let $ABCD$ be a cyclic convex quadrilateral. Let $E$ be the intersection of lines $AB, CD$. $P$ is the intersection of line passing $B$ and perpendicular to $AC$, and line passing $C$ and perpendicular to $BD$. $Q$ is the intersection of line passing $D$ and perpendicular to $AC$, and line passing $A$ and perpendicular to $BD$. Prove that three points $E, P, Q$ are collinear.

2011 Korean MO p1
Two circles $O, O'$ having same radius meet at two points, $A,B (A \not = B)$. Point $P,Q$ are each on circle $O$ and $O'$ $(P \not = A,B ~ Q\not = A,B )$. Select the point $R$ such that $PAQR$ is a parallelogram. Assume that $B, R, P, Q$ is cyclic. Now prove that $PQ = OO'$.

2011 Korean MO p6
Let $ABC$ be a triangle and its incircle meets $BC, AC, AB$ at $D, E$ and $F$ respectively. Let point $P$ on the incircle and inside $\triangle AEF$. Let $X=PB \cap DF , Y=PC \cap DE, Q=EX \cap FY$. Prove that the points $A$ and $Q$ lies on $DP$ simultaneously or located opposite sides from $DP$.

2012 Korean MO p1
Let $ABC$ be an obtuse triangle with $\angle A > 90^{\circ}$. Let circle $O$ be the circumcircle of $ABC$. $D$ is a point lying on segment $AB$ such that $AD = AC$. Let $AK$ be the diameter of circle $O$. Two lines $AK$ and $CD$ meet at $L$. A circle passing through $D, K, L$ meets with circle $O$ at $P ( \ne K )$ . Given that $AK = 2, \angle BCD = \angle BAP = 10^{\circ}$, prove that $DP = \sin ( \frac{ \angle A}{2} )$.

2012 Korean MO p6
Let $w$ be the incircle of triangle $ABC$. Segments $BC, CA$ meet with $w$ at points $D, E$. A line passing through $B$ and parallel to $DE$ meets $w$ at $F$ and $G$. ($F$ is nearer to $B$ than $G$.) Line $CG$ meets $w$ at $H ( \ne G )$. A line passing through $G$ and parallel to $EH$ meets with line $AC$ at $I$. Line $IF$ meets with circle $w$ at $J (\ne F )$. Lines $CJ$ and $EG$ meets at $K$. Let $l$ be the line passing through $K$ and parallel to $JD$. Prove that $l, IF, ED$ meet at one point.

2013 Korean MO p1
Let $P$ be a point on segment $BC$. $Q, R$ are points on $AC, AB$ such that $PQ \parallel AB$ and $PR \parallel AC$. $O, O_{1}, O_{2}$ are the circumcenters of triangle $ABC, BPR, PCQ$. The circumcircles of $BPR, PCQ$ meet at point $K (\ne P)$. Prove that $OO_{1} = KO_{2}$.

2013 Korean MO p6
Let $O$ be circumcenter of triangle $ABC$. For a point $P$ on segmet $BC$, the circle passing through $P, B$ and tangent to line $AB$ and the circle passing through $P, C$ and tangent to line $AC$ meet at point $Q ( \ne P )$. Let $D, E$ be foot of perpendicular from $Q$ to $AB, AC$. ($D \ne B, E \ne C$) Two lines $DE$ and $BC$ meet at point $R$. Prove that $O, P, Q$ are collinear if and only if $A, R, Q$ are collinear.

2014 Korean MO p3
$AB$ is a chord of $O$ and $AB$ is not a diameter of $O$. The tangent lines to $O$ at $A$ and $B$ meet at $C$. Let $M$ and $N$ be the midpoint of the segments $AC$ and $BC$, respectively. A circle passing through $C$ and tangent to $O$ meets line $MN$ at $P$ and $Q$. Prove that $\angle PCQ = \angle CAB$.

2014 Korean MO p5
There is a convex quadrilateral $ABCD$ which satisfies $\angle A=\angle D$. Let the midpoints of $AB, AD, CD$ be $L,M,N$. Let's say the intersection point of $AC, BD$ be $E$ . Let's say point $F$ which lies on $\overrightarrow{ME}$ satisfies $\overline{ME}\times \overline{MF}=\overline{MA}^{2}$. Prove that $\angle LFM=\angle MFN$.

2015 Korean MO p2
Let the circumcircle of $\triangle ABC$ be $\omega$. A point $D$ lies on segment $BC$, and $E$ lies on segment $AD$. Let ray $AD \cap \omega = F$. A point $M$, which lies on $\omega$, bisects $AF$ and it is on the other side of $C$ with respect to $AF$. Ray $ME \cap \omega = G$, ray $GD \cap \omega = H$, and $MH \cap AD = K$. Prove that $B, E, C, K$ are cyclic.

2015 Korean MO p6
An isosceles trapezoid $ABCD$, inscribed in $\omega$, satisfies $AB=CD, AD<BC, AD<CD$.
A circle with center $D$ and passing $A$ hits $BD, CD, \omega$ at $E, F, P(\not= A)$, respectively.
Let $AP \cap EF = Q$, and $\omega$ meet $CQ$ and the circumcircle of $\triangle BEQ$ at $R(\not= C), S(\not= B)$, respectively.  Prove that $\angle BER= \angle FSC$.

2016 Korean MO p2
A non-isosceles triangle $\triangle ABC$ has its incircle tangent to $BC, CA, AB$ at points $D, E, F$.  Let the incenter be $I$. Say $AD$ hits the incircle again at $G$, at let the tangent to the incircle at $G$ hit $AC$ at $H$. Let $IH \cap AD = K$, and let the foot of the perpendicular from $I$ to $AD$ be $L$. Prove that $IE \cdot IK= IC \cdot IL$.

2016 Korean MO p3

Acute triangle $\triangle ABC$ has area $S$ and perimeter $L$. A point $P$ inside $\triangle ABC$ has $dist(P,BC)=1, dist(P,CA)=1.5, dist(P,AB)=2$. Let $BC \cap AP = D$, $CA \cap BP = E$, $AB \cap CP= F$. Let $T$ be the area of $\triangle DEF$. Prove the following inequality.

$$\left( \frac{AD \cdot BE \cdot CF}{T} \right)^2 > 4L^2 + \left( \frac{AB \cdot BC \cdot CA}{24S} \right)^2$$

2016 Korean MO p5
A non-isosceles triangle $\triangle ABC$ has incenter $I$ and the incircle hits $BC, CA, AB$ at $D, E, F$. Let $EF$ hit the circumcircle of $CEI$ at $P \not= E$. Prove that $\triangle ABC = 2 \triangle ABP$.

2017 Korean MO p3
Let there be a scalene triangle $ABC$, and its incircle hits $BC, CA, AB$ at $D, E, F$. The perpendicular bisector of $BC$ meets the circumcircle of $ABC$ at $P, Q$, where $P$ is on the same side with $A$ with respect to $BC$. Let the line parallel to $AQ$ and passing through $D$ meet $EF$ at $R$. Prove that the intersection between $EF$ and $PQ$ lies on the circumcircle of $BCR$.

2017 Korean MO p6
In a quadrilateral $ABCD$, we have $\angle ACB = \angle ADB = 90$ and $CD < BC$. Denote $E$ as the intersection of $AC$ and $BD$, and let the perpendicular bisector of $BD$ hit $BC$ at $F$. The circle with center $F$ which passes through $B$ hits $AB$ at $P (\neq B)$ and $AC$ at $Q$. Let $M$ be the midpoint of $EP$. Prove that the circumcircle of $EPQ$ is tangent to $AB$ if and only if $B, M, Q$ are colinear.

2018 Korean MO p1
Let there be an acute triangle $\triangle ABC$ with incenter $I$. $E$ is the foot of the perpendicular from $I$ to $AC$. The line which passes through $A$ and is perpendicular to $BI$ hits line $CI$ at $K$. The line which passes through $A$ and is perpendicular to $CI$ hits the line which passes through $C$ and is perpendicular to $BI$ at $L$. Prove that $E, K, L$ are colinear.

2018 Korean MO p5

Let there be a convex quadrilateral $ABCD$. The angle bisector of $\angle A$ meets the angle bisector of $\angle B$, the angle bisector of $\angle D$ at $P, Q$ respectively. The angle bisector of $\angle C$ meets the angle bisector of $\angle D$, the angle bisector of $\angle B$ at $R, S$ respectively. $P, Q, R, S$ are all distinct points. $PR$ and $QS$ meets perpendicularly at point $Z$. Denote $l_A, l_B, l_C, l_D$ as the exterior angle bisectors of $\angle A, \angle B, \angle C, \angle D$. Denote $E = l_A \cap l_B$, $F= l_B \cap l_C$, $G = l_C \cap l_D$, and $H= l_D \cap l_A$. Let $K, L, M, N$ be the midpoints of $FG, GH, HE, EF$ respectively.

Prove that the area of quadrilateral $KLMN$ is equal to $ZM \cdot ZK + ZL \cdot ZN$

2019 Korean MO p2

Triangle $ABC$ is an acute triangle with distinct sides. Let $I$ the incenter, $\Omega$ the circumcircle, $E$ the $A$-excenter of triangle $ABC$. Let $\Gamma$ the circle centered at $E$ and passes $A$. $\Gamma$ and $\Omega$ intersect at point $D(\neq A)$, and the perpendicular line of $BC$ which passes $A$ meets $\Gamma$ at point $K(\neq A)$. $L$ is the perpendicular foot from $I$ to $AC$. Now if $AE$ and $DK$ intersects at $F$, prove that $BE\cdot CI=2\cdot CF\cdot CL$.

2019 Korean MO p6

In acute triangle $ABC$, $AB>AC$. Let $I$ the incenter, $\Omega$ the circumcircle of triangle $ABC$, and $D$ the foot of perpendicular from $A$ to $BC$. $AI$ intersects $\Omega$ at point $M(\neq A)$, and the line which passes $M$ and perpendicular to $AM$ intersects $AD$ at point $E$. Now let $F$ the foot of perpendicular from $I$ to $AD$.
Prove that $ID\cdot AM=IE\cdot AF$.

2020 Korean MO p2

$H$ is the orthocenter of an acute triangle $ABC$, and let $M$ be the midpoint of $BC$. Suppose $(AH)$ meets $AB$ and $AC$ at $D,E$ respectively. $AH$ meets $DE$ at $P$, and the line through $H$ perpendicular to $AH$ meets $DM$ at $Q$. Prove that $P,Q,B$ are collinear.

2020 Korean MO p6

Let $ABCDE$ be a convex pentagon such that quadrilateral $ABDE$ is a parallelogram and quadrilateral $BCDE$ is inscribed in a circle. The circle with center $C$ and radius $CD$ intersects the line $BD, DE$ at points $F, G(\neq D)$, and points $A, F, G$ is on line l. Let $H$ be the intersection point of line $l$ and segment $BC$.

Consider the set of circle $\Omega$ satisfying the following condition:

Circle $\Omega$ passes through $A, H$ and intersects the sides $AB, AE$ at point other than $A$.

Let $P, Q(\neq A)$ be the intersection point of circle $\Omega$ and sides $AB, AE$.Prove that $AP+AQ$ is constant.

2021 Korean MO p1

Let $ABC$ be an acute triangle and $D$ be an intersection of the angle bisector of $A$ and side $BC$. Let $\Omega$ be a circle tangent to the circumcircle of triangle $ABC$ and side $BC$ at $A$ and $D$, respectively. $\Omega$ meets the sides $AB, AC$ again at $E, F$, respectively. The perpendicular line to $AD$, passing through $E, F$ meets $\Omega$ again at $G, H$, respectively. Suppose that $AE$ and $GD$ meet at $P$, $EH$ and $GF$ meet at $Q$, and $HD$ and $AF$ meet at $R$. Prove that $\dfrac{\overline{QF}}{\overline{QG}}=\dfrac{\overline{HR}}{\overline{PG}}$.

2021 Korean MO p6

Let $ABC$ be an obtuse triangle with $\angle A > \angle B > \angle C$, and let $M$ be a midpoint of the side $BC$. Let $D$ be a point on the arc $AB$ of the circumcircle of triangle $ABC$ not containing $C$. Suppose that the circle tangent to $BD$ at $D$ and passing through $A$ meets the circumcircle of triangle $ABM$ again at $E$ and $\overline{BD}=\overline{BE}$. $\omega$, the circumcircle of triangle $ADE$, meets $EM$ again at $F$.

Prove that lines $BD$ and $AE$ meet on the line tangent to $\omega$ at $F$.

2022 Korean MO p2

In a scalene triangle $ABC$, let the angle bisector of $A$ meets side $BC$ at $D$. Let $E, F$ be the circumcenter of the triangles $ABD$ and $ADC$, respectively. Suppose that the circumcircles of the triangles $BDE$ and $DCF$ intersect at $P(\neq D)$, and denote by $O, X, Y$ the circumcenters of the triangles $ABC, BDE, DCF$, respectively. Prove that $OP$ and $XY$ are parallel.

2022 Korean MO p5

For a scalene triangle $ABC$ with an incenter $I$, let its incircle meets the sides $BC, CA, AB$ at $D, E, F$, respectively. Denote by $P$ the intersection of the lines $AI$ and $DF$, and $Q$ the intersection of the lines $BI$ and $EF$. Prove that $\overline{PQ}=\overline{CD}$