geometry problems from South Korean Junior Mathematical Olympiads (KJMO)
with aops links in the names
with aops links in the names
collected inside aops here
2005 - 2026
For triangle $ABC, P$ and $Q$ satisfy $\angle BPA + \angle AQC = 90^o$. 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 $N$ ($A \ne N$, 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 KJMO P5
n $\triangle ABC$, let the bisector of $\angle BAC$ hit the circumcircle at $M$. Let $P$ be the intersection of $CM$ and $AB$. Denote by $(V,WX,YZ)$ the intersection of the line passing $V$ perpendicular to $WX$ with the line $YZ$. Prove that the points $(P,AM,AC), (P,AC,AM), (P,BC,MB)$ are collinear.
In a circle $O$, there are six points, $A,B,C,D,E, F$ in a counterclockwise order. $BD \perp CF$, and $CF,BE,AD$ are concurrent. Let the perpendicular from $B$ to $AC$ be $M$, and the perpendicular from $D$ to $CE$ be $N$. Prove that $AE // MN$.
2006 KJMO P7
A line through point $P$ outside of circle $O$ meets the said circle at $B,C$ ($PB < PC$). Let $PO$ meet circle $O$ at $Q,D$ (with $PQ < PD$). Let the line passing $Q$ and perpendicular to $BC$ meet circle $O$ at $A$. If $BD^2 = AD\cdot CP$, prove that $PA$ is a tangent to $O$.
2007 KJMO P4
Let $P$ be a point inside $\triangle ABC$. Let the perpendicular bisectors of $PA,PB,PC$ be $\ell_1,\ell_2,\ell_3$. Let $D =\ell_1 \cap \ell_2 , E=\ell_2 \cap \ell_3, F=\ell_3 \cap \ell_1$. If $A,B,C,D,E,F$ lie on a circle, prove that $C, P,D$ are collinear.
2007 KJMO P7
Let the incircle of $\triangle ABC$ meet $BC,CA,AB$ at $J,K,L$. Let $D(\ne B, J),E(\ne C,K), F(\ne A,L)$ be points on $BJ,CK,AL$. If the incenter of $\triangle ABC$ is the circumcenter of $\triangle DEF$ and $\angle BAC = \angle DEF$, prove that $\triangle ABC$ and $\triangle DEF$ are isosceles triangles.
In a $\triangle XYZ$, points $A,B$ lie on segment $ZX, C,D$ lie on segment $XY , E, F$ lie on segment $YZ$. $A, B, C, D$ lie on a circle, and $\frac{AZ \cdot EY \cdot ZB \cdot Y F}{EZ \cdot CY \cdot ZF \cdot Y D}= 1$ . Let $L = ZX \cap DE,M = XY \cap AF,N = Y Z \cap BC$.
Prove that $L,M,N$ are collinear.
Let there be a pentagon $ABCDE$ inscribed in a circle $O$. The tangent to $O$ at $E$ is parallel to $AD$. A point $F$ lies on $O$ and it is in the opposite side of $A$ with respect to $CD$, and satisfies $AB \cdot BC \cdot DF = AE \cdot ED \cdot CF$ and $\angle CFD = 2\angle BFE$. Prove that the tangent to $O$ at $B,E$ and line $AF$ concur at one point.
2009 KJMO P2
In an acute triangle $\triangle ABC$, let $A',B',C'$ be the reflection of $A,B,C$ with respect to $BC,CA,AB$. Let $D = B'C \cap BC',E = CA' \cap C'A,F = A'B \cap AB'$. Prove that $AD,BE,CF$ are concurrent
2009 KJMO P5
Acute triangle $\triangle ABC$ satises $AB < AC$. Let the circumcircle of this triangle be $O$, and the midpoint of $BC,CA,AB$ be $D,E,F$. Let $P$ be the intersection of the circle with $AB$ as its diameter and line $DF$, which is in the same side of $C$ with respect to $AB$. Let $Q$ be the intersection of the circle with $AC$ as its diameter and the line $DE$, which is in the same side of $B$ with respect to $AC$. Let $PQ \cap BC = R$, and let the line passing through $R$ and perpendicular to $BC$ meet $AO$ at $X$. Prove that $AX = XR$.
2010 KJMO P3
In an acute triangle $\triangle ABC$, let there be point $D$ on segment $AC, E$ on segment $AB$ such that $\angle ADE = \angle ABC$. Let the bisector of $\angle A$ hit $BC$ at $K$. Let the foot of the perpendicular from $K$ to $DE$ be $P$, and the foot of the perpendicular from $A$ to $DE$ be $L$. Let $Q$ be the midpoint of $AL$. If the incenter of $\triangle ABC$ lies on the circumcircle of $\triangle ADE$, prove that $P,Q$ and the incenter of $\triangle ADE$ are collinear.
2010 KJMO P7
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 KJMO P2
Let $ABCD$ be a cyclic quadrilateral inscirbed in circle $O$. Let the tangent to $O$ at $A$ meet $BC$ at $S$, and the tangent to $O$ at $B$ meet $CD$ at $T$. Circle with $S$ as its center and passing $A$ meets $BC$ at $E$, and $AE$ meets $O$ again at $F(\ne A)$. The circle with $T$ as its center and passing $B$ meets $CD$ at $K$. Let $P = BK \cap AC$. Prove that $P,F,D$ are collinear if and only if $AB = AP$.
2011 KJMO P5
In triangle $ABC$, ($AB \ne AC$), let the orthocenter be $H$, circumcenter be $O$, and the midpoint of $BC$ be $M$. Let $HM \cap AO = D$. Let $P,Q,R,S$ be the midpoints of $AB,CD,AC,BD$. Let $X = PQ\cap RS$. Find $AH/OX$.
2012 KJMO P2
A pentagon $ABCDE$ is inscribed in a circle $O$, and satis fies $\angle A = 90^o, AB = CD$. Let $F$ be a point on segment $AE$. Let $BF$ hit $O$ again at $J(\ne B), CE \cap DJ = K, BD\cap FK = L$. Prove that $B,L,E,F$ are cyclic.
2012 KJMO P5
Let $ABCD$ be a cyclic quadrilateral inscirbed in a circle $O$ ($AB> AD$), and let $E$ be a point on segment $AB$ such that $AE = AD$. Let $AC \cap DE = F$, and $DE \cap O = K(\ne D)$. The tangent to the circle passing through $C,F,E$ at $E$ hits $AK$ at $L$. Prove that $AL = AD$ if and only if $\angle KCE = \angle ALE$.
2013 KJMO P2
A pentagon $ABCDE$ is inscribed in a circle $O$, and satises $AB = BC , AE = DE$. The circle that is tangent to $DE$ at $E$ and passing $A$ hits $EC$ at $F$ and $BF$ at $G (\ne F)$. Let $DG\cap O = H (\ne D)$. Prove that the tangent to $O$ at $E$ is perpendicular to $HA$.
In an acute triangle $\triangle ABC, \angle A > \angle B$. Let the midpoint of $AB$ be $D$, and let the foot of the perpendicular from $A$ to $BC$ be $E$, and $B$ from $CA$ be $F$. Let the circumcenter of $\triangle DEF$ be $O$. A point $J$ on segment $BE$ satisfies $\angle ODC = \angle EAJ$. Prove that $AJ \cap DC$ lies on the circumcircle of $\triangle BDE$.
2014 KJMO P1
In a triangle $\triangle ABC$ with incenter $I$. Let $D$ = $AI$ $\cap$ $BC$ ,$E$ = incenter of $\triangle ABD$, $F$ = incenter of $\triangle ACD$, $P$ = intersection of $\odot BCE$ and $\overline {ED}$, $Q$ = intersection of $\odot BCF$ and $\overline {FD}$, $M$ = midpoint of $\overline {BC}$ . Prove that $D, M, P, Q$ concycle.
2014 KJMO P7
In a parallelogram $ABCD$ $(AB < BC)$ . The incircle of $\triangle ABC$ meets $\overline {BC}$ and $\overline {CA}$ at $P, Q$. The incircle of $\triangle ACD$ and $\overline {CD}$ meets at $R$. Let $S$ = $PQ$ $\cap$ $AD$. $U$ = $AR$ $\cap$ $CS$. $T$, a point on $\overline {BC}$ such that $\overline {AB} = \overline {BT}$ . Prove that $AT, BU, PQ$ are concurrent
2015 KJMO P1
In an acute, scalene triangle $\triangle ABC$, let $O$ be the circumcenter. Let $M$ be the midpoint of $AC$. Let the perpendicular from $A$ to $BC$ be $D$. Let the circumcircle of $\triangle OAM$ hit $DM$ at $P(\not= M)$. Prove that $B, O, P$ are colinear.
2015 KJMO P5
Let $I$ be the incenter of an acute triangle $\triangle ABC$, and let the incircle be $\Gamma$.
Let the circumcircle of $\triangle IBC$ hit $\Gamma$ at $D, E$, where $D$ is closer to $B$ and $E$ is closer to $C$. Let $\Gamma \cap BE = K (\not= E)$, $CD \cap BI = T$, and $CD \cap \Gamma = L (\not= D)$. Let the line passing $T$ and perpendicular to $BI$ meet $\Gamma$ at $P$, where $P$ is inside $\triangle IBC$. Prove that the tangent to $\Gamma$ at $P$, $KL$, $BI$ are concurrent.
2016 KJMO P2 (also KMO)
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 KJMO P6
Circle $O_1$ is tangent to $AC$, $BC$(side of triangle $ABC$) at point $D, E$. Circle $O_2$ include $O_1$, is tangent to $BC$, $AB$(side of triangle $ABC$) at point $E, F$ . The tangent of $O_2$ at $P(DE \cap O_2, P \neq E)$ meets $AB$ at $Q$. A line passing through $O_1$(center of $O_1$) and parallel to $BO_2$($O_2$ is also center of $O_2$) meets $BC$ at $G$, $EQ \cap AC=K, KG \cap EF=L$, $EO_2$ meets circle $O_2$ at $N(\neq E)$, $LO_2 \cap FN=M$.
IF $N$ is a middle point of $FM$, prove that $BG=2EG$
2017 KJMO P2 (also KMO)
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 $BCR$.
Let triangle $ABC$ be an acute scalene triangle, and denote $D,E,F$ by the midpoints of $BC,CA,AB$, respectively. Let the circumcircle of $DEF$ be $O_1$, and its center be $N$. Let the circumcircle of $BCN$ be $O_2$. $O_1$ and $O_2$ meet at two points $P, Q$. $O_2$ meets $AB$ at point $K(\neq B)$ and meets $AC$ at point $L(\neq C)$. Show that the three lines $EF,PQ,KL$ are concurrent.
2018 KJMO P3
Let there be a scalene triangle $ABC$, and denote $M$ by the midpoint of $BC$. The perpendicular bisector of $BC$ meets the circumcircle of $ABC$ at point $P$, on the same side with $A$ with respect to $BC$. Let the incenters of $ABM$ and $AMC$ be $I,J$, respectively. Let $\angle BAC=\alpha$, $\angle ABC=\beta$, $\angle BCA=\gamma$. Find $\angle IPJ$.
2018 KJMO P5
Let there be an acute scalene triangle $ABC$ with circumcenter $O$. Denote $D,E$ be the reflection of $O$ with respect to $AB,AC$, respectively. The circumcircle of $ADE$ meets $AB$, $AC$, the circumcircle of $ABC$ at points $K,L,M$, respectively, and they are all distinct from $A$. Prove that the lines $BC,KL,AM$ are concurrent.
In an acute triangle $ABC$, point $D$ is on the segment $AC$ such that $\overline{AD}=\overline{BC}$
and $\overline{AC}^2-\overline{AD}^2=\overline{AC}\cdot\overline{AD}$. The line that is parallel to
the bisector of $\angle{ACB}$ and passes the point $D$ meets the segment $AB$ at point $E$. Prove,
if $\overline{AE}=\overline{CD}$, $\angle{ADB}=3\angle{BAC}$.
2019 KJMO P7
Let $O$ be the outer center of an acute triangle $ABC$. Let $D$ be the intersection of the bisector of the
angle $A$ and $BC$. Suppose that $\angle ODC = 2 \angle DAO$. The circumcircle of $ABD$ meets
the line segment $OA$ and the line $OD$ at $E (\neq A,O)$, and $F(\neq D)$, respectively. Let $X$ be
the intersection of the line $DE$ and the line segment $AC$. Let $Y$ be the intersection of the bisector
of the angle $BAF$ and the segment $BE$. Prove that $\frac{\overline{AY}}{\overline{BY}}= \frac{\overline{EX}}{\overline{EO}}$.
2020 KJMO P2
Let $ABC$ be an acute triangle with circumcircle $\Omega$ and $\overline{AB} < \overline{AC}$.
The angle bisector of $A$ meets $\Omega$ again at $D$, and the line through $D$, perpendicular to
$BC$ meets $\Omega$ again at $E$. The circle centered at $A$, passing through $E$ meets the line
$DE$ again at $F$. Let $K$ be the circumcircle of triangle $ADF$. Prove that $AK$ is perpendicular
to $BC$.
2020 KJMO P4
In an acute triangle $ABC$ with $\overline{AB} > \overline{AC}$, let $D, E, F$ be the feet of the
altitudes from $A, B, C$, respectively. Let $P$ be an intersection of lines $EF$ and $BC$, and let $Q$
be a point on the segment $BD$ such that $\angle QFD = \angle EPC$. Let $O, H$ denote the
circumcenter and the orthocenter of triangle $ABC$, respectively. Suppose that $OH$ is perpendicular
to $AQ$. Prove that $P, O, H$ are collinear.
Let $ABCD$ be a cyclic quadrilateral with circumcircle $\Omega$ and let diagonals $AC$ and $BD$
intersect at $X$. Suppose that $AEFB$ is inscribed in a circumcircle of triangle $ABX$ such that $EF$
and $AB$ are parallel. $FX$ meets the circumcircle of triangle $CDX$ again at $G$. Let $EX$ meets
$AB$ at $P$, and $XG$ meets $CD$ at $Q$. Denote by $S$ the intersection of the perpendicular
bisector of $\overline{EG}$ and $\Omega$ such that $S$ is closer to $A$ than $B$. Prove that line
through $S$ parallel to $PQ$ is tangent to $\Omega$.
2021 KJMO P4
In an acute triangle $ABC$ with $\overline{AB} < \overline{AC}$, angle bisector of $A$ and
perpendicular bisector of $\overline{BC}$ intersect at $D$. Let $P$ be an interior point of triangle
$ABC$. Line $CP$ meets the circumcircle of triangle $ABP$ again at $K$. Prove that $B, D, K$ are
collinear if and only if $AD$ and $BC$ meet on the circumcircle of triangle $APC$.
2022 KJMO P1
The inscribed circle of an acute triangle $ABC$ meets the segments $AB$ and $BC$ at $D$ and $E$
respectively. Let $I$ be the incenter of the triangle $ABC$. Prove that the intersection of the line $AI$and $DE$ is on the circle whose diameter is $AC$(passing through A, C).
2022 KJMO P6Let $ABC$ be a isosceles triangle with $\overline{AB}=\overline{AC}$. Let $D(\neq A, C)$ be a point on the side $AC$, and circle $\Omega$ is tangent to $BD$ at point $E$, and $AC$ at point $C$.
Denote by $F(\neq E)$ the intersection of the line $AE$ and the circle $\Omega$, and $G(\neq a)$ the
intersection of the line $AC$ and the circumcircle of the triangle $ABF$. Prove that points $D, E, F,$
and $G$ are concyclic.
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