Korea Final 1993 - 2019 (FKMO) 47p

geometry problems from South Korean Mathematical Olympiads  (KMO) - Final Round
(usually mentioned as FKMO)
with aops links in the names

1993 - 2019

Let be given a triangle $ABC$ with $BC = a, CA = b, AB = c$. Find point $P$ in the plane for which $aAP^{2}+bBP^{2}+cCP^{2}$ is minimum, and compute this minimum.

Consider a triangle $ABC$ with $BC = a, CA = b, AB = c.$ Let $D$ be the midpoint of $BC$ and $E$ be the intersection of the bisector of $A$ with $BC$ . The circle through $A, D, E$ meets $AC, AB$ again at $F, G$ respectively. Let $H\not = B$ be a point on $AB$ with $BG = GH$ . Prove that triangles $EBH$ and $ABC$ are similar and ﬁnd the ratio of their areas.

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) $IO' = 2IO$

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 $r_1r_2$.

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 dropped from $P$ to $BC,CA,AB$. Express $S_{A_1B_1C_1}/S_{ABC}$ in terms of $R$ and $d = OP$, where $S_{XYZ}$ is the area of $\triangle XYZ$.

Let $\ell$ be a line having no common points with a triangle $ABC$. Let $L,M,N$ be  the projections of $A,B,C$ onto $\ell$, and let $LX,MY,NZ$ be the perpendiculars from $L,M,N$ to $BC,CA,AB$, respectively. Prove that these three perpendiculars are concurrent.

Two radii $OA$ and $OB$ of a unit circle form an angle $\alpha$, $0 <\alpha <\pi /2$. Let $P$ be an arbitrary point on the arc $AB$. A ray of light from $P$ is reflected from the segments $OB,OA$ and the arc $AB$ so that it moves along the sides of a fixed triangle $PQR$, with $Q$ on $OB$ and R on $OA$. Prove that the perimeter of $\triangle PQR$ does not depend on $P$, and find it.

1997 FKMO problem 2
The incircle of a triangle $A_1A_2A_3$ is centered at $O$ and meets the segment $OA_j$ at $B_j$ , $j =1, 2, 3$. A circle with center $B_j$ is tangent to the two sides of the triangle having $A_j$ as an endpoint and intersects the segment $OB_j$ at $C_j$. Prove that
$\frac{OC_1 + OC_2 + OC_3}{A_1A_2 + A_2A_3 + A_3A_1} \leq \frac{1}{4\sqrt{3}}$ and find the conditions for equality.

1998 FKMO problem 2
Let $D$,$E$,$F$ be points on the sides $BC$,$CA$,$AB$ respectively of a triangle $ABC$. Lines $AD$,$BE$,$CF$ intersect the circumcircle of $ABC$ again at $P$,$Q$,$R$, respectively.Show that:  $\frac{AD}{PD}+\frac{BE}{QE}+\frac{CF}{RF}\geq 9$ and find the cases of equality.

1998 FKMO problem 5
Let $I$ be the incenter of triangle $ABC$, $O_1$ a circle through $B$ tangent to $CI$, and $O_2$ a circle through $C$ tangent to $BI$. Prove that $O_1$,$O_2$ and the circumcircle of $ABC$ have a common point.

1999 FKMO problem 1
We are given two triangles. Prove, that if $\angle{C}=\angle{C'}$ and $\frac{R}{r}=\frac{R'}{r'}$, then they are similar.

2000 FKMO problem 3
A rectangle $ABCD$ is inscribed in a circle with centre $O$. The exterior bisectors of $\angle ABD$ and $\angle ADB$ intersect at $P$; those of $\angle DAB$ and $\angle DBA$ intersect at $Q$; those of $\angle ACD$ and $\angle ADC$ intersect at $R$; and those of $\angle DAC$ and $\angle DCA$ intersect at $S$. Prove that $P,Q,R$, and $S$ are concyclic.

2001 FKMO problem 2
Let $P$ be a given point inside a convex quadrilateral $O_1O_2O_3O_4$. For each $i = 1,2,3,4$, consider the lines $l$ that pass through $P$ and meet the rays $O_iO_{i-1}$ and $O_iO_{i+1}$ (where $O_0 = O_4$ and $O_5 = O_1$) at distinct points $A_i(l)$ and $B_i(l)$, respectively. Denote $f_i(l) = PA_i(l) \cdot PB_i(l)$. Among all such lines $l$, let $l_i$ be the one that minimizes $f_i$. Show that if $l_1 = l_3$ and $l_2 = l_4$, then the quadrilateral $O_1O_2O_3O_4$ is a parallelogram.

2001 FKMO problem 5
In a triangle $ABC$ with $\angle B < 45^{\circ}$, $D$ is a point on $BC$ such that the incenter of $\triangle ABD$ coincides with the circumcenter $O$ of $\triangle ABC$. Let $P$ be the intersection point of the tangent lines to the circumcircle $\omega$ of $\triangle AOC$ at points $A$ and $C$. The lines $AD$ and $CO$ meet at $Q$. The tangent to $\omega$ at $O$ meets $PQ$ at $X$. Prove that $XO=XD$.

2002 FKMO problem 5
Let $ABC$ be an acute triangle and let $\omega$ be its circumcircle. Let the perpendicular line from $A$ to $BC$ meet $\omega$ at $D$. Let $P$ be a point on $\omega$, and let $Q$ be the foot of the perpendicular line from $P$ to the line $AB$. Prove that if $Q$ is on the outside of $\omega$ and $2\angle QPB = \angle PBC$, then $D,P,Q$ are collinear.

2003 FKMO problem 2
Let $M$ be the intersection of two diagonal, $AC$ and $BD$, of a rhombus $ABCD$, where angle $A<90^\circ$. Construct $O$ on segment $MC$ so that $OB<OC$ and let $t=\frac{MA}{MO}$, provided that $O \neq M$. Construct a circle that has $O$ as centre and goes through $B$ and $D$. Let the intersections between the circle and $AB$ be $B$ and $X$. Let the intersections between the circle and $BC$ be $B$ and $Y$. Let the intersections of $AC$ with $DX$ and $DY$ be $P$ and $Q$, respectively. Express $\frac{OQ}{OP}$ in terms of $t$.

2003 FKMO problem 4
Let $P$, $Q$, and $R$ be the points where the incircle of a triangle $ABC$ touches the sides $AB$, $BC$, and $CA$, respectively. Prove the inequality $\frac{BC} {PQ} + \frac{CA} {QR} + \frac{AB} {RP} \geq 6$

2004 FKMO problem 1
An isosceles triangle with $AB=AC$ has an inscribed circle $O$, which touches its sides $BC,CA,AB$ at $K,L,M$ respectively. The lines $OL$ and $KM$ intersect at $N$; the lines $BN$ and $CA$ intersect at $Q$. Let $P$ be the foot of the perpendicular from $A$ on $BQ$. Suppose that $BP=AP+2\cdot PQ$. Then, what values can the ratio $\frac{AB}{BC}$ assume?

2004 FKMO problem 5
An acute triangle $ABC$ has circumradius $R$, inradius $r$. $A$ is the biggest angle among $A,B,C$. Let $M$ be the midpoint of $BC$, and $X$ be the intersection of two lines  that touches circumcircle of $ABC$ and goes through $B,C$ respectively. Prove the following inequality : $\frac{r}{R} \geq \frac{AM}{AX}$.

2005 FKMO problem 3
In a trapezoid $ABCD$ with $AD \parallel BC , O_{1}, O_{2}, O_{3}, O_{4}$ denote the circles with diameters $AB, BC, CD, DA$, respectively. Show that there exists a circle with center inside the trapezoid which is tangent to all the four circles $O_{1},..., O_{4}$ if and only if $ABCD$ is a parallelogram.

2005 FKMO problem 4
In the following, the point of intersection of two lines $g$ and $h$ will be abbreviated as $g\cap h$.
Suppose $ABC$ is a triangle in which $\angle A =90^{\circ}$ and $\angle B > \angle C$. Let $O$ be the circumcircle of the triangle $ABC$. Let $l_{A}$ and $l_{B}$ be the tangents to the circle $O$ at $A$ and $B$, respectively.
Let $BC \cap l_{A} =S$ and $AC \cap l_{B} = D$. Furthermore, let $AB \cap DS =E$, and let $CE \cap l_{A}=T$. Denote by $P$ the foot of the perpendicular from $E$ on $l_{A}$. Denote by $Q$ the point of intersection of the line $CP$ with the circle $O$ (different from $C$). Denote by $R$ be the point of intersection of the line $QT$ with the circle $O$ (different from $Q$). Finally, define $U = BR \cap l_{A}$. Prove that
$\frac {SU \cdot SP}{TU \cdot TP} = \frac {SA^{2}}{TA^{2}}.$

2006 FKMO problem 1
In a triangle $ABC$ with $AB\not = AC$, the incircle touches the sides $BC, CA, AB$ at $D, E, F$ , respectively. Line $AD$ meets the incircle again at $P$ . The line $EF$ and the line through $P$ perpendicular to $AD$ meet at $Q$. Line $AQ$ intersects $DE$ at $X$ and $DF$ at $Y$ . Prove that $A$ is the midpoint of $XY$.

2006 FKMO problem 5
In a convex hexagon $ABCDEF$ triangles $ABC , CDE , EFA$ are similar. Find conditions on these triangles under which triangle $ACE$ is equilateral if and only if so is $BDF.$

2007 FKMO problem 1
Let $O$ be the circumcenter of an acute triangle $ABC$ and let $k$ be the circle with center $P$ that is tangent to $O$ at $A$ and tangent to side $BC$ at $D$. Circle $k$ meets $AB$ and $AC$ again at $E$ and $F$ respectively. The lines $OP$ and $EP$ meet $k$ again at $I$ and $G$. Lines $BO$ and $IG$ intersect at $H$. Prove that $\frac{{DF}^2}{AF}=GH$.

2007 FKMO problem 5
For the vertex $A$ of a triangle $ABC$, let $l_a$ be the distance between the projections on $AB$ and $AC$ of the intersection of the angle bisector of ∠$A$ with side $BC$. Define $l_b$ and $l_c$ analogously. If $l$ is the perimeter of triangle $ABC$, prove that $\frac{l_a l_b l_c}{l^3}\le\frac{1}{64}$.

2008 FKMO problem 1
Hexagon $ABCDEF$ is inscribed in a circle $O$. Let $BD \cap CF = G, AC \cap BE = H, AD \cap CE = I$ . Following conditions are satisfied.$BD \perp CF , CI=AI$ . Prove that $CH=AH+DE$ is equivalent to $GH \times BD = BC \times DE$

2008 FKMO problem 5
Quadrilateral $ABCD$ is inscribed in a circle $O$. Let $AB\cap CD=E$ and $P\in BC, EP\perp BC$, $R\in AD, ER\perp AD$, $EP\cap AD=Q, ER\cap BC=S$ .Let $K$ be the midpoint of $QS$ . Prove that $E, K, O$ are collinear.

2009 FKMO problem 2
$ABC$ is an obtuse triangle. (angle $B$ is obtuse) Its circumcircle is $O$. A tangent line for $O$ passing $C$ meets with $AB$ at $B_1$. Let $O_1$ be a circumcenter of triangle $AB_1C$. $B_2$ is a point on the segment $BB_1$. Let $C_1$ be a contact point of the tangent line for $O$ passing $B_2$, which is more closer to $C$. Let $O_2$ be a circumcenter of triangle $AB_2C_1$. Prove that if $OO_2$ and $AO_1$ is perpendicular, then five points $O,O_2,O_1,C_1,C$ are concyclic.

2009 FKMO problem 4
$ABC$ is an acute triangle. (angle $C$ is bigger than angle $B$) Let $O$ be a center of the circle which passes $B$ and tangents to $AC$ at $C$. $O$ meets the segment $AB$ at $D$. $CO$ meets the circle $(O)$ again at $P$, a line, which passes $P$ and parallel to $AO$, meets $AC$ at $E$, and $EB$ meets the circle $(O)$ again at $L$. A perpendicular bisector of $BD$ meets $AC$ at $F$ and $LF$ meets $CD$ at $K$. Prove that two lines $EK$ and $CL$ are parallel.

2010 FKMO problem 2
Let $I$ be the incentre and $O$ the circumcentre of a given acute triangle $ABC$. The incircle is tangent to $BC$ at $D$. Assume that $\angle B < \angle C$ and the segments $AO$ and $HD$ are parallel, where $H$ is the orthocentre of triangle $ABC$. Let the intersection of the line $OD$ and $AH$ be $E$. If the midpoint of $CI$ is $F$, prove that $E,F,I,O$ are concyclic.

2010 FKMO problem 4
Given is a trapezoid $ABCD$ where $AB$ and $CD$ are parallel, and $A,B,C,D$ are clockwise in this order. Let $\Gamma_1$ be the circle with center $A$ passing through $B$, $\Gamma_2$ be the circle with center $C$ passing through $D$. The intersection of line $BD$ and $\Gamma_1$ is $P$ $( \ne B,D)$. Denote by $\Gamma$ the circle with diameter $PD$, and let $\Gamma$ and $\Gamma_1$ meet at $X$$( \ne P)$. $\Gamma$ and $\Gamma_2$ meet at $Y$. If the circumcircle of triangle $XBY$ and $\Gamma_2$ meet at $Q$, prove that $B,D,Q$ are collinear.

2011 FKMO problem 2
$ABC$ is an acute triangle. $P$(different from $B,C$) is a point on side $BC$.  $H$ is an orthocenter, and $D$ is a foot of perpendicular from $H$ to $AP$. The circumcircle of the triangle $ABD$ and $ACD$ is $O _1$ and $O_2$, respectively. A line $l$ parallel to $BC$ passes $D$ and meet $O_1$ and $O_2$ again at $X$ and $Y$, respectively. $l$ meets $AB$ at $E$, and $AC$ at $F$. Two lines $XB$ and $YC$ intersect at $Z$. Prove that $ZE=ZF$ is a necessary and sufficient condition for $BP=CP$.

2011 FKMO problem 5
$ABC$ is a triangle such that $AC<AB<BC$ and $D$ is a point on side $AB$ satisfying $AC=AD$.  The circumcircle of $ABC$ meets with the bisector of angle $A$ again at $E$ and meets with $CD$ again at $F$. $K$ is an intersection point of $BC$ and $DE$. Prove that $CK=AC$ is a necessary and sufficient condition for $DK \cdot EF = AC \cdot DF$.

2012 FKMO problem 2
For a triangle $ABC$ which $\angle B \ne 90^{\circ}$ and $AB \ne AC$, define $P_{ABC}$ as follows ;
Let $I$ be the incenter of triangle $ABC$, and let  $D, E, F$ be the intersection points with the incircle and segments $BC, CA, AB$. Two lines $AB$ and $DI$ meet at  $S$ and let $T$ be the intersection point of line $DE$ and the line which is perpendicular with $DF$ at $F$. The line $ST$ intersects line $EF$ at $R$. Now define $P_{ABC}$ be one of the intersection points of the incircle and the circle with diameter $IR$, which is located in other side with $A$ about $IR$.
Now think of an isosceles triangle $XYZ$ such that $XZ = YZ > XY$. Let $W$ be the point on the side $YZ$ such that $WY < XY$ and Let $K = P_{YXW}$ and $L = P_{ZXW}$. Prove that $2 KL \le XY$

2012 FKMO problem 4
Let $ABC$ be an acute triangle. Let $H$ be the foot of perpendicular from $A$ to $BC$. $D, E$ are the points on $AB, AC$ and let $F, G$ be the foot of perpendicular from $D, E$ to $BC$. Assume that $DG \cap EF$ is on $AH$. Let $P$ be the foot of perpendicular from $E$ to $DH$. Prove that $\angle APE = \angle CPE$.

2013 FKMO problem 1
For a triangle $\triangle ABC (\angle B > \angle C)$, $D$ is a point on $AC$ satisfying $\angle ABD = \angle C$. Let $I$ be the incenter of $\triangle ABC$, and circumcircle of $\triangle CDI$ meets $AI$ at $E ( \ne I )$. The line passing $E$ and parallel to $AB$ meets the line $BD$ at $P$. Let $J$ be the incenter of $\triangle ABD$, and $A'$ be the point such that $AI = IA'$. Let $Q$ be the intersection point of $JP$ and $A'C$. Prove that $QJ = QA'$.

2013 FKMO problem 4
For a triangle $ABC$, let $B_1 ,C_1$ be the excenters of $B, C$. Line $B_1 C_1$ meets with the circumcircle of $\triangle ABC$ at point $D (\ne A)$. $E$ is the point which satisfies $B_1 E \bot CA$ and $C_1 E \bot AB$. Let $w$ be the circumcircle of $\triangle ADE$. The tangent to the circle $w$ at $D$ meets $AE$ at $F$. $G , H$ are the points on $AE, w$ such that $DGH \bot AE$. The circumcircle of $\triangle HGF$ meets $w$ at point $I ( \ne H )$, and $J$ be the foot of perpendicular from $D$ to $AH$. Prove that $AI$ passes the midpoint of $DJ$

2014 FKMO problem 2
Let $ABC$ be a isosceles triangle with $AC = BC > AB$. Let $E, F$ be the midpoints of segments $AC, AB$, and let $l$ be the perpendicular bisector of $AC$. Let $l$ meets $AB$ at $K$, the line through $B$ parallel to $KC$ meets $AC$ at point $L$, and line $FL$ meets $l$ at $W$. Let $P$ be a point on segment $BF$. Let $H$ be the orthocenter of triangle $ACP$ and line $BH$ and $CP$ meet at point $J$. Line $FJ$ meets $l$ at $M$. Prove that $AW = PW$ if and only if $B$ lies on the circumcircle of $EFM$.

2014 FKMO problem 4
Let $ABC$ be a isosceles triangle with $AC=BC$. Let $D$ a point on a line $BA$ such that $A$ lies between $B, D$. Let $O_1$ be the circumcircle of triangle $DAC$. $O_1$ meets $BC$ at point $E$. Let $F$ be the point on $BC$ such that $FD$ is tangent to circle $O_1$, and let $O_2$ be the circumcircle of $DBF$. Two circles $O_1 , O_2$ meet at point $G ( \ne D)$. Let $O$ be the circumcenter of triangle $BEG$. Prove that the line $FG$ is tangent to circle $O$ if and only if $DG \bot FO$.

2015 FKMO problem 2
In a triangle $\triangle ABC$ with incenter $I$, the incircle meets lines $BC, CA, AB$ at $D, E, F$ respectively. Define the circumcenter of $\triangle IAB$ and $\triangle IAC$ $O_1$ and $O_2$ respectively. Let the two intersections of the circumcircle of $\triangle ABC$ and line $EF$ be $P, Q$. Prove that the circumcenter of $\triangle DPQ$ lies on the line $O_1O_2$.

2015 FKMO problem 4
$\triangle ABC$ is an acute triangle and its orthocenter is $H$. The circumcircle of $\triangle ABH$ intersects line $BC$ at $D$. Lines $DH$ and $AC$ meets at $P$, and the circumcenter of $\triangle ADP$ is $Q$. Prove that the circumcenter of $\triangle ABH$ lies on the circumcircle of $\triangle BDQ$.

2016 FKMO problem 1
In a acute triangle $\triangle ABC$, denote $D, E$ as the foot of the perpendicular from $B$ to $AC$ and $C$ to $AB$. Denote the reflection of $E$ with respect to $AC, BC$ as $S, T$. The circumcircle of $\triangle CST$ hits $AC$ at point $X (\not= C)$.  Denote the circumcenter of $\triangle CST$ as $O$. Prove that $XO \perp DE$.

2016 FKMO problem 5
An acute triangle $\triangle ABC$ has incenter $I$, and the incircle hits $BC, CA, AB$ at $D, E, F$.
Lines $BI, CI, BC, DI$ hits $EF$ at $K, L, M, Q$ and the line connecting the midpoint of segment $CL$ and $M$ hits the line segment $CK$ at $P$. Prove that $PQ=\frac{AB \cdot KQ}{BI}$

2017 FKMO problem 1
A acute triangle $\triangle ABC$ has circumcenter $O$. The circumcircle of $OAB$, called $O_1$, and the circumcircle of $OAC$, called $O_2$, meets $BC$ again at $D ( \not=B )$ and $E ( \not= C )$ respectively. The perpendicular bisector of $BC$ hits $AC$ again at $F$. Prove that the circumcenter of $\triangle ADE$ lies on $AC$ if and only if the centers of $O_1, O_2$ and $F$ are colinear.

2017 FKMO problem 5
Let there be cyclic quadrilateral $ABCD$ with $L$ as the midpoint of $AB$ and $M$ as the midpoint of $CD$. Let $AC \cap BD = E$, and let rays $AB$ and $DC$ meet again at $F$. Let $LM \cap DE = P$. Let $Q$ be the foot of the perpendicular from $P$ to $EM$. If the orthocenter of $\triangle FLM$ is $E$, prove the following equality:  $\frac{EP^2}{EQ} = \frac{1}{2} \left( \frac{BD^2}{DF} - \frac{BC^2}{CF} \right)$

2018 FKMO problem 2
Triangle $ABC$ satisfies $\angle C=90^{\circ}$. A circle passing $A,B$ meets segment $AC$ at $G(\neq A,C)$ and it meets segment $BC$ at point $D(\neq B)$. Segment $AD$ cuts segment $BG$ at $H$, and let $l$, the perpendicular bisector of segment $AD$, cuts the perpendicular bisector of segment $AB$ at point $E$. A line passing $D$ is perpendicular to $DE$ and cuts $l$ at point $F$. If the circumcircle of triangle $CFH$ cuts $AC$, $BC$ at $P(\neq C),Q(\neq C)$ respectively, then prove that $PQ$ is perpendicular to $FH$.

2018 FKMO problem 4
Triangle $ABC$ satisfies $\angle ABC < \angle BCA < \angle CAB < 90^{\circ}$. $O$ is the circumcenter of triangle $ABC$, and $K$ is the reflection of $O$ in $BC$. $D,E$ is the foot of perpendicular line from $K$ to line $AB$, $AC$, respectively. Line $DE$ meets $BC$ at $P$, and a circle with diameter $AK$ meets the circumcircle of triangle $ABC$ at $Q(\neq A)$. If   $PQ$ cuts the perpendicular bisector of $BC$ at $S$, then prove that $S$ lies on the circle with diameter $AK$.

For a rectangle $ABCD$ which is not a square, there is $O$ such that $O$ is on the perpendicular bisector of $BD$ and $O$ is in the interior of $\triangle BCD$. Denote by $E$ and $F$ the second intersections of the circle centered at $O$ passing through $B, D$ and $AB, AD$. $BF$ and $DE$ meets at $G$, and $X, Y, Z$ are the foots of the perpendiculars from $G$ to $AB, BD, DA$. $L, M, N$ are the foots of the perpendiculars from $O$ to $CD, BD, BC$. $XY$ and $ML$ meets at $P$, $YZ$ and $MN$ meets at $Q$. Prove that $BP$ and $DQ$ are parallel.

2019 FKMO problem 4
Let triangle $ABC$ be an acute scalene triangle with orthocenter $H$. The foot of perpendicular from $A$ to $BC$ is $O$, and denote $K,L$ by the midpoints of $AB, AC$, respectively. For a point $D(\neq O,B,C)$ on segment $BC$, let $E,F$ be the orthocenters of triangles $ABD, ACD$, respectively, and denote $M,N$ by the midpoints of $DE,DF$. The perpendicular line from $M$ to $KH$ cuts the perpendicular line from $N$ to $LH$ at $P$. If $Q$ is the midpoint of $EF$, and $S$ is the orthocenter of triangle $HPQ$, then prove that as $D$ varies on $BC$, the ratio $\frac{OS}{OH}$, $\frac{OQ}{OP}$ remains constant.