### Indonesia 2002-18 (INAMO) (OSN) + SHL 2014-15 43p

geometry problems from Indonesian National Science Olympiads (INAMO) (OSN)
with aops links in the names

2002 - 2018
and Shortlist Geometry 2014-15

INAMO Shortlist in aops: 2014, 2015

2002 Indonesia MO P4
Given a triangle $ABC$ where $AC > BC$, $D$ is located on the circumcircle of $ABC$ such that $D$ is the midpoint of the arc $AB$ that contains $C$. $E$ is a point on $AC$ such that $DE$ is perpendicular to $AC$. Prove that $AE = EC + CB$.

2002 Indonesia MO P7
Let $ABCD$ be a rhombus where $\angle DAB = 60^\circ$, and $P$ be the intersection between $AC$ and $BD$. Let $Q,R,S$ be three points on the boundary of $ABCD$ such that $PQRS$ is a rhombus. Prove that exactly one of $Q,R,S$ lies on one of $A,B,C,D$.

2003 Indonesia MO P2
Let $ABCD$ be a quadrilateral, and $P,Q,R,S$ are the midpoints of $AB, BC, CD, DA$ respectively. Let $O$ be the intersection between $PR$ and $QS$. Prove that $PO = OR$ and $QO = OS$.

2004 Indonesia MO P4
There exists 4 circles, $a,b,c,d$, such that $a$ is tangent to both $b$ and $d$, $b$ is tangent to both $a$ and $c$, $c$ is both tangent to $b$ and $d$, and $d$ is both tangent to $a$ and  $c$. Show that all these tangent points are located on a circle.

2004 Indonesia MO P7
Given triangle $ABC$ with $C$ a right angle, show that the diameter of the incenter is $a+b-c$, where $a=BC$, $b=CA$, and $c=AB$.

2005 Indonesia MO P4
Let $M$ be a point in triangle $ABC$ such that $\angle AMC=90^{\circ}$, $\angle AMB=150^{\circ}$, $\angle BMC=120^{\circ}$. The centers of circumcircles of triangles $AMC,AMB,BMC$ are $P,Q,R$, respectively. Prove that the area of $\triangle PQR$ is greater than the area of $\triangle ABC$.

2005 Indonesia MO P7
Let $ABCD$ be a convex quadrilateral. Square $AB_1A_2B$ is constructed such that the two vertices $A_2,B_1$ is located outside $ABCD$. Similarly, we construct squares $BC_1B_2C$, $CD_1C_2D$, $DA_1D_2A$. Let $K$ be the intersection of $AA_2$ and $BB_1$, $L$ be the intersection of $BB_2$ and $CC_1$, $M$ be the intersection of $CC_2$ and $DD_1$, and $N$ be the intersection of $DD_2$ and $AA_1$. Prove that $KM$ is perpendicular to $LN$.

2006 Indonesia MO P5
In triangle $ABC$, $M$ is the midpoint of side $BC$ and $G$ is the centroid of triangle $ABC$. A line $l$ passes through $G$, intersecting line $AB$ at $P$ and line $AC$ at $Q$, where $P\ne B$ and $Q\ne C$. If $[XYZ]$ denotes the area of triangle $XYZ$, show that $\frac{[BGM]}{[PAG]}+\frac{[CMG]}{[QGA]}=\frac32$.

2007 Indonesia MO P1
Let $ABC$ be a triangle with $\angle ABC=\angle ACB=70^{\circ}$. Let point $D$ on side $BC$ such that $AD$ is the altitude, point $E$ on side $AB$ such that $\angle ACE=10^{\circ}$, and point $F$ is the intersection of $AD$ and $CE$. Prove that $CF=BC$.

2007 Indonesia MO P7
Points $A,B,C,D$ are on circle $S$, such that $AB$ is the diameter of $S$, but $CD$ is not the diameter. Given also that $C$ and $D$ are on different sides of $AB$. The tangents of $S$ at $C$ and $D$ intersect at $P$. Points $Q$ and $R$ are the intersections of line $AC$ with line $BD$ and line $AD$ with line $BC$, respectively.
(a) Prove that $P$, $Q$, and $R$ are collinear.
(b) Prove that $QR$ is perpendicular to line $AB$.

2008 Indonesia MO P1
Given triangle $ABC$. Points $D,E,F$ outside triangle $ABC$ are chosen such that triangles $ABD$, $BCE$, and $CAF$ are equilateral triangles. Prove that cicumcircles of these three triangles are concurrent.

2008 Indonesia MO P7
Given triangle $ABC$ with sidelengths $a,b,c$. Tangents to incircle of $ABC$ that parallel with triangle's sides form three small triangle (each small triangle has 1 vertex of $ABC$). Prove that the sum of area of incircles of these three small triangles and the area of incircle of triangle $ABC$ is equal to
$\frac{\pi (a^{2}+b^{2}+c^{2})(b+c-a)(c+a-b)(a+b-c)}{(a+b+c)^{3}}$

2009 Indonesia MO P3
For every triangle $ABC$, let $D,E,F$ be a point located on segment $BC,CA,AB$, respectively. Let $P$ be the intersection of $AD$ and $EF$. Prove that:
$\frac{AB}{AF}\cdot DC+\frac{AC}{AE}\cdot DB=\frac{AD}{AP}\cdot BC$

2009 Indonesia MO P4
Given an acute triangle $ABC$. The incircle of triangle $ABC$ touches $BC,CA,AB$ respectively at $D,E,F$. The angle bisector of $\angle A$ cuts $DE$ and $DF$ respectively at $K$ and $L$. Suppose $AA_1$ is one of the altitudes of triangle $ABC$, and $M$ be the midpoint of $BC$.
(a) Prove that $BK$ and $CL$ are perpendicular with the angle bisector of $\angle BAC$.
(b) Show that $A_1KML$ is a cyclic quadrilateral.

2010 Indonesia MO P2
Given an acute triangle $ABC$ with $AC>BC$ and the circumcenter of triangle $ABC$ is $O$. The altitude of triangle $ABC$ from $C$ intersects $AB$ and the circumcircle at $D$ and $E$, respectively. A line which passed through $O$ which is parallel to $AB$ intersects $AC$ at $F$. Show that the line $CO$, the line which passed through $F$ and perpendicular to $AC$, and the line which passed through $E$ and parallel with $DO$ are concurrent.

by Fajar Yuliawan, Bandung

2010 Indonesia MO P8
Given an acute triangle $ABC$ with circumcenter $O$ and orthocenter $H$. Let $K$ be a point inside $ABC$ which is not $O$ nor $H$. Point $L$ and $M$ are located outside the triangle $ABC$ such that $AKCL$ and $AKBM$ are parallelogram. At last, let $BL$ and $CM$ intersects at $N$, and let $J$ be the midpoint of $HK$. Show that $KONJ$ is also a parallelogram.

by Raja Oktovin, Pekanbaru

2011 Indonesia MO P3
Given an acute triangle $ABC$, let $l_a$ be the line passing $A$ and perpendicular to $AB$, $l_b$ be the line passing $B$ and perpendicular to $BC$, and $l_c$ be the line passing $C$ and perpendicular to $CA$. Let $D$ be the intersection of $l_b$ and $l_c$, $E$ be the intersection of $l_c$ and $l_a$, and $F$ be the intersection of $l_a$ and $l_b$. Prove that the area of the triangle $DEF$ is at least three times of the area of $ABC$.[/quote]

2011 Indonesia MO P8
Given a triangle $ABC$. Its incircle is tangent to $BC, CA, AB$ at $D, E, F$ respectively. Let $K, L$ be points on $CA, AB$ respectively such that $K \neq A \neq L, \angle EDK = \angle ADE, \angle FDL = \angle ADF$. Prove that the circumcircle of $AKL$ is tangent to the incircle of $ABC$.

2012 Indonesia MO P3
Given an acute triangle $ABC$ with $AB>AC$ that has circumcenter $O$. Line $BO$ and $CO$ meet the bisector of $\angle BAC$ at $P$ and $Q$, respectively. Moreover, line $BQ$ and $CP$ meet at $R$. Show that $AR$ is perpendicular to $BC$.

by Soewono and Fajar Yuliawan

2012 Indonesia MO P8
Given a triangle $ABC$, let the bisector of $\angle BAC$ meets the side $BC$ and circumcircle of triangle $ABC$ at $D$ and $E$, respectively. Let $M$ and $N$ be the midpoints of $BD$ and $CE$, respectively. Circumcircle of triangle $ABD$ meets $AN$ at $Q$. Circle passing through $A$ that is tangent to $BC$ at $D$ meets line $AM$ and side $AC$ respectively at $P$ and $R$. Show that the four points $B,P,Q,R$ lie on the same line.

by Fajar Yuliawan

2013 Indonesia MO P2
Let $ABC$ be an acute triangle and $\omega$ be its circumcircle. The bisector of $\angle BAC$ intersects $\omega$ at [another point] $M$. Let $P$ be a point on $AM$ and inside $\triangle ABC$. Lines passing $P$ that are parallel to $AB$ and $AC$ intersects $BC$ on $E, F$ respectively. Lines $ME, MF$ intersects $\omega$ at points $K, L$ respectively. Prove that $AM, BL, CK$ are concurrent.

2013 Indonesia MO P7
Let $ABCD$ be a parallelogram. Construct squares $ABC_1D_1, BCD_2A_2, CDA_3B_3, DAB_4C_4$ on the outer side of the parallelogram. Construct a square having $B_4D_1$ as one of its sides and it is on the outer side of $AB_4D_1$ and call its center $O_A$. Similarly do it for $C_1A_2, D_2B_3, A_3C_4$ to obtain $O_B, O_C, O_D$. Prove that $AO_A = BO_B = CO_C = DO_D$.

2014 Indonesia MO P3
Let $ABCD$ be a trapezoid (quadrilateral with one pair of parallel sides) such that $AB < CD$. Suppose that $AC$ and $BD$ meet at $E$ and $AD$ and $BC$ meet at $F$. Construct the parallelograms $AEDK$ and $BECL$. Prove that $EF$ passes through the midpoint of the segment $KL$.

2014 Indonesia MO P6
Let $ABC$ be a triangle. Suppose $D$ is on $BC$ such that $AD$ bisects $\angle BAC$. Suppose $M$ is on $AB$ such that $\angle MDA = \angle ABC$, and $N$ is on $AC$ such that $\angle NDA = \angle ACB$. If $AD$ and $MN$ intersect on $P$, prove that $AD^3 = AB \cdot AC \cdot AP$.

2015 Indonesia MO P3
Given an acute triangle $ABC$. $\Gamma _{B}$ is a circle that passes through $AB$, tangent to $AC$ at $A$ and centered at $O_{B}$. Define $\Gamma_C$ and $O_C$ the same way. Let the altitudes of $\triangle ABC$ from $B$ and $C$ meets the circumcircle of $\triangle ABC$ at $X$ and $Y$, respectively. Prove that $A$, the midpoint of $XY$ and the midpoint of $O_{B}O_{C}$ is collinear.

2015 Indonesia MO P3
Given an acute triangle $ABC$. $\Gamma _{B}$ is a circle that passes through $AB$, tangent to $AC$ at $A$ and centered at $O_{B}$. Define $\Gamma_C$ and $O_C$ the same way. Let the altitudes of $\triangle ABC$ from $B$ and $C$ meets the circumcircle of $\triangle ABC$ at $X$ and $Y$, respectively. Prove that $A$, the midpoint of $XY$ and the midpoint of $O_{B}O_{C}$ is collinear.

Let $ABC$ be an acute angled triangle with circumcircle $O$. Line $AO$ intersects the circumcircle of triangle  $ABC$ again at point $D$. Let $P$ be a point on the side  $BC$. Line passing through $P$ perpendicular to $AP$ intersects lines  $DB$ and $DC$ at $E$ and $F$ respectively . Line passing through $D$ perpendicular to $BC$ intersects $EF$ at point $Q$. Prove that $EQ = FQ$ if and only if $BP = CP$.

2016 Indonesia MO P1
Let $ABCD$ be a cyclic quadrilateral wih both diagonals perpendicular to each other and intersecting at point $O$. Let $E,F,G,H$ be the orthogonal projections of $O$ on sides $AB,BC,CD,DA$ respectively.
a. Prove that $\angle EFG + \angle GHE = 180^o$
b. Prove that $OE$ bisects angle  $\angle FEH$ .

2016 Indonesia MO P6
For a quadrilateral $ABCD$, we call a square $amazing$ if all of its sides(extended if necessary) pass through distinct vertices of $ABCD$(no side passing through 2 vertices). Prove that for an arbitrary $ABCD$ such that its diagonals are not perpendicular, there exist at least 6 $amazing$ squares

2017 Indonesia MO P1
$ABCD$ is a parallelogram. $g$ is a line passing $A$. Prove that the distance from $C$ to $g$ is either the sum or the difference of the distance from $B$ to $g$, and the distance from $D$ to $g$.

2017 Indonesia MO P7
Let $ABCD$ be a parallelogram. $E$ and $F$ are on $BC, CD$ respectively such that the triangles $ABE$ and $BCF$ have the same area. Let $BD$ intersect $AE, AF$ at $M, N$ respectively. Prove there exists a triangle whose side lengths are $BM, MN, ND$.

2018 Indonesia MO P2
Let $\Gamma_1, \Gamma_2$ be circles that touch at a point $A$, and $\Gamma_2$ is inside $\Gamma_1$. Let $B$ be on $\Gamma_2$, and let $AB$ intersect $\Gamma_1$ on $C$. Let $D$ be on $\Gamma_1$ and $P$ be on the line $CD$ (may be outside of the segment $CD$). $BP$ intersects $\Gamma_2$ at $Q$. Prove that $A,D,P,Q$ lie on a circle.

2018 Indonesia MO P8
Let $I, O$ be the incenter and circumcenter of the triangle $ABC$ respectively. Let the excircle $\omega_A$ of $ABC$ be tangent to the side $BC$ on $N$, and tangent to the extensions of the sides $AB, AC$ on $K, M$ respectively. If the midpoint of $KM$ lies on the circumcircle of $ABC$, prove that $O, I, N$ are collinear.

2014-15 INAMO Geometry Shortlist

2014 INAMO Shortlist G1
The inscribed circle of the $ABC$ triangle has center  $I$ and touches to $BC$ at $X$. Suppose the $AI$ and $BC$ lines intersect at $L$, and $D$ is the reflection of $L$ wrt $X$. Points $E$ and $F$ respectively are the result of a reflection of $D$ wrt to lines $CI$ and $BI$ respectively. Show that quadrilateral $BCEF$ is cyclic .

2014 INAMO Shortlist G2 (problem 6)
Let $ABC$ be a triangle. Suppose $D$ is on $BC$ such that $AD$ bisects $\angle BAC$. Suppose $M$ is on $AB$ such that $\angle MDA = \angle ABC$, and $N$ is on $AC$ such that $\angle NDA = \angle ACB$. If $AD$ and $MN$ intersect on $P$, prove that $AD^3 = AB \cdot AC \cdot AP$.

2014 INAMO Shortlist G3 (problem 3)
Let $ABCD$ be a trapezoid (quadrilateral with one pair of parallel sides) such that $AB < CD$. Suppose that $AC$ and $BD$ meet at $E$ and $AD$ and $BC$ meet at $F$. Construct the parallelograms $AEDK$ and $BECL$. Prove that $EF$ passes through the midpoint of the segment $KL$.

2014 INAMO Shortlist G4
Given an acute triangle $ABC$ with $AB <AC$. Points $P$ and $Q$ lie on the angle bisector of $\angle BAC$ so that $BP$ and $CQ$ are perpendicular on that angle bisector. Suppose that point $E, F$ lie respectively at sides $AB$ and $AC$ respectively, in such a way that $AEPF$ is a kite. Prove that the lines $BC, PF$, and $QE$ intersect at one point.

2014 INAMO Shortlist G5
Given quadrilateral $ABCD$. Suppose $E, F, G, H$ are respectively the midpoint of the sides $AB, BC, CD, DA$. The line passing through $G$ and perpendicular on $AB$ intersects the line passing through $H$ and perpendicular on $BC$ at point $K$. Prove that $\angle EKF = \angle ABC$.

2014 INAMO Shortlist G6
Given an $ABC$ acute triangle with $O$ the center of the circumscribed circle. Suppose that $\omega$ is a circle that is tangent to the line $AO$ at point $A$ and also tangent to the line $BC$. Prove that  $\omega$ is also tangent to the circumcircle of the triangle $BOC$.

2015 INAMO Shortlist G1
Given a cyclic quadrilateral $ABCD$ so that $AB = AD$ and $AB + BC <CD$. Prove that the angle $ABC$ is more than $120$ degrees.

2015 INAMO Shortlist G2
Two circles that are not equal are tangent externally at point $R$. Suppose point $P$ is the intersection of the external common tangents of the two circles. Let $A$ and $B$ are two points on different circles so that $RA$ is perpendicular to $RB$. Show that the line $AB$ passes through $P$.
2015 INAMO Shortlist G3
Given $ABC$ triangle with incircle $L_1$ and circumcircle $L_2$. If points $X, Y, Z$ lie on $L_2$, such that $XY, XZ$ are tangent to $L_1$, then prove that $YZ$ is also tangent to $L_1$.

2015 INAMO Shortlist G4
Given an isosceles triangle $ABC$ with $AB = AC$, suppose $D$ is the midpoint of the $AC$. The circumcircle of the $DBC$ triangle intersects the altitude from $A$ at point $E$ inside the triangle  $ABC$, and the circumcircle of the triangle $AEB$ cuts the side  $BD$ at point $F$. If $CF$ cuts $AE$ at point $G$, prove that $AE = EG$.

2015 INAMO Shortlist G5
Let $ABC$ be an acute triangle. Suppose that circle $\Gamma_1$ has it's center on the side $AC$ and is tangent to the sides $AB$ and $BC$, and circle $\Gamma_2$ has it's center on the side  $AB$ and is tangent to the sides $AC$ and $BC$. The circles  $\Gamma_1$ and $\Gamma_2$ intersect at two points $P$ and $Q$. Show that if $A, P, Q$ are collinear, then $AB = AC$.

2015 INAMO Shortlist G6 (problem 6)
Let $ABC$ be an acute angled triangle with circumcircle $O$. Line $AO$ intersects the circumcircle of triangle  $ABC$ again at point $D$. Let $P$ be a point on the side  $BC$. Line passing through $P$ perpendicular to $AP$ intersects lines  $DB$ and $DC$ at $E$ and $F$ respectively . Line passing through $D$ perpendicular to $BC$ intersects $EF$ at point $Q$. Prove that $EQ = FQ$ if and only if $BP = CP$.

2015 INAMO Shortlist G7 (problem 3)
Given an acute triangle $ABC$. $\Gamma _{B}$ is a circle that passes through $AB$, tangent to $AC$ at $A$ and centered at $O_{B}$. Define $\Gamma_C$ and $O_C$ the same way. Let the altitudes of $\triangle ABC$ from $B$ and $C$ meets the circumcircle of $\triangle ABC$ at $X$ and $Y$, respectively. Prove that $A$, the midpoint of $XY$ and the midpoint of $O_{B}O_{C}$ is collinear.

$ABC$ is an acute triangle with $AB> AC$.  $\Gamma_B$ is a circle that passes through $A,B$ and is tangent to $AC$ on $A$. Define similar for  $\Gamma_C$. Let $D$ be the intersection  $\Gamma_B$ and  $\Gamma_C$ and $M$ be the midpoint of $BC$. $AM$ cuts  $\Gamma_C$ at $E$. Let $O$ be the center of the circumscibed circle of the triangle ABC. Prove that the circumscibed circle of the triangle $ODE$ is tangent to $\Gamma_B$.