Indonesia 2002-21 (INAMO) (OSN) + SHL 82p

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

Olimpiade Sains Nasional

                                      Indonesia MO 2002 - 2021

and Shortlist Geometry 2008-10, 2014-15, 2017

INAMO Shortlist in aops: 2014, 2015

Indonesia Shortlist Geometry inside aops: here

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.

2015 Indonesia MO P6

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.

2019 Indonesia MO P3
Given that $ABCD$ is a rectangle such that $AD > AB$, where $E$ is on $AD$ such that $BE \perp AC$. Let $M$ be the intersection of $AC$ and $BE$. Let the circumcircle of $\triangle ABE$ intersects $AC$ and $BC$ at $N$ and $F$. Moreover, let the circumcircle of $\triangle DNE$ intersects $CD$ at $G$. Suppose $FG$ intersects $AB$ at $P$. Prove that $PM = PN$.

2019 Indonesia MO P6
Given a circle with center $O$, such that $A$ is not on the circumcircle. Let $B$ be the reflection of $A$ with respect to $O$. Now let $P$ be a point on the circumcircle. The line perpendicular to $AP$ through $P$ intersects the circle at $Q$.Prove that $AP \times BQ$ remains constant as $P$ varies.

2020 Indonesia MO P1

Given an acute triangle $ABC$ and the point $D$ on segment $BC$. Circle $c_1$ passes through $A, D$ and its centre lies on $AC$. Whereas circle $c_2$ passes through $A, D$ and its centre lies on $AB$. Let $P \neq A$ be the intersection of $c_1$ with $AB$ and $Q \neq A$ be the intersection of $c_2$ with $AC$. Prove that $AD$ bisects $\angle{PDQ}$.

2020 Indonesia MO P6

Given a cyclic quadrilateral $ABCD$. Let $X$ be a point on segment $BC$ ($X \not= BC$) such that

line $AX$ is perpendicular to the angle bisector of $\angle CBD$, and $Y$ be a point on segment $AD$

($Y \not= D)$ such that $BY$ is perpendicular to the angle bisector of $\angle CAD$. Prove that $XY$

is parallel to $CD$.

2021 Indonesia MO P2

Let $ABC$ be an acute triangle. Let $D$ and $E$ be the midpoint of segment $AB$ and $AC$

respectively. Suppose $L_1$ and $L_2$ are circumcircle of triangle $ABC$ and $ADE$ respectively.

$CD$ intersects $L_1$ and $L_2$ at $M (M \not= C)$ and $N (N \not= D)$. If $DM = DN$, prove that

$\triangle ABC$ is isosceles. 2021 Indonesia MO P7

Given $\triangle ABC$ with circumcircle $\ell$. Point $M$ in $\triangle ABC$ such that $AM$ is the

angle bisector of $\angle BAC$. Circle with center $M$ and radius $MB$ intersects $\ell$ and $BC$

at $D$ and $E$ respectively, $(B \not= D, B \not= E)$. Let $P$ be the midpoint of arc $BC$ in $\ell$

that didn't have $A$. Prove that $AP$ angle bisector of $\angle DPE$ if and only if $\angle B = 90^{\circ}$.

2008, 2010, 2014-15

INAMO Geometry Shortlist

2008 INAMO Shortlist G1  (problem 1)

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

2008 INAMO Shortlist G2

Let $ABC$ be an isosceles triangle right at $C$ and $P$ any point on $CB$. Let also $Q$ be the midpoint of $AB$ and $R, S$ be the points on $AP$ such that $CR$ is perpendicular to $AP$ and $|AS|=|CR|$. Prove that the $|RS|  =  \sqrt2 |SQ|$.

2008 INAMO Shortlist G3

Given triangle $ABC$. A circle $\Gamma$ is tangent to the circumcircle of triangle $ABC$ at $A$ and tangent to $BC$ at $D$. Let $E$ be the intersection of circle $\Gamma$ and $AC$. Prove that

$$R^2=OE^2+CD^2\left(1- \frac{BC^2}{AB^2+AC^2}\right)$$ where $O$ is the center of the circumcircle of triangle $ABC$, with radius $R$.

2008 INAMO Shortlist G4

Given that two circles $\sigma_1$ and $\sigma_2$ internally tangent at $N$ so that $\sigma_2$ is inside $\sigma_1$. The points $Q$ and $R$ lies at $\sigma_1$ and $\sigma_2$, respectively, such that $N,R,Q$ are collinear. A line through $Q$ intersects $\sigma_2$ at $S$ and intersects $\sigma_1$ at $O$. The line through $N$ and $S$ intersects $\sigma_1$ at $P$. Prove that $$\frac{PQ^3}{PN^2} = \frac{PS \cdot RS}{NS}.$$

2008 INAMO Shortlist G5

Let $ABCD$ be quadrilateral inscribed in a circle. Let $M$ be the midpoint of the segment $BD$. If the tangents of the circle at $B$, and at $D$ are also concurrent with the extension of $AC$, prove that $\angle AMD = \angle CMD$.

2008 INAMO Shortlist G6 (problem 7)

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

$\frac{\pi (a^{2}+b^{2}+c^{2})(b+c-a)(c+a-b)(a+b-c)}{(a+b+c)^{3}}$

2008 INAMO Shortlist G7

Given an isosceles trapezoid $ABCD$ with base $AB$. The diagonals $AC$ and $BD$ intersect at point $S$. Let $M$ the midpoint of $BC$ and the bisector of the angle $BSC$ intersect $BC$ at $N$. Prove that $\angle AMD = \angle AND$.

2008 INAMO Shortlist G8

Prove that there is only one triangle whose sides are consecutive natural numbers and one of the angles is twice the other angle.

2008 INAMO Shortlist G9

Given a triangle $ABC$, the points $D$, $E$, and $F$ lie on the sides $BC$, $CA$, and $AB$, respectively, are such that

$$DC + CE = EA + AF = FB + BD.$$ Prove that $$DE + EF + FD \ge \frac12 (AB + BC + CA).$$

2008 INAMO Shortlist G10

Given a triangle $ABC$ with $AB = AC$, angle $\angle A = 100^o$ and $BD$ bisector of angle $\angle B$. Prove that $$BC = BD + DA.$$

2009 INAMO Shortlist G1

Given triangle $ABC$, $AL$ bisects angle $\angle BAC$ with $L$ on side $BC$. Lines $LR$ and $LS$ are parallel to $BA$ and $CA$ respectively, $R$ on side $AC$ and$S$ on side $AB$, respectively. Through point $B$ draw a perpendicular on $AL$, intersecting $LR$ at $M$. If point $D$ is the midpoint of $BC$, prove that that the three points $A, M, D$ lie on a straight line.

2009 INAMO Shortlist G2 (problem 3)

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:

$$\frac{AB}{AF}\times DC+\frac{AC}{AE}\times DB=\frac{AD}{AP}\times BC$$

2009 INAMO Shortlist G3

Given a quadrilateral $ABCD$ inscribed in circle $\Gamma$.From a point P outside $\Gamma$, draw tangents $PA$ and $PB$ with $A$ and $B$ as touspoints. The line $PC$ intersects $\Gamma$ at point $D$. Draw a line through $B$ parallel to $PA$, this line intersects $AC$ and $AD$ at points $E$ and $F$ respectively. Prove that $BE = BF$.

2009 INAMO Shortlist G4

Let $D, E, F$, be the touchpoints of the incircle in triangle $ABC$ with sides $BC, CA, AB$, respectively, . Also, let $AD$ and $EF$ intersect at $P$. Prove that $$\frac{AP}{AD} \ge 1 - \frac{BC}{AB + CA}$$ .

2009 INAMO Shortlist G5

Two circles intersect at points $A$ and $B$. The line $\ell$ through A intersects the circles at $C$ and $D$, respectively. Let $M, N$ be the midpoints of arc $BC$ and arc $BD$. which does not contain $A$, and suppose that $K$ is the midpoint of the segment $CD$ . Prove that $\angle MKN=90^o$.

2009 INAMO Shortlist G6

Suppose the points $D, E, F$ lie on sides $BC, CA, AB$, respectively, so that $AD, BE, CF$ are the altitudes. Also, let $AD$ and $EF$ intersect at $P$. Prove that $$\frac{AP}{AD} \ge 1 - \frac{BC^2}{AB^2 + CA^2}$$

2009 INAMO Shortlist G7

Given a convex quadrilateral $ABCD$, such that $OA = \frac{OB \cdot OD}{OC + CD}$ where $O$ is the intersection of the two diagonals. The circumcircle of triangle $ABC$ intersects $BD$ at point $Q$. Prove that $CQ$ bisects $\angle ACD$

2009 INAMO Shortlist G8

Suppose the points $D, E, F$ lie on sides $BC, CA, AB$, respectively, so that $AD, BE, CF$ are angle bisectors. Define $P_1$, $P_2$, $P_3$ respectively as the intersection point of $AD$ with $EF$, $BE$ with $DF$, $CF$ with $DE$ respectively. Prove that

$$\frac{AD}{AP_1}+\frac{BE}{BP_2}+\frac{CF}{CP_3} \ge 6$$

2009 INAMO Shortlist G9

Given triangle $ABC$. Let $A_1B_1$, $A_2B_2$,$...$, $A_{2008}B_{2008}$ be $2008$ lines parallel to $AB$ which divide triangle $ABC$ into $2009$ equal areas. Calculate the value of $$\big \lfloor \frac{A_1B_1}{2A_2B_2} + \frac{A_1B_1}{2A_3B_3} + ... + \frac{A_1B_1}{2A_{2008}B_{2008}} \rfloor$$

2009 INAMO Shortlist G10

Given a triangle $ABC$ with incenter $I$ . It is known that $E_A$ is center of the ex-circle tangent to $BC$. Likewise, $E_B$ and $E_C$ are the centers of the ex-circles tangent to $AC$ and $AB$, respectively. Prove that $I$ is the orthocenter of the triangle $E_AE_BE_C$.

2009 INAMO Shortlist G11 (problem 8)

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.

2009 INAMO Shortlist G12

In triangle $ABC$, the incircle is tangent to $BC$ at $D$, to $AC$ at $E$, and to $AB$ at $F$. Prove that:

$$\frac{CE-EA}{\sqrt{AB}}+\frac{AF-FB}{\sqrt{BC}} +\frac{BD-DC}{\sqrt{CA}} \ge \frac{BD-DC}{\sqrt{AB}}

+\frac{CE-EA}{\sqrt{BC}} +\frac{AF-FB}{\sqrt{CA}}$$

2010 INAMO Shortlist G1

In triangle $ABC$, let $D$ be the midpoint of $BC$, and $BE$, $CF$ are the altitudes. Prove that $DE$ and $DF$ are both tangents to the circumcircle of triangle $AEF$

2010 INAMO Shortlist G2

Given an acute triangle $ABC$. The inscribed circle of triangle $ABC$ is tangent to $AB$ and $AC$ at $X$ and $Y$ respectively. Let $CH$ be the altitude. The perpendicular bisector of the segment $CH$ intersects the line $XY$ at $Z$. Prove that $\angle BZC = 90^o.$

2010 INAMO Shortlist G3

Suppose $L_1$ is a circle with center $O$, and $L_2$ is a circle with center $O'$. The circles intersect at $A$ and $B$ such that $\angle OAO' = 90^o$. Suppose that point $X$ lies on the circumcircle of triangle $OAB$, but lies inside $L_2$. Let the extension of $OX$ intersect $L_1$ at $Y$ and $Z$. Let the extension of $O'X$ intersect $L_2$ at $W$ and $V$ . Prove that $\vartriangle XWZ$ is congruent with $\vartriangle  XYV$.

2010 INAMO Shortlist G4 (problem 8)

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.

Raja Oktovin, Pekanbaru

2010 INAMO Shortlist G5

Given an arbitrary triangle $ABC$, with $\angle A = 60^o$ and $AC < AB$. A circle with diameter $BC$, intersects $AB$ and $AC$ at $F$ and $E$, respectively. Lines $BE$ and $CF$ intersect at $D$. Let $\Gamma$ be the circumcircle of $BCD$, where the center of $\Gamma$ is $O$. Circle $\Gamma$ intersects the line $AB$ and the extension of $AC$ at $M$ and $N$, respectively. $MN$ intersects $BC$ at $P$. Prove that points $A$, $P$, $O$ lie on the same line.

2010 INAMO Shortlist G6 (problem 2)

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.

Fajar Yuliawan, Bandung

2010 INAMO Shortlist G7

In triangle $ABC$, find the smallest possible value of $$|(\cot A + \cot B)(\cot B +\cot C)(\cot C + \cot A)|$$

2010 INAMO Shortlist G8

Given an acute triangle $ABC$ and points $D$, $E$, $F$ on sides $BC$, $CA$ and $AB$, respectively. If the lines $DA$, $EB$ and $FC$ are the angle bisectors of triangle $DEF$, prove that the three lines are the altitudes of triangle $ABC$.

2010 INAMO Shortlist G9

Given two circles $\Gamma_1$ and $\Gamma_2$ which intersect at points $A$ and $B$. A line through $A$ intersects $\Gamma_1$ and $\Gamma_2$ at points $C$ and $D$, respectively. Let $M$ be the midpoint of arc $BC$ in $\Gamma_1$ ,which does not contains $A$, and $N$ is the midpoint of the arc $BD$ in $\Gamma_2$, which does not contain $A$. If $K$ is the midpoint of $CD$, prove that $\angle MKN = 90^o.$

2010 INAMO Shortlist G10

Given two circles with one of the centers of the circle is on the other circle. The two circles intersect at two points $C$ and $D$. The line through $D$ intersects the two circles again at $A$ and $B$. Let $H$ be the midpoint of the arc $AC$ that does not contain $D$ and the segment $HD$ intersects circle that does not contain $H$ at point $E$. Show that $E$ is the center of the incircle of the triangle $ACD$.

2010 INAMO Shortlist G11

Given triangle $ABC$ and point $P$ on the circumcircle of triangle $ABC$. Suppose the line $CP$ intersects line $AB$ at point $E$ and line $BP$ intersect line $AC$ at point $F$. Suppose also the perpendicular bisector of $AB$ intersects $AC$ at point $K$ and the perpendicular bisector of $AC$ intersects $AB$ at point $J$. Prove that $$\left( \frac{CE}{BF}\right)^2= \frac{AJ \cdot  JE }{ AK \cdot  KF}$$ .

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 a cyclic 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.

2015 INAMO Shortlist G8

$ABC$ 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 $\Gamma_B$.

2017 INAMO Shortlist G1

Let $ABCD$ be a square with side $1$, with $P$ and $Q$ being points on the sides $AB$ and $BC$, respectively, such that $PB + BQ = 1$. If $PC$ intersects $AQ$ at $E$, prove that the line $DE$ is perpendicular to the line $PQ$.

2017 INAMO Shortlist G2

$ABC$ is an acute triangle with $D, E$ and $F$ being the feet of the altitudes of the triangle $ABC$ on sides $BC, AC$ and $AB$ respectively. Let $P, Q$ and $R$ be the midpoints of $DE, EF$ and $FD$ respectively. Then show that the lines passing through $P, Q$, and $R$ perpendicular on sides $AB, BC$, and $CA$, respectively, intersect at a single point.

2017 INAMO Shortlist G3 (problem 1)

Let $\vartriangle ABC$ be an isosceles triangle with $\angle BAC = 100^o$. Let $D, E$ be points on ray $\overrightarrow{AB}$ so that $BC = AD = BE$. Show that $BC \cdot  DE = BD \cdot CE$

2017 INAMO Shortlist G4

Let $ABC$ be an isosceles triangle with $A$ being the apex, less than $60^o$ with $D$ the point on the side $AC$ , such that $\angle DBC =  \angle BAC$. Let $L_1$ be a line passing through point $A$ and parallel to side $BC$. Let $L_2$ be the perpendicular bisector of side $BD$. $L_1$ and $L_2$ intersect at point $E$. show that the $EC$ is bisected by $AB$.

2017 INAMO Shortlist G5

Let $ABC$ be a triangle with circumcircle $\Gamma$. Let the tangents of circle $\Gamma$ at points $B$ and $C$ intersect at point $D$ . Let $M$ be the point on the side $BC$ such that $\angle BAM = \angle CAD$. Prove that the center of circle $\Gamma$ lies on the line $MD$.

2017 INAMO Shortlist G6

Let A$BC$ be a triangle with $A$ right angle and $D$ is a point on the side $BC$ such that $AD$ is perpendicular to the side $BC$. Let $W_1$ and $W_2$ are the centers of the incircles of the triangles $ABD$ and $ADC$ respectively. Line $W_1W_2$ intersects $AB$ at $X$ and $AC$ at $Y$. Prove that $AX = AD = AY$.

2017 INAMO Shortlist G7

Let $ABC$ be an acute triangle with $E$ and $F$ on sides $AB$ and $AC$ respectively, and $O$ be it's circumcenter. Let $AO$ intersect $BC$ at point $D$. Let the perpendicular from point $D$ on sides $AB$ and $AC$ intersect them at points $M,N$ respectively. Let the perpendicular on the side $BC$ from points $E,F,M,N$ intersect it at points $E',F',M',N'$ respectively. Prove that $A, D, E, F$ lie on the same circle if and only if $E'F '= M'N'$.

2017 INAMO Shortlist G8 (problem 7)

Let $ABCD$ be a convex quadrilateral with shortest side $AB$ and longest side $CD$, and suppose that $AB < CD$. Show that there is a point $E \ne C, D$ on segment $CD$ with the following property:

For all points $P \ne E$ on side $CD$, if we define $O_1$ and $O_2$ to be the circumcenters of $\vartriangle APD$ and $\vartriangle  BPE$ respectively, then the length of $O_1O_2$ does not depend on $P$.

source: http://tomi.or.id/