Indonesia TST 2007-12, 2015-16,-22 61p

 geometry problems from Indonesian Team Selection Tests (TST) with aops links in the names

 TST 2007-12 , 2015-16, 2022

+ random TST problems posted in aops

collected inside aops here

2007 Indonesia  TST 1/ 1.

2007 Indonesia  TST 1/ 2.

2007 Indonesia  TST 1/ 3.

2007 Indonesia  TST 1/ 4.

2007 Indonesia  TST 1/ 5.

2007 Indonesia  TST 2/ 1.1

Let $P$ be a point in triangle $ABC$, and define $\alpha,\beta,\gamma$ as follows: $$\alpha= \angle BPC- \angle BAC, \quad \beta = \angle CPA - \angle \angle CBA, \quad \gamma = \angle APB- \angle ACB.$$ Prove that $$PA\dfrac{\sin \angle BAC}{\sin \alpha}= PB\dfrac{\sin \angle CBA}{\sin \beta}= PC\dfrac{\sin \angle ACB}{\sin \gamma}.$$  

2007 Indonesia  TST 2/ 2.1

Define an $n$-gon to be lattice if their vertices are lattice points. Prove that inside every lattice convex pentagon, there exists a lattice point.

2007 Indonesia  TST 2/ 3.1

Given triangle $ABC$ and its circumcircle $\Gamma$, let $M$ and $N$ be the midpoints of arcs $BC$ (that does not contain $A$) and $CA$ (that does not contain $B$), repsectively. Let $X$ be a variable point on arc $AB$ that does not contain $C$. Let $O_1$ and $O_2$ be the incenter of triangle $XAC$ and $XBC$, respectively. Let the circumcircle of triangle $XO_1O_2$ meets $\Gamma$ at $Q$.

(a) Prove that $QNO_1$ and $QMO_2$ are similar.

(b) Find the locus of $Q$ as $X$ varies.

2007 Indonesia  TST 2/ 4.2

Let $ABCD$ be a convex quadrtilateral such that $AB$ is not parallel with $CD$. Let $\Gamma_1$ be a circle that passes through $A$ and $B$ and is tangent to $CD$ at $P$. Also, let $\Gamma_2$ be a circle that passes through $C$ and $D$ and is tangent to $AB$ at $Q$. Let the circles $\Gamma_1$ and $\Gamma_2$ intersect at $E$ and $F$. Prove that $EF$ passes through the midpoint of $PQ$ iff $BC \parallel AD$.

2007 Indonesia  TST 2/ 5.1

Let $ABCD$ be a cyclic quadrilateral and $O$ be the intersection of diagonal $AC$ and $BD$. The circumcircles of triangle $ABO$ and the triangle $CDO$ intersect at $K$. Let $L$ be a point such that the triangle $BLC$ is similar to $AKD$ (in that order). Prove that if $BLCK$ is a convex quadrilateral, then it has an incircle.

2008 Indonesia  TST 1/ 1.

2008 Indonesia  TST 1/ 2.

2008 Indonesia  TST 1/ 3.

2008 Indonesia  TST 1/ 4.

2008 Indonesia  TST 1/ 5.

2008 Indonesia  TST 2/ 1.1

Let $ABCD$ be a square with side $20$ and $T_1, T_2, ..., T_{2000}$ are points in $ABCD$ such that no $3$ points in the set $S = \{A, B, C, D, T_1, T_2, ..., T_{2000}\}$ are collinear. Prove that there exists a triangle with vertices in $S$, such that the area is less than $1/10$.

2008 Indonesia  TST 2/ 2.3

Let $\Gamma_1$ and $\Gamma_2$ be two circles that tangents each other at point $N$, with $\Gamma_2$ located inside $\Gamma_1$. Let $A, B, C$ be distinct points on $\Gamma_1$ such that $AB$ and $AC$ tangents $\Gamma_2$ at $D$ and $E$, respectively. Line $ND$ cuts $\Gamma_1$ again at $K$, and line $CK$ intersects line $DE$ at $I$.

(i) Prove that $CK$ is the angle bisector of $\angle ACB$.

(ii) Prove that $IECN$ and $IBDN$ are cyclic quadrilaterals.

2008 Indonesia  TST 2/ 3.1

Let $ABCD$ be a cyclic quadrilateral, and angle bisectors of $\angle BAD$ and $\angle BCD$ meet at point $I$. Show that if $\angle BIC = \angle  IDC$, then $I$ is the incenter of triangle $ABD$.

2008 Indonesia  TST 2/ 4.3

Let $ABCD$ be a convex quadrilateral with $AB$ is not parallel to $CD$ Circle $\Gamma_{1}$ with

center $O_1$ passes through $A$ and $B$, and touches segment $CD$ at $P$. Circle $\Gamma_{2}$ with center $O_2$ passes through $C$ and $D$, and touches segment $AB$ at $Q$. Let $E$ and $F$ be the intersection of circles $\Gamma_{1}$ and $\Gamma_{2}$. Prove that $EF$ bisects segment $PQ$ if and only if $BC$ is parallel to $AD$.

2009 Indonesia  Training 

Let $P$ be an interior point of triangle $ABC$, and let $x,y,z$ denote the distance from $P$ to $BC,AC,$ and $AB$ respectively. Where should $P$ be located to maximize the product $xyz$?

2009 Indonesia  TST  1/ 1.4

Given triangle $ABC$. Let the tangent lines of the circumcircle of $AB$ at $B$ and $C$ meet at $A_0$. Define $B_0$ and $C_0$ similarly.

a) Prove that $AA_0,BB_0,CC_0$ are concurrent.

b) Let $K$ be the point of concurrency. Prove that $KG\parallel BC$ if and only if $2a^2=b^2+c^2$.

2009 Indonesia  TST 1 /2.2  ( IMO 1991, P1, ISL 1991, P6 USS 4)

Given a triangle $\,ABC,\,$ let $\,I\,$ be the center of its inscribed circle. The internal bisectors of the angles $\,A,B,C\,$ meet the opposite sides in $\,A^{\prime },B^{\prime },C^{\prime }\,$ respectively. Prove that

$$\frac {1}{4} < \frac {AI\cdot BI\cdot CI}{AA^{\prime }\cdot BB^{\prime }\cdot CC^{\prime }} \leq \frac {8}{27}.$$

2009 Indonesia  TST  1/ 3.3

Let $ABC$ be an acute triangle with $\angle BAC=60^{\circ}$. Let $P$ be a point in triangle $ABC$ with $\angle APB=\angle BPC=\angle CPA=120^{\circ}$. The foots of perpendicular from $P$ to $BC,CA,AB$ are $X,Y,Z$, respectively. Let $M$ be the midpoint of $YZ$.

a) Prove that $\angle YXZ=60^{\circ}$

b) Prove that $X,P,M$ are collinear.

2009 Indonesia  TST  1/ 4.2

Let $ABC$ be a triangle with $\angle BAC=60^{\circ}$. The incircle of $ABC$ is tangent to $AB$ at $D$. Construct a circle with radius $DA$ and cut the incircle of $ABC$ at $E$. If $AF$ is an altitude, prove that $AE\ge AF$.

2009 Indonesia  TST  1/ 5.4

Let $ABCD$ be a convex quadrilateral. Let $M,N$ be the midpoints of $AB,AD$ respectively. The foot of perpendicular from $M$ to $CD$ is $K$, the foot of perpendicular from $N$ to $BC$ is $L$. Show that if $AC,BD,MK,NL$ are concurrent, then $KLMN$ is a cyclic quadrilateral.

2009 Indonesia  TST 2 / 1.4

Two cirlces $C_1$ and $C_2$, with center $O_1$ and $O_2$ respectively, intersect at $A$ and $B$. Let $O_1$ lies on $C_2$. A line $l$ passes through $O_1$ but does not pass through $O_2$. Let $P$ and $Q$ be the projection of $A$ and $B$ respectively on the line $l$ and let $M$ be the midpoint of $\overline{AB}$. Prove that $MPQ$ is an isoceles triangle.

2009 Indonesia  TST 2/ 2.3

Let $C_1$ be a circle and $P$ be a fixed point outside the circle $C_1$. Quadrilateral $ABCD$ lies on the circle $C_1$ such that rays $AB$ and $CD$ intersect at $P$. Let $E$ be the intersection of $AC$ and $BD$. 

(a) Prove that the circumcircle of triangle $ADE$ and the circumcircle of triangle $BEC$ pass through a fixed point.

(b) Find the the locus of point $E$.

2009 Indonesia  TST 2/ 3.4

Given triangle $ABC$ with $AB>AC$. $l$ is tangent line of the circumcircle of triangle $ABC$ at $A$. A circle with center $A$ and radius $AC$, intersect $AB$ at $D$ and $l$ at $E$ and $F$. Prove that the lines $DE$ and $DF$ pass through the incenter and excenter of triangle $ABC$.

2009 Indonesia  TST  2/ 4.3

Let $ABC$ be an isoceles triangle with $AC= BC$. A point $P$ lies inside $ABC$ such that $$\angle PAB = \angle PBC, \angle PAC = \angle PCB.$$ Let $M$ be the midpoint of $AB$ and $K$ be the intersection of $BP$ and $AC$. Prove that $AP$ and $PK$ trisect $\angle MPC$.

2009 Indonesia  TST 2/ 5.1

Let $ABC$ be a triangle. A circle $P$ is internally tangent to the circumcircle of triangle $ABC$ at $A$ and tangent to $BC$ at $D$. Let $AD$ meets the circumcircle of $ABC$ agin at $Q$. Let $O$ be the circumcenter of triangle $ABC$. If the line $AO$ bisects $\angle DAC$, prove that the circle centered at $Q$ passing through $B$, circle $P$, and the perpendicular line of $AD$ from $B$, are all concurrent.

2010 Indonesia  TST 1/ 1.4

Let $ABC$ be a non-obtuse triangle with $CH$ and $CM$ are the altitude and median, respectively. The angle bisector of $\angle BAC$ intersects $CH$ and $CM$ at $P$ and $Q$, respectively. Assume that $$\angle ABP=\angle PBQ=\angle QBC,$$  

(a) prove that $ABC$ is a right-angled triangle, and

(b) calculate $\dfrac{BP}{CH}$.

2010 Indonesia  TST 1/ 2.1

Let $ABCD$ be a trapezoid such that $AB \parallel CD$ and assume that there are points $E$ on the line outside the segment $BC$ and $F$ on the segment $AD$ such that $\angle DAE = \angle CBF$. Let $I,J,K$ respectively be the intersection of line $EF$ and line $CD$, the intersection of line $EF$ and line $AB$, and the midpoint of segment $EF$. Prove that $K$ is on the circumcircle of triangle $CDJ$ if and only if $I$ is on the circumcircle of triangle $ABK$.

2010 Indonesia  TST 1/ 3.4

Let $ABC$ be an acute-angled triangle such that there exist points $D,E,F$ on side $BC,CA,AB$, respectively such that the inradii of triangle $AEF,BDF,CDE$ are all equal to $r_0$. If the inradii of triangle $DEF$ and $ABC$ are $r$ and $R$, respectively, prove that $$r+r_0=R.$$

2010 Indonesia  TST 1/ 4.2

Circles $\Gamma_1$ and $\Gamma_2$ are internally tangent to circle $\Gamma$ at $P$ and $Q$, respectively. Let $P_1$ and $Q_1$ are on $\Gamma_1$ and $\Gamma_2$ respectively such that $P_1Q_1$ is the common tangent of $P_1$ and $Q_1$. Assume that $\Gamma_1$ and $\Gamma_2$ intersect at $R$ and $R_1$. Define $O_1,O_2,O_3$ as the intersection of $PQ$ and $P_1Q_1$, the intersection of $PR$ and $P_1R_1$, and the intersection $QR$ and $Q_1R_1$. Prove that the points $O_1,O_2,O_3$ are collinear.

2010 Indonesia  TST 1/ 5.1

Is there a triangle with angles in ratio of $1: 2: 4$ and the length of its sides are integers with at least one of them is a prime number?

2010 Indonesia  TST 2/ 1.2 (IMO ISL 2003 G4)

Given circles $\Gamma_1, \Gamma_2, \Gamma_3, \Gamma_4$ such that $\Gamma_1$ and $\Gamma_3$ are externally tangent at $P$, and $\Gamma_2$ and $\Gamma_4$ are externally tangent at $P$. If $\Gamma_1$ and $\Gamma_2$ intersects each other again at $A$, $\Gamma_2$ and $\Gamma_3$ at $B$, $\Gamma_3$ and $\Gamma_4$ and $C$, and $\Gamma_4$ and $\Gamma_1$ at $D$. Prove that

$$\frac{AB \cdot  BC}{CD \cdot  DA} =\frac{P B^2}{P C^2}$$

2010 Indonesia  TST 2/ 2.3 (ITAMO 2004)

Two parallel lines $r,s$ and two points $P \in r$ and $Q \in s$ are given in a plane. Consider all pairs of circles $(C_P, C_Q)$ in that plane such that $C_P$ touches $r$ at $P$ and $C_Q$ touches $s$ at $Q$ and which touch each other externally at some point $T$. Find the locus of $T$.

2010 Indonesia  TST 2/ 3.3

Given acute triangle $ABC$ with circumcenter $O$ and the center of nine-point circle $N$. Point $N_1$ are given such that $\angle NAB = \angle N_1AC$ and $\angle NBC = \angle N_1BA$. Perpendicular bisector of segment $OA$ intersects the line $BC$ at $A_1$. Analogously define $B_1$ and $C_1$. Show that all three points $A_1,B_1,C_1$ are collinear at a line that is perpendicular to $ON_1$.

2010 Indonesia  TST 2/ 4.3

Given a non-isosceles triangle $ABC$ with incircle $k$ with center $S$. $k$ touches the side $BC,CA,AB$ at $P,Q,R$  respectively. The line $QR$ and line $BC$ intersect at $M$. A circle which passes through $B$ and $C$ touches $k$ at $N$. The circumcircle of triangle $MNP$ intersects $AP$ at $L$. Prove that $S,L,M$ are collinear.

2010 Indonesia  TST 2/ 5.3

Let $ABCD$ be a convex quadrilateral with $AB$ is not parallel to $CD$. Circle $\omega_1$ with center $O_1$ passes through $A$ and $B$, and touches segment $CD$ at $P$. Circle $\omega_2$ with center $O_2$ passes through $C$ and $D$, and touches segment $AB$ at $Q$. Let $E$ and $F$ be the intersection of circles $\omega_1$ and $\omega_2$. Prove that $EF$ bisects segment $PQ$ if and only if $BC$ is parallel to $AD$.

2011 Indonesia  TST 1/ 1.

2011 Indonesia  TST 1/ 2.

2011 Indonesia  TST 1/ 3.

2011 Indonesia  TST 1/ 4.

2011 Indonesia  TST 1/ 5.

2011 Indonesia  TST 2/ 1.3  (IMO ISL 2001, G4)

Let $M$ be a point in the interior of triangle $ABC$. Let $A'$ lie on $BC$ with $MA'$ perpendicular to $BC$. Define $B'$ on $CA$ and $C'$ on $AB$ similarly. Define

$p(M) = \frac{MA' \cdot MB' \cdot MC'}{MA \cdot MB \cdot MC}.$

Determine, with proof, the location of $M$ such that $p(M)$ is maximal. Let $\mu(ABC)$ denote this maximum value. For which triangles $ABC$ is the value of $\mu(ABC)$ maximal?

2011 Indonesia  TST 2/ 2.3

Circle $\omega$ is inscribed in quadrilateral $ABCD$ such that $AB$ and $CD$ are not parallel and intersect at point $O.$ Circle $\omega_1$ touches the side $BC$ at $K$ and touches line $AB$ and $CD$ at points which are located outside quadrilateral $ABCD;$ circle $\omega_2$ touches side $AD$ at $L$ and touches line $AB$ and $CD$ at points which are located outside quadrilateral $ABCD.$ If $O,K,$ and $L$ are collinear$,$ then show that the midpoint of side $BC,AD,$ and the center of circle $\omega$ are also collinear.

2011 Indonesia  TST 2/ 3.2

On a line $\ell$ there exists $3$ points $A, B$, and $C$ where $B$ is located between $A$ and $C$. Let $\Gamma_1, \Gamma_2, \Gamma_3$ be circles with $AC, AB$, and $BC$ as diameter respectively; $BD$ is a segment, perpendicular to $\ell$ with $D$ on $\Gamma_1$. Circles $\Gamma_4, \Gamma_5, \Gamma_6$ and $\Gamma_7$ satisfies the following conditions:

$\bullet$ $\Gamma_4$ touches $\Gamma_1, \Gamma_2$, and$BD$.

$\bullet$ $\Gamma_5$ touches $\Gamma_1, \Gamma_3$, and $BD$.

$\bullet$ $\Gamma_6$ touches $\Gamma_1$ internally, and touches $\Gamma_2$ and $\Gamma_3$ externally.

$\bullet$ $\Gamma_7$ passes through $B$ and the tangent points of $\Gamma_2$ with $\Gamma_6$, and $\Gamma_3$ with $\Gamma_6$.

Show that the circles $\Gamma_4, \Gamma_5$, and $\Gamma_7$ are congruent.

2011 Indonesia  TST 2/ 4.3

Let $\Gamma$ is a circle with diameter $AB$. Let $\ell$ be the tangent of $\Gamma$ at $A$, and $m$ be the tangent of $\Gamma$ through $B$. Let $C$ be a point on $\ell$, $C \ne A$, and let $q_1$ and $q_2$ be two lines that passes through $C$. If $q_i$ cuts $\Gamma$ at $D_i$ and $E_i$ ($D_i$ is located between $C$ and $E_i$) for $i = 1, 2$. The lines $AD_1, AD_2, AE_1, AE_2$ intersects $m$ at $M_1, M_2, N_1, N_2$ respectively. Prove that $M_1M_2 = N_1N_2$.

2011 Indonesia  TST 2/ 5.3

Let $ABC$ and $PQR$ be two triangles such that

(a) $P$ is the mid-point of $BC$ and $A$ is the midpoint of $QR$.

(b) $QR$ bisects $\angle BAC$ and $BC$ bisects $\angle QPR$

Prove that $AB+AC=PQ+PR$.

2012 Indonesia  TST 1/ 1.

2012 Indonesia  TST 1/ 2.

2012 Indonesia  TST 1/ 3.

2012 Indonesia  TST 1/ 4.

2012 Indonesia  TST 1/ 5.

2012 Indonesia  TST 2/ 1.3 

Given a convex quadrilateral $ABCD$, let $P$ and $Q$ be points on $BC$ and $CD$ respectively such that $\angle BAP = \angle DAQ$. Prove that the triangles $ABP$ and $ADQ$ have the same area if the line connecting their orthocenters is perpendicular to $AC$.

2012 Indonesia  TST 2/ 2.2

Let $ABC$ be a triangle, and its incenter touches the sides $BC,CA,AB$ at $D,E,F$ respectively. Let $AD$ intersects the incircle of $ABC$ at $M$ distinct from $D$. Let $DF$ intersects the circumcircle of $CDM$ at $N$ distinct from $D$. Let $CN$ intersects $AB$ at $G$. Prove that $EC = 3GF$.

2012 Indonesia  TST 2/ 3.3

Given a cyclic quadrilateral $ABCD$ with the circumcenter $O$, with $BC$ and $AD$ not parallel. Let $P$ be the intersection of $AC$ and $BD$. Let $E$ be the intersection of the rays $AB$ and $DC$. Let $I$ be the incenter of $EBC$ and the incircle of $EBC$ touches $BC$ at $T_1$. Let $J$ be the excenter of $EAD$ that touches $AD$ and the excircle of $EAD$ that touches $AD$ touches $AD$ at $T_2$. Let $Q$ be the intersection between $IT_1$ and $JT_2$. Prove that $O,P,Q$ are collinear.

2012 Indonesia  TST 2/ 4.2

Let $\omega$ be a circle with center $O$, and let $l$ be a line not intersecting $\omega$. $E$ is a point on $l$ such that $OE$ is perpendicular with $l$. Let $M$ be an arbitrary point on $M$ different from $E$. Let $A$ and $B$ be distinct points on the circle $\omega$ such that $MA$ and $MB$ are tangents to $\omega$. Let $C$ and $D$ be the foot of perpendiculars from $E$ to $MA$ and $MB$ respectively. Let $F$ be the intersection of $CD$ and $OE$. As $M$ moves, determine the locus of $F$.

2012 Indonesia  TST 2.5/ 1.3

The incircle of a triangle $ABC$ is tangent to the sides $AB,AC$ at $M,N$ respectively. Suppose $P$ is the intersection between $MN$ and the bisector of $\angle ABC$. Prove that $BP$ and $CP$ are perpendicular. 

2012 Indonesia  TST 2.5/ 2.3

The [i]cross[/i] of a convex $n$-gon is the quadratic mean of the lengths between the possible pairs of vertices. For example, the cross of a $3 \times 4$ rectangle is $\sqrt{ \dfrac{3^2 + 3^2 + 4^2 + 4^2 + 5^2 + 5^2}{6} } = \dfrac{5}{3} \sqrt{6}$.

Suppose $S$ is a dodecagon ($12$-gon) inscribed in a unit circle. Find the greatest possible cross of $S$.

2012 Indonesia  TST 2.5/ 3.2

Suppose $l(M, XYZ)$ is a Simson line of the triangle $XYZ$ that passes through $M$.

Suppose $ABCDEF$ is a cyclic hexagon such that $l(A, BDF), l(B, ACE), l(D, ABF), l(E, ABC)$ intersect at a single point. Prove that $CDEF$ is a rectangle.

2012 Indonesia  TST 2.5/ 4.3

Let $P_1P_2\ldots P_n$ be an $n$-gon such that for all $i,j \in \{1,2,\ldots,n\}$ where $i \neq j$, there exists $k \neq i,j$ such that $\angle P_iP_kP_j = 60^\circ$. Prove that $n=3$.

2013 Indonesia  TST 1/ 1.

2013 Indonesia  TST 1/ 2.

2013 Indonesia  TST 1/ 3.

2013 Indonesia  TST 1/ 4.

2013 Indonesia  TST 1/ 5.

2013 Indonesia  TST 2/ 1.

2013 Indonesia  TST 2/ 2.

2013 Indonesia  TST 2/ 3.

2013 Indonesia  TST 2/ 4.

2013 Indonesia  TST 2/ 5.

2014 Indonesia  TST 1/ 1.

2014 Indonesia  TST 1/ 2.

2014 Indonesia  TST 1/ 3.

2014 Indonesia  TST 1/ 4.

2014 Indonesia  TST 1/ 5.

2014 Indonesia  TST 2/ 1.

2014 Indonesia  TST 2/ 2.

2014 Indonesia  TST 2/ 3.

2014 Indonesia  TST 2/ 4.

2014 Indonesia  TST 2/ 5.

2014 Indonesia  TST

Let $\triangle ABC$. $AP$ and $CQ$ are the altitude. $M$ is the midpoint of $AC$. Point $E$ and $F$ is on $BM$ such that $\angle APE=\angle BAC$ and $\angle ACB=\angle CQF$. Let $H$ be the orthocentre. If $E$ and $F$ are inside $\triangle ABP$ and $\triangle CBQ$ respectively, prove that $AE,CF,BH$ are collinear.

2014 Indonesia  TST

Let $\triangle ABC$, the perpendicular from $A$ meet $BC$ at $E$. $D$ is the midpoint of $BC$. line $l$ is the angular bisector of $\angle BAC$. Let point $P$ be the intersection of perpendicular bisector $ED$ and the line perpendicular from $D$ to $l$. Prove that $P$ is in the nine-point circle of triangle $\triangle ABC$

2015 Indonesia  TST 1/ 1.1

Let $H$ be the orthocenter point of an acute triangle $ABC,$ with altitudes $AP$ and $CQ$. In the median $BM$, the points $E$ and $F$ are chosen such that $\angle APE=\angle BAC$ and $\angle CQF=\angle BCA$, where point $E$ lies in triangle $APB$, and point $F$ lies in triangle $CQB$. Prove that the lines $AE$, $CF$, and $BH$ intersect at one point.

2015 Indonesia  TST 1/ 1.2

Given a triangle $ABC$ with $AB\ne AC$. Let point $D$ be the midpoint of side $BC$ and $E$ be the foot of the altitude drawn from $D$ perpendicular on the bisector of the angle $BAC$. Prove that $E$ lies on the circle of nine points of triangle $ABC$.

2015 Indonesia  TST 1/ 1.3

The circles $\Gamma_1$ and $\Gamma_2$ intersect at $C$ and $D$. The line $\ell$ through $D$ intersects $\Gamma_1$ and $\Gamma_2$ at points $A$ and $B$, respectively. The points $P$ and $Q$ lie on $\Gamma_1$ and $\Gamma_2$, respectively. The lines $PD$ and $AC$ intersect at $M$, the lines $QD$ and $BC$ intersect at N. Suppose $O$ is the center of the circumcircle of triangle $ABC$, prove that $OD$ is perpendicular to $MN$ if and only if $P, Q, M$, and $N$ lie in the same circle.

2015 Indonesia  TST 1/ 1.4

A line $g$ intersects lines $BC$, $CA$ and $AB$ at $A_1$, $B_1$ and $C_1$, respectively. A line $h$, with $h\ne g$, intersects lines $BC$,$CA$ and $AB$ at $A_2$,$B_2$ and $C_2$, respectively. Next, suppose $A_3$, $B_3$ and $C_3$ are the intersections of $BC$ with $B_1C_2$, $CA$ with $C_1A_2$ and $AB$ with $A_1B_2$, respectively. Should $A_3$, $B_3$ and $C_3$ lie in one line?

2015 Indonesia  TST 2/ 1.

2015 Indonesia  TST 2/ 2.

2015 Indonesia  TST 2/ 3.

2015 Indonesia  TST 2/ 4.

2015 Indonesia  TST 2/ 5.

2016 Indonesia  TST 1/ 1.2

Given a parallelogram $ABCD$ with base $AB$ . Let $S$ be the point of intersection between the two diagonals and point $I$ is the center of the incircle in triangle $ABD$, and point $T$ is the touchpoint of the incircle in triangle ABD with diagonal $BD$. Prove that IS is parallel to $CT$.

2016 Indonesia  TST 1/ 2.2

In the quadrilateral $ABCD$, the point $F$ is the intersection of the two diagonals. Let $P, Q$, and $M$ be the midpoints of $BF,$ $CF$, and $BC$, respectively. Let $E$ be the intersection of rays $BA$ and $CD$ and $N$ be the midpoint of $EF$. Prove that the quadrilateral $AEDF$ is congruent with the quadrilateral $QNPM$ .

2016 Indonesia  TST 1/ 3.3

Given a cyclic quadrilateral $ABCD$, where $AD = BD$. Let $M$ be the intersection between segment $AC$ and segment $BD$. Let $I$ be the incenter of the circle in the triangle $BCM$. Let $N$ be the second point of intersection between $AC$ and the circumcircle of the triangle $BMI$. Prove that $AN \cdot NC=CD\cdot  BN$.

2016 Indonesia  TST 1/ 4..4

In a scalene triangle $ABC$, the points $D$ and $E$ lie on the lines $AB$ and $AC$, respectively, such that the circumcircles of triangles $ACD$ and $ABE$ are tangent to the line $BC$. Let $F$ be the intersection of $BC$ and $DE$. Prove that $AF$ is perpendicular to the Euler line of triangle $ABC$.

2016 Indonesia  TST 2/ 1.

2016 Indonesia  TST 2/ 2.

2016 Indonesia  TST 2/ 3.

2016 Indonesia  TST 2/ 4.

2016 Indonesia  TST 2/ 5.

2016 Indonesia  TST

Given a non isosceles $\triangle ABC$ with altitude $AD$. Let $E$ and $F$ be the midpoints of $AC$ and $AB$. For each point $P$ on the plane, let $Y$ and $Z$ be the point on the plane such that  $E$ and $F$ be the midpoints of $PY$ and $PZ$, respectively. Let $P'$ be the midpoint $DP$ and $M$ be the intersection of $BY$ and $CZ$. Prove that the line $MP'$ passes a fixed point.

2016 Indonesia  TST

Let $I$ be the incenter of an acute $\triangle ABC$. Let $\Gamma$ be the $C$ excircle. Let $\Gamma$ touches $AB$ at the point $D$ and let $DI$ meet $\Gamma$ at the point $S$ ($S\neq D$). Prove that $DI$ bisects the $\angle ASB$.

2016 Indonesia  TST

Given non isosceles $\triangle ABC$ and the altitude $AD$. The point $P$, $P\neq A$ and $P\neq D$ on the segment $AD$. Let $E$ be the intersection of $BP$, $AC$ and let $F$ be the intersection of $CP$,$AB$. Assume that $$BF\cdot CD=BD\cdot CE.$$ Let $G$ be the intersection of cimcurcircle $DEF$ and the segment $BC$ such that the point $G$ is located between $D$ and $C$. Prove that $AB+AC=BC+AE$ if and only if $BF+CG=CE+BD$.

2017 Indonesia  TST 1/ 1.

2017 Indonesia  TST 1/ 2.

2017 Indonesia  TST 1/ 3.

2017 Indonesia  TST 1/ 4.

2017 Indonesia  TST 1/ 5.

2017 Indonesia  TST 2/ 1.

2017 Indonesia  TST 2/ 2.

2017 Indonesia  TST 2/ 3.

2017 Indonesia  TST 2/ 4.

2017 Indonesia  TST 2/ 5.

2018 Indonesia  TST 1/ 1.

2018 Indonesia  TST 1/ 2.

2018 Indonesia  TST 1/ 3.

2018 Indonesia  TST 1/ 4.

2018 Indonesia  TST 1/ 5.

2018 Indonesia  TST 2/ 1.

2018 Indonesia  TST 2/ 2.

2018 Indonesia  TST 2/ 3.

2018 Indonesia  TST 2/ 4.

2018 Indonesia  TST 2/ 5.

2018 Indonesia CSMO TST 4

Given an acute triangle $ABC$ where $AB<AC$. The excircle $w_B$ and $w_C$ opposite $B$ and $C$ of $\triangle ABC$ centered at $B_1$ and $C_1$ respectively. Let $D$ be the midpoint of $B_1 C_1$, $E$ be the intersection of the lines $AB$ and $CD$ and $F$ be the intersection of the lines $AC$ and $BD$. Suppose $M$ is the midpoint of $BC$, prove that $EF$, the angle bisector of $\angle BAC$ and the circumcircle of $\triangle B_1C_1 M$ are concurrent.

2019 Indonesia  TST 1/ 1.

2019 Indonesia  TST 1/ 2.

2019 Indonesia  TST 1/ 3.

2019 Indonesia  TST 1/ 4.

2019 Indonesia  TST 1/ 5.

2019 Indonesia  TST 2/ 1.

2019 Indonesia  TST 2/ 2.

2019 Indonesia  TST 2/ 3.

2019 Indonesia  TST 2/ 4.

2019 Indonesia  TST 2/ 5.

2020 Indonesia  TST 1/ 1.

2020 Indonesia  TST 1/ 2.3

Given an acute triangle $ABC$ with $AB \not= BC$. The angle bisector of $\angle BAC$ intersects $BC$ and $(ABC)$ at $D$ and $E$ respectively. $P$ is on the angle bisector of $\angle BAC$ such that $P$ is between $A$ and $D$. $M$ and $N$ are on $(ABC)$ such that $\angle ANP = \angle EMP = 90^{\circ}$. If $(PMN)$ and $(DEM)$ intersects the second time at point $Q$.

Prove that $B,Q,C$ are collinear if and only if $P$ is the incenter of $\triangle ABC$.

2020 Indonesia  TST 1/ 3.

2020 Indonesia  TST 1/ 4.

2020 Indonesia  TST 1/ 5.

2020 Indonesia  TST 2/ 1.

2020 Indonesia  TST 2/ 2.

2020 Indonesia  TST 2/ 3.

2020 Indonesia  TST 2/ 4.

2020 Indonesia  TST 2/ 5.

2021 Indonesia  TST 1/ 1.

2021 Indonesia  TST 1/ 2.

2021 Indonesia  TST 1/ 3.

2021 Indonesia  TST 1/ 4.

Does there exist a rectangle that could be partitioned into an equilateral hexagon of side length $1$ and several right-triangles of size $1 , \sqrt{3}, 2$.

2021 Indonesia  TST 1/ 5.3

The circle $k_1$ and the circle $k_2$ intersect on points $A$ and $B$, such that $k_1$ goes through the centre point of $k_2$, which is $O$. Line $P$ intersects $k_1$ on points $O$ and $K$, moreover it intersects $k_2$ at points $L$ and $M$, such that $L$ is situated between $K$ and $O$. The point $P$ lies on $AB$ such that $LP \perp AB$. Prove that the line $KP$ is parallel to the median of $\bigtriangleup{ABM}$ which goes through $M$.

2022 Indonesia  TST 1.G

Given an acute triangle $ABC$. with $H$ as its orthocenter, lines $\ell_1$ and $\ell_2$ go through $H$ and are perpendicular to each other. Line $\ell_1$ cuts $BC$ and the extension of $AB$ on $D$ and $Z$ respectively. Whereas line $\ell_2$ cuts $BC$ and the extension of $AC$ on $E$ and $X$ respectively. If the line through $D$ and parallel to $AC$ and the line through $E$ parallel to $AB$ intersects at $Y$, prove that $X,Y,Z$ are collinear.

2022 Indonesia  TST 2.G

Given that $ABC$ is a triangle, points $A_i, B_i, C_i \hspace{0.15cm} (i \in \{1,2,3\})$ and $O_A, O_B, O_C$ satisfy the following criteria:

a) $ABB_1A_2, BCC_1B_2, CAA_1C_2$ are rectangles not containing any interior points of the triangle $ABC$,

b) $\displaystyle \frac{AB}{BB_1} = \frac{BC}{CC_1} = \frac{CA}{AA_1}$,

c) $AA_1A_3A_2, BB_1B_3B_2, CC_1C_3C_2$ are parallelograms, and

d) $O_A$ is the centroid of rectangle $BCC_1B_2$, $O_B$ is the centroid of rectangle $CAA_1C_2$, and $O_C$ is the centroid of rectangle $ABB_1A_2$.

Prove that $A_3O_A, B_3O_B,$ and $C_3O_C$ concur at a point.

2022 Indonesia  TST 3.G

Let $AB$ be the diameter of circle $\Gamma$ centred at $O$. Point $C$ lies on ray $\overrightarrow{AB}$. The line through $C$ cuts circle $\Gamma$ at $D$ and $E$, with point $D$ being closer to $C$ than $E$ is. $OF$ is the diameter of the circumcircle of triangle $BOD$. Next, construct $CF$, cutting the circumcircle of triangle $BOD$ at $G$. Prove that $O,A,E,G$ are concyclic.

2022 Indonesia  TST 4.G

In a nonisosceles triangle $ABC$, point $I$ is its incentre and $\Gamma$ is its circumcircle. Points $E$ and $D$ lie on $\Gamma$ and the circumcircle of triangle $BIC$ respectively such that $AE$ and $ID$ are both perpendicular to $BC$. Let $M$ be the midpoint of $BC$, $N$ be the midpoint of arc $BC$ on $\Gamma$ containing $A$, $F$ is the point of tangency of the $A-$excircle on $BC$, and $G$ is the intersection of line $DE$ with $\Gamma$. Prove that lines $GM$ and $NF$ intersect at a point located on $\Gamma$.

source: https://sites.google.com/site/imoidn/