Taiwan TST 2014-17 31p

geometry problems from Taiwanese Team Selection Tests (TST)
with aops links in the names
(only those not in IMO Shortlist)

2014 - 2017

2014 Taiwan TST Round 1 Quiz 3.1
Let $O_1$, $O_2$ be two circles with radii $R_1$ and $R_2$, and suppose the circles meet at $A$ and $D$. Draw a line $L$ through $D$ intersecting $O_1$, $O_2$ at $B$ and $C$. Now allow the distance between the centers as well as the choice of $L$ to vary. Find the length of $AD$ when the area of $ABC$ is maximized.

2014 Taiwan TST Round 1 Mock IMO 3
Let $ABC$ be a triangle with incenter $I$, and suppose the incircle is tangent to $CA$ and $AB$ at $E$ and $F$. Denote by $G$ and $H$ the reflections of $E$ and $F$ over $I$. Let $Q$ be the intersection of $BC$ with $GH$, and let $M$ be the midpoint of $BC$. Prove that $IQ$ and $IM$ are perpendicular.

2014 Taiwan TST Round 1 Mock IMO 4
Let $ABC$ be an acute triangle and let $D$ be the foot of the $A$-bisector. Moreover, let $M$ be the midpoint of $AD$. The circle $\omega_1$ with diameter $AC$ meets $BM$ at  $E$, while the circle $\omega_2$ with diameter $AB$ meets $CM$ at $F$. Assume that $E$ and $F$ lie inside $ABC$. Prove that $B$, $E$, $F$, $C$ are concyclic.

2014 Taiwan TST Round 2 Quiz 2.1
Let $ABC$ be a triangle with incenter $I$ and circumcenter $O$. A straight line $L$ is parallel to $BC$ and tangent to the incircle. Suppose $L$ intersects $IO$ at $X$, and select $Y$ on $L$ such that $YI$ is perpendicular to $IO$. Prove that $A$, $X$, $O$, $Y$ are cyclic.

by Telv Cohl
2014 Taiwan TST Round 2 Mock IMO P6
Let $P$ be a point inside triangle $ABC$, and suppose lines $AP$, $BP$, $CP$ meet the circumcircle again at $T$, $S$, $R$ (here $T \neq A$, $S \neq B$, $R \neq C$). Let $U$ be any point in the interior of $PT$. A line through $U$ parallel to $AB$ meets $CR$ at $W$, and the line through $U$ parallel to $AC$ meets $BS$ again at $V$. Finally, the line through $B$ parallel to $CP$ and the line through $C$ parallel to $BP$ intersect at point $Q$. Given that $RS$ and $VW$ are parallel, prove that $\angle CAP = \angle BAQ$.

2014 Taiwan TST Round 3 Quiz 2.1
In convex hexagon $ABCDEF$, $AB \parallel DE$, $BC \parallel EF$, $CD \parallel FA$, and $AB+DE = BC+EF = CD+FA.$ The midpoints of sides $AB$, $BC$, $DE$, $EF$ are $A_1$, $B_1$, $D_1$, $E_1$, and segments $A_1D_1$ and $B_1E_1$ meet at $O$. Prove that $\angle D_1OE_1 = \frac12 \angle DEF$.

2014 Taiwan TST Round 3  Mock IMO P3
Let $M$ be any point on the circumcircle of triangle $ABC$. Suppose the tangents from $M$ to the incircle meet $BC$ at two points $X_1$ and $X_2$. Prove that the circumcircle of triangle $MX_1X_2$ intersects the circumcircle of $ABC$ again at the tangency point of the $A$-mixtilinear incircle.

2015 Taiwan TST Round 1 Quiz 2.2
Given any triangle $ABC.$ Let $O_1$ be it's circumcircle, $O_2$ be it's nine point circle, $O_3$ is a circle with orthocenter of $ABC$, $H$, and centroid $G$, be it's diameter. Prove that: $O_1,O_2,O_3$ share axis. (i.e. chose any two of them, their axis will be the same one, if $ABC$ is an obtuse triangle, the three circle share two points.)

2015 Taiwan TST Round 1 Quiz 3.2
Given a triangle $ABC$, $A',B',C'$ are the midpoints of $\overline{BC},\overline{AC},\overline{AB}$, respectively. $B^*,C^*$ lie in $\overline{AC},\overline{AB}$, respectively, such that $\overline{BB^*},\overline{CC^*}$ are the altitudes of the triangle $ABC$. Let $B^{\#},C^{\#}$ be the midpoints of $\overline{BB^*},\overline{CC^*}$, respectively. $\overline{B'B^{\#}}$ and $\overline{C'C^{\#}}$ meet at $K$, and $\overline{AK}$ and $\overline{BC}$ meet at $L$. Prove that $\angle{BAL}=\angle{CAA'}$

2015 Taiwan TST Round 1 Mock IMO P4
Let $ABC$ be a triangle and $M$ be the midpoint of $BC$, and let $AM$ meet the circumcircle of $ABC$ again at $R$. A line passing through $R$ and parallel to $BC$ meet the circumcircle of $ABC$ again at $S$. Let $U$ be the foot from $R$ to $BC$, and $T$ be the reflection of $U$ in $R$. $D$ lies in $BC$ such that $AD$ is an altitude. $N$ is the midpoint of $AD$. Finally let $AS$ and $MN$ meets at $K$. Prove that $AT$ bisector $MK$.

2015 Taiwan TST Round 2 Quiz 1.2
Let $\omega$ be the incircle of triangle $ABC$ and $\omega$ touches $BC$ at $D$. $AD$ meets $\omega$ again at $L$. Let $K$ be $A$-excenter, and $M,N$ be the midpoint of $BC,KM$, respectively. Prove that $B,C,N,L$ are concyclic.

2015 Taiwan TST Round 2 Mock IMO P4
Let $ABC$ be a triangle with incircle $\omega$, incenter $I$ and circumcircle $\Gamma$. Let $D$ be the tangency point of $\omega$ with $BC$, let $M$ be the midpoint of $ID$, and let $A'$ be the diametral opposite of $A$ with respect to $\Gamma$. If we denote $X=A'M\cap \Gamma$, then prove that the circumcircle of triangle $AXD$ is tangent to $BC$.

2015 Taiwan TST Round 3 Quiz 1.2
Let $O$ be the circumcircle of the triangle $ABC$. Two circles $O_1,O_2$ are tangent to each of the circle $O$ and the rays $\overrightarrow{AB},\overrightarrow{AC}$, with $O_1$ interior to $O$, $O_2$ exterior to $O$. The common tangent of $O,O_1$ and the common tangent of $O,O_2$ intersect at the point $X$. Let $M$ be the midpoint of the arc $BC$ (not containing the point $A$) on the circle $O$, and the segment $\overline{AA'}$ be the diameter of $O$. Prove that $X,M$, and $A'$ are collinear.

2015 Taiwan TST Round 3 Quiz 3.2
In a scalene triangle $ABC$ with incenter $I$, the incircle is tangent to sides $CA$ and $AB$ at points $E$ and $F$. The tangents to the circumcircle of triangle $AEF$ at $E$ and $F$ meet at $S$. Lines $EF$ and $BC$ intersect at $T$. Prove that the circle with diameter $ST$ is orthogonal to the nine-point circle of triangle $BIC$.

by Evan Chen
2015 Taiwan TST Round 3 Mock IMO P4
Let $ABC$ be a fixed acute-angled triangle. Consider some points $E$ and $F$ lying on the sides $AC$ and $AB$, respectively, and let $M$ be the midpoint of $EF$. Let the perpendicular bisector of $EF$ intersect the line $BC$ at $K$, and let the perpendicular bisector of $MK$ intersect the lines $AC$ and $AB$ at $S$ and $T$, respectively. If the quadrilateral $KSAT$ is cycle, prove that $\angle{KEF}=\angle{KFE}=\angle{A}$.

2016 Taiwan TST Round 1 Quiz 1.2
Circles $O_1$ and $O_2$ intersects at two points $B$ and $C$, and $BC$ is the diameter of circle $O_1$. Construct a tangent line of circle $O_1$ at $C$ and intersecting circle $O_2$ at another point $A$. We join $AB$ to intersect $O_1$ at point $E$, then join $CE$ and extend it to intersect circle $O_2$ at point $F$. Assume that $H$ is an arbitrary point on the line segment $AF$. We join $HE$ and extend it to intersect circle $O_1$ at point $G$, and join $BG$ and extend it to intersect the extended line of $AC$ at point $D$. Prove that $\frac{AH}{HF}=\frac{AC}{CD}$.

Let $AB$ be a chord on a circle $O$, $M$ be the midpoint of the smaller arc $AB$. From a point $C$ outside the circle $O$ draws two tangents to the circle $O$ at the points $S$ and $T$. Suppose $MS$ intersects with $AB$ at the point $E$, $MT$ intersects with $AB$ at the point $F$. From $E,F$ draw a line perpendicular to $AB$ that intersects with $OS,OT$ at the points $X,Y$, respectively. Draw another line from $C$ which intersects with the circle $O$ at the points $P$ and $Q$. Let $R$ be the intersection point of $MP$ and $AB$. Finally, let $Z$ be the circumcenter of triangle $PQR$. Prove that $X$,$Y$ and $Z$ are collinear.

2016 Taiwan TST Round 2 Quiz 2.1
Let $O$ be the circumcenter of triangle $ABC$, and $\omega$ be the circumcircle of triangle $BOC$. Line $AO$ intersects with circle $\omega$ again at the point $G$. Let $M$ be the midpoint of side $BC$, and the perpendicular bisector of $BC$ meets circle $\omega$ at the points $O$ and $N$. Prove that the midpoint of the segment $AN$ lies on the radical axis of the circumcircle of triangle $OMG$, and the circle whose diameter is $AO$.

Let $AXYZB$ be a convex pentagon inscribed in a semicircle with diameter $AB$, and let $K$ be the foot of the altitude from $Y$ to $AB$. Let $O$ denote the midpoint of $AB$ and $L$ be the intersection of $XZ$ with $YO$. Select a point $M$ on line $KL$ with $MA=MB$ , and finally, let $I$ be the reflection of $O$ across $XZ$. Prove that if quadrilateral $XKOZ$ is cyclic then so is quadrilateral $YOMI$.

by Evan Chen
Let $ABC$ be an acute-angled triangle, with $\angle B \neq \angle C$ . Let $M$ be the midpoint of side $BC$, and $E,F$ be the feet of the altitude from $B,C$ respectively. Denote by $K,L$ the midpoints of segments $ME,MF$, respectively. Suppose $T$ is a point on the line $KL$ such that $AT//BC$. Prove that $TA=TM$ .

2017 Taiwan TST Round 1 Quiz 2.1
Let ${\cal C}_1$ and ${\cal C}_2$ be concentric circles, with ${\cal C}_2$ in the interior of  ${\cal C}_1$. From a point $A$ on ${\cal C}_1$ one draws the tangent $AB$ to ${\cal C}_2$ ($B\in {\cal C}_2$). Let $C$ be the second point of intersection of $AB$ and ${\cal C}_1$, and let $D$ be the midpoint of $AB$. A line passing through $A$ intersects ${\cal C}_2$ at $E$ and $F$ in such a way that the perpendicular bisectors of $DE$ and $CF$ intersect at a point $M$ on $AB$. Find, with proof, the ratio $AM/MC$.

2017 Taiwan TST Round 1 Mock IMO P4
Two line $BC$ and $EF$ are parallel. Let $D$ be a point on segment $BC$ different from $B$,$C$. Let $I$ be the intersection of $BF$ ans $CE$. Denote the circumcircle of $\triangle CDE$ and $\triangle BDF$ as $K$,$L$. Circle $K$,$L$ are tangent with $EF$ at $E$,$F$,respectively. Let $A$ be the other intersection of circle $K$ and $L$. Let $DF$ and circle $K$ intersect again at $Q$, and $DE$ and circle $L$ intersect again at $R$. Let $EQ$ and $FR$ intersect at $M$. Prove that $I$, $A$, $M$ are collinear.

2017 Taiwan TST Round 2 Quiz 2.2
Given a $\triangle ABC$ and three points $D, E, F$ such that $DB = DC,$ $EC = EA,$ $FA = FB,$ $\measuredangle BDC = \measuredangle CEA = \measuredangle AFB.$ Let $\Omega_D$ be the circle with center $D$ passing through $B, C$ and similarly for $\Omega_E, \Omega_F.$ Prove that the radical center of $\Omega_D, \Omega_E, \Omega_F$ lies on the Euler line of $\triangle DEF.$

by Telv Cohl
2017 Taiwan TST Round 2 Quiz 3.2
Let $ABC$ be a triangle such that $BC>AB$, $L$ be the internal angle bisector of $\angle ABC$. Let $P,Q$ be the feet from $A,C$ to $L$, respectively. Suppose $M,N$ are the midpoints of $\overline{AC}$ and $\overline{BC}$, respectively. Let $O$ be the circumcenter of triangle $PQM$, and the circumcircle intersects $AC$ at point $H$. Prove that $O,M,N,H$ are concyclic.

2017 Taiwan TST Round 2 Mock IMO P1
Given a circle and four points $B,C,X,Y$ on it. Assume $A$ is the midpoint of $BC$, and $Z$ is the midpoint of $XY$. Let $L_1,L_2$ be lines perpendicular to $BC$ and pass through $B,C$ respectively. Let the line pass through $X$ and perpendicular to $AX$ intersects $L_1,L_2$ at $X_1,X_2$ respectively. Similarly, let the line pass through $Y$ and perpendicular to $AY$ intersects $L_1,L_2$ at $Y_1,Y_2$ respectively. Assume $X_1Y_2$ intersects $X_2Y_1$ at $P$. Prove that $\angle AZP=90^o.$

by William Chao
2017 Taiwan TST Round 3 Quiz 2.2
$\triangle ABC$ satisfies $\angle A=60^{\circ}$. Call its circumcenter and orthocenter $O, H$, respectively. Let $M$ be a point on the segment $BH$, then choose a point $N$ on the line $CH$ such that $H$ lies between $C, N$, and $\overline{BM}=\overline{CN}$. Find all possible value of $\frac{\overline{MH}+\overline{NH}}{\overline{OH}}$

2017 Taiwan TST Round 3 Mock IMO P3
Given a $\triangle ABC$ with circumcircle $\Gamma.$ Let $A'$ be the antipode of $A$ in $\Gamma$ and $D$ be the point s.t. $\triangle BCD$ is an equilateral triangle ($A$ and $D$ are on the opposite side of $BC$). Let the perpendicular from $A'$ to $A'D$ cuts $CA,$ $AB$ at $E,$ $F,$ resp. and $T$ be the point s.t. $\triangle ETF$ is an isosceles triangle with base $EF$ and base angle $30^{\circ}$ ($A$ and $T$ are on the opposite side of $EF$). Prove that $AT$ passes through the 9-point center of $\triangle ABC.$

by Telv Cohl

misc problems
Given six points $A, B, C, D, E, F$ such that $\triangle BCD \stackrel{+}{\sim} \triangle ECA \stackrel{+}{\sim} \triangle BFA$ and let $I$ be the incenter of $\triangle ABC.$ Prove that the circumcenter of $\triangle AID, \triangle BIE, \triangle CIF$ are collinear.

by Telv Cohl
Let $ABC$ be a triangle with circumcircle $\Omega$, circumcenter $O$ and orthocenter $H$. Let $S$ lie on $\Omega$ and $P$ lie on $BC$ such that $\angle ASP=90^\circ$, line $SH$ intersects the circumcircle of $\triangle APS$ at $X\neq S$. Suppose $OP$ intersects $CA,AB$ at $Q,R$, respectively, $QY,RZ$ are the altitude of $\triangle AQR$. Prove that $X,Y,Z$ are collinear.

by Shuang-Yen Lee
Given a $\triangle ABC$ with circumcircle $\Omega$ and a point $P.$ Let $D$ be the second intersection of $AP$ with $\Omega,$ $E, F$ be the intersection of $BP, CP$ with $CA, AB,$ respectively$,$ $M$ be the intersection of $\odot (AEF)$ with $\Omega,$ $T$ be the intersection of the tangent of $\Omega$ at $B,C$ and $U$ be the second intersection of $TD$ with $\Omega.$ Prove that the reflection of $U$ in $BC$ lies on $\odot (DMP).$

by Telv Cohl
Let $I$ be the incenter of $\triangle{ABC}$ and $l$ the perpendicular bisector of $\overline{AI}$. let $P$ be on the circumcircle of $\triangle{ABC}$, and let $Q$ be the intersection of $l$ and $AP$. Let $R$ be a point on $l$ such that $\angle{IPR}=90^{\circ}$. Let the midline of $\triangle{ABC}$ that is parallel to $BC$ intersects $IQ$ at $M$. Prove that $\angle{AMR}=90^{\circ}$.