geometry problems from Thailand Team Selection Tests (TST) and October Camps with aops links in the names
(only those not in IMO Shortlist)
TSTS only collected inside aops here
TST 2011 - 2020
Let $\vartriangle ABC$ be an acute-angled triangle with $AB > AC$, and let $AD$ be its altitude from $A$ to $BC$. Let the circle $\Omega$ with diameter $BC$ intersect $AC$ at $E$, and let the tangents to $\Omega$ at $B$ and $E$ meet at $X$. Suppose that the lines $AX$ and $BC$ meet at $Y$ . Prove that $\frac{1}{DC} -\frac{1}{BD}=\frac{2}{DY}$.
Let $\vartriangle ABC$ be a triangle with $AB \le AC$ and let $P$ be an interior point lying on the angle bisector of $\angle BAC$. Let $D$ and $E$ be points on segments $P C$ and $P B$, respectively, so that $\angle PBD = \angle PCE$. Lines $BD$ and $AC$ intersect at $X$, and lines $CE$ and $AB$ intersect at $Y$ . Prove that $BX \le CY$.
2011 Thailand TST 3.2 (also 2016 TST 10.3)
Let $\vartriangle ABC$ be an acute triangle with orthocenter $H$. Points $Y$ and $Z$ are chosen on $AC, AB$ such that $\angle HYC = \angle HZB = 60^o$. Let $U$ be the circumcenter of $\vartriangle HYZ$, and let $N$ be the nine-point center of $\vartriangle ABC$. Prove that $A, U, N$ are collinear.
A triangle $\vartriangle ABC$ is given, with circumcircle $\Gamma$ and an incircle of unit radius. Let $\Gamma_A$ be the circle tangent to the lines $AB$ and $AC$ and the circle $\Gamma$ internally. Define $\Gamma_B$ and $\Gamma_C$ similarly. Let $R_A, R_B$ and $R_C$ be the radii of $\Gamma_A, \Gamma_B$ and $ \Gamma_C$, respectively. Prove that $R_A + R_B + R_C \ge 4$.
A circle $\Omega$ through the vertex $A$ of $\vartriangle ABC$ is tangent to segment $BC$ at $D$. Let $E$ be the projection from $D$ onto $AC$, and let $F$ be a point on $\Omega$ lying on the opposite side of $D$ with respect to line $EC$ so that $\angle EFC = 2 \angle ECD$. Suppose that $CF$ meets $\Omega$ again at $G$. Prove that $AG$ is parallel to $BC$.
Let $D$ and $E$ be interior points of sides $AB$ and $AC$ of a triangle $ABC$, respectively, sonthat the line $DE$ intersects the extension of $BC$ at $F$. Prove that $\vartriangle ADE$ is similar to the triangle whose vertices are the circumcenters of $\vartriangle ABC$, $\vartriangle DFB$ and $\vartriangle EFC$.
Let $I$ be the incenter of $\vartriangle ABC$ whose incircle touches sides $AB, BC$ and $CA$ at points $X, Y$ and $Z$, respectively. Let $H'$ be the orthocenter of $\vartriangle XYZ$. Suppose that lines $IH'$ and $BC$ intersect at $D$ and lines $AD$ and $XH'$ intersect at $P$. Prove that $P Y = Y X$.
Let $I$ be the incenter of 4ABC, and let $\ell$ be a tangent line to the incircle of $\vartriangle ABC$, which is not one of its sides. The line $\ell$ meets the sides $AB, BC$ and the extension of $CA$ at points $X, Y$ and $Z$ respectively. Let $AY$ intersect $CX$ at $P$, and let lines $IP$ and $BZ$ intersect at $Q$. Prove that if $A, C, Y, X$ are concyclic, then $ZI^2 = ZQ \cdot ZB$.
Let $A_1, A_2, A_3, A_4$ be four points in the plane, chosen so that $A_4$ is the centroid of $\vartriangle A_1A_2A_3$. Find a point $A_5$ on the plane that maximizes the ratio of areas $$\frac{\underset{1\le i<j<k \le 5}{min}A_iA_jA_k}{\underset{1\le i<j<k \le 5}{max}A_iA_jA_k}$$
Let $M$ be the midpoint of side $BC$ of a triangle $\vartriangle ABC$. Let $S$ and $T$ be points on $BM$ and $CM$, respectively, such that $SM = MT$. Let $P$ and $Q$ be points on $AT$ and $AS$, respectively, such that $\angle PST = \angle BAS$ and $\angle QTS = \angle CAT$. Suppose that lines $BQ$ and $PC$ intersect at a point $R$. Show that $RM \perp BC$.
Let $\vartriangle ABC$ be a non-equilateral triangle, and let $I$ and $O$ be its incenter and circumcenter, respectively. Prove that $\angle AIO \le 90^o$ if and only if $2BC \le AB + AC$, and that both inequalities become equalities simultaneously.
Let $\vartriangle ABC$ be an acute-angled triangle with circumcircle $\odot O$ centered at $O$. Let $P$ be a point inside $\vartriangle ABC$ and let lines $AP, BP, CP$ intersect $\odot O$ again at $A', B', C'$, respectively. Let $B_c, C_b, A_b, B_a, C_a$ and $A_c$, respectively, denote the circumcenters of $\vartriangle PBC', \vartriangle PCB', \vartriangle PAB', \vartriangle PBA', \vartriangle PCA'$ and $\vartriangle PAC'$. Prove that lines $B_cC_b, A_bB_a$ and $C_aA_c$ concur at the midpoint of segment $OP$.
Does there exist a convex equilateral $n$-gon whose vertices all lie on the curve $y = x^2$ ,where
a) $n = 2012$,
b) $n = 2013$ ?
In $\vartriangle ABC$, the incircle centered at $I$ touches sides $BC, CA$ and $AB$ at $D, E$ and $F$, respectively. A circle $k$ cuts segments $EF, FD$ and $DE$ at $\{X_1, X_2\}, \{X_3, X_4\}$ and $\{X_5, X_6\}$,respectively. Suppose that lines $X_1X_4, X_2X_5$ and $X_3X_6$ all pass through the center $G$ of $k$.
a) Prove that points $A, D$ and $G$ are collinear.
b) Let the two lines through $G$ parallel to $DE$ and $DF$ intersect line $BC$ at $P$ and $Q$. Prove that $IP = IQ$
Let $M$ be the midpoint of arc $BC$ not containing $A$ on the circumcircle of a given triangle $\vartriangle ABC$. Let $I$ be the incenter of $\vartriangle ABC$, and let $E$ and $F$ be the projections of $I$ onto $MB$ and $MC$, respectively. Prove that $IE + IF \le AM$.
Let $O$ and $I$ be the circumcenter and incenter of a scalene triangle $\vartriangle ABC$ respectively. The incircle of $\vartriangle ABC$ touches sides $BC, CA$ and $AB$ at $D, E$ and $F$, respectively. Let $AP, BQ$ and $CR$ be the angle bisectors of $\vartriangle ABC$ so that $P, Q$ and $R$ lie on $BC, CA$ and $AB$ respectively. Let the reflections of line $OI$ across lines $DE$ and $DF$ intersect at $X$. Prove that $P, Q, R, X$ lie on a circle.
Let $\vartriangle ABC$ be a triangle with $AB > AC$. The incircle of $\vartriangle ABC$ touches its sides $BC, CA$ and $AB$ at $D, E$ and $F$, respectively. The angle bisector of $\angle BAC$ cuts $DE$ and $DF$ at $K$ and $L$, respectively. Let $M$ be the midpoint of $BC$ and let $H$ be the foot of altitude from $A$ to $BC$. Prove that $\angle MLK =\angle MHK$.
2013 Thailand TST 6.1 (also) (Miklos Schweitzer 1991 p2)
A number of points are given on a unit circle so that the product of the distances from any point on the circle to the given points does not exceed $2.$ Prove that the given points are the vertices of a regular polygon.
In $\vartriangle ABC$, the incircle $\gamma$ is tangent to the sides $BC, CA$ and $AB$ at $A_1, B_1$ and $C_1,$ respectively. Let $\omega_A$ be the unique circle through $B$ and $C$ which is tangent to $\gamma$, the line through $A_1$ and the tangency point of $\gamma$ and $\omega_A$ meets $\omega_A$ again at $M_A$. Points $M_B$ and $M_C$ are defined analogously. Prove that
a) $A_1MA, B_1MB, C_1MC$ concur at a single point, which we call $K$, and
b) $K, I, O$ are collinear and the distances between them satisfy $\frac{KO}{KI}=\frac{R(M_AM_BM_C)}{r(ABC)}$, where $R(XYZ)$ and $r(XY )$, respectively, denote the circumradius and inradius of $\vartriangle XYZ$.
Let $\vartriangle ABC$ be a scalene triangle with incenter $I$, and let $I_b$ and $I_c$ be its excenters opposite vertices $B$ and $C$, respectively. Let $D$ be the intersection point of the perpendiculars from $I_b$ to $AC$ and from $I_c$ to $AB$. The angle bisectors of $\angle BI_bD$ and$ \angle CI_cD$ are drawn intersecting at $G$, and the line through $G$ parallel to $AI$ intersects $I_bI_c$ at $H$. Prove that the circle centered at $G$ with radius $GH$ is tangent to the circumcircle of $\vartriangle ABC$.
2014 Thailand TST 3.1 (also Taiwan TST 2014)
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.
Let $\vartriangle ACB$ be an acute triangle with circumcenter $O$, orthocenter $H$, and nine-point center $N$. Let $P$ be the second intersection of $AO$ and the circumcircle of $\vartriangle BOC$, and let $Q$ be the reflection of $A$ over $BC$. Show that the midpoint of segment $PQ$ lies on line $AN$.
Let $\vartriangle ABC$ with a scalene triangle whose incircle $\odot (I, r)$ (i.e. centered at $I$ with radius $r$) touches the sides $BC, CA, AB$ at $X, Y, Z$ respectively. Let $X_1, Y_1, Z_1$, respectively, be the images of $X, Y, Z$ under the homothety $h(I, 2r)$. (In other words, $X_1, Y_1, Z_1$ lie on the rays $\overrightarrow{IX},\overrightarrow{IY} ,\overrightarrow{IZ}$, respectively, such that $IX_1 = IY_1 = IZ_1 = 2r$.) Prove that
a) the lines $AX_1, BY_1, CZ_1$ pass through a single point, which we call $Q$, and
b) if $O$ is the circumcircle of $\vartriangle ABC$, and $P$ is the intersection point line $OI$ and the reflection of line $AQ$ over line $AI$, then $\angle PCI = \angle ICQ$.
Show that the Miquel point of a complete quadrilateral lies on the nine-point circle of the triangle determined by its 3 diagonals.
Prove that every convex polyhedron without a quadrilateral or pentagonal face must have at least $4$ triangular faces.
A tangential quadrilateral $ABCD$ which is not a trapezoid is given. The extensions of sides $AD$ and $BC$ intersect at $E$ and the extensions of sides AB and $CD$ intersect at $F$, so that exactly one of $\vartriangle AEF$ and $\vartriangle CEF$ is outside $ABCD$. Let the incircle of $\vartriangle AEF$ be tangent to lines $AD$ and $AB$ at $K$ and $L$, respectively, and let the incircle of $\vartriangle CEF$ be tangent to lines $BC$ and $CD$ at $M$ and $N$, respectively.
a) Prove that $K, L, M, N$ lie on a circle.
b) Prove that $A, B, C, D$ lie on a circle if and only if $KN \perp LM$.
A cyclic quadrilateral $ABCD$ is given. Let $M$ be the set of the $16$ centers of all incircles and excircles of $\vartriangle BCD$, $\vartriangle ACD$, $\vartriangle ABD$ and $\vartriangle ABC$. Prove that there exist two sets $K$ and $L$, each consisting of four parallel lines, such that any line in $K \cup L$ contains exactly four points of $M$.
A scalene triangle $\vartriangle ABC$ is given with circumcircle $\omega$ and circumcenter $O$. Let $M$ be the midpoint of $BC$, and assume $M \ne O$. The circumcircle of $\vartriangle OAM$ intersects ω again at $D \ne A$. Prove that:
a) The intersection point of the tangent lines to $\omega$ at $A$ and $D$ lies on line $BC$.
b) The triangles $\vartriangle AMB, \vartriangle ACD$ and $\vartriangle DMB$ are similar.
A given convex polygon can cover any triangle whose side lengths are at most $1$. Prove that the area of the convex polygon is at least $\frac12 \cos 10^o$ .
Assume that $ O$,$ I$ and $ I_a$ are the circumcenter,incenter and the excenter corresponding to the edge $ BC$ of the triangle $ ABC$,respectively.Let $ II_a$ intersect the segment $ BC$ and $ (O)$ at $ A_1$ and $ M$ respectively. ($ M \in II_a$) .Let $ N$ be the midpoint of the arc $ MCA$ and $ S,T$ be the intersections of lines $ NI,NI_a$ with $ (O)$,respectively.Prove that $ S,T,A_1$ are collinear
Let $ABCD$ be a quadrilateral inscribed in a circle $\omega$. The symmedians through $B$ of $\vartriangle ABD$ and $\vartriangle CBD$ intersect $\omega$. again at points $P$ and $Q$, respectively. Lines $CP$ and $AB$ intersect at $X$, and lines $AQ$ and $BC$ intersect at $Y$ . Prove that the points $X, D, Y$ are collinear
Let $P$ be a point in the interior of a triangle $ABC$. The three cevians $AA', BB', CC'$ of $P$ divide the triangle into six triangles. Prove that the circumcenters of the six triangles are concyclic if and only if $P$ is the centroid of $ABC$.
Let $H,I,M$ be the orthocenter, incenter and circumcenter of triangle $ABC$ respectively. Let the incircle touch $BC$ at $K$. Given that $IO//BC$, prove that $AO//HK$
A triangle $\vartriangle ABC$ with $AB > BC$ is inscribed into a circle $\omega$. Points $M$ and $N$ are on the sides $AB$ and $BC$, respectively, such that $AM = CN$. Let lines $MN$ and $AC$ intersect at $K$, and let $S$ be the midpoint of arc $AC$ containing $B$. The circle with diameter$ KS$ intersects line $MN$ at $T$ and intersects the angle bisector of $\angle MKA$ at $D$.
a) Prove that $MT = TN$.
b) Let $P$ be the incenter of $\vartriangle CKN$ and let $Q$ be the $K$-excenter of $\vartriangle AKM$. Prove that $DP = DQ$.
Let $ABCD$ be a convex quadrilateral, and let $M$ be a point inside the quadrilateral. Suppose that the projections of $M$ onto the sides $AB, BC, CD, DA$ lie on a circle with center $O$. Let $N$ be the reflection of $M$ over $O$. Prove that the projections of$ B$ onto the lines $AM, AN, CM, CN$ also lie on a circle
Let $ABCD$ be a convex quadrilateral such that $\angle DAB = 180^o - 2\angle BCD$. The incircle of $\vartriangle ABD$ is tangent to the sides $AB$ and $AD$ at $P$ and $Q$, respectively. Prove that the circumcircle of $\vartriangle APQ$ is tangent to the circumcircle of $\vartriangle BCD$.
Let $\vartriangle ABC$ be an acute-angled triangle whose altitudes $AA_1$ and$ BB_1$ intersect at $H$. Let $\omega_1$ be the circle centered at $H$ passing through $B_1$ and let $\omega_2$ be the circle centered at $B$ passing through $B_1$. Let $CN$ and $CK$ be the tangent lines from $C$ to circles $\omega_1$ and $\omega_2$ respectively ($N$ and $K$ are distinct from $B_1$). Prove that $A_1, N$ and $K$ are collinear.
Let $\omega$ be the circumcircle of triangle $ABC$, and let $M$ be a midpoint of arc $BC$, not containing $A$. The incircle of $\vartriangle ABC$ is centered at $I$ and touches BC at $D$. The $A$-excircle of $\vartriangle ABC$ is centered at $I_A$ and touches $BC$ at $E$. Lines $MD$ and $ME$ are drawn intersecting $\omega$ again at points $T \ne D$ and $R \ne E$, respectively. Let $RI_A$ intersect $\omega$ at $S \ne R$. Show that $T, I$, and $S$ are collinear.
Two circles $\omega_1$ and $\omega_2$ intersect at points $A$ and $B$. A line $\ell_1$ through $A$ intersects $\omega_1$ and $\omega_2$ again at $C$ and $E$, respectively, and a point $G$ is chosen on this line between $A$ and $E$. A line $\ell_2$ through $B$ intersects $\omega_1$ and $\omega_2$ again at $D$ and $F$, respectively, and a point $H$ is chosen on this line between $B$ and $F$. Line $CH$ intersects $FG$ at $I$ and intersects $\omega_1$ again at $J$. Line $DG$ intersects $EH$ at $K$ and intersects $\omega_1$ again at $L$. Lines $EH$ and $FG$ intersect $\omega_2$ again at $M$ and $N$, respectively. Assume that points $A, B, ... , N$ are all distinct. Prove that $I, J, K, L, M, N$ lie on a circle.
Let $ABC$ be a triangle with circumcircle $\omega$. Let $D$ be a point on $AB$. Let $\Gamma$ be the circle which is tangent to line $DB$ at $M$, line $DC$ at $N$, and also to $\omega$ externally. Suppose that the external angle bisector of $\angle ABC$ intersects $MN$ at $X$. Show that $AX$ bisects $\angle BAC$
Let $\vartriangle ABC$ be a triangle with circumcircle $\omega$. The circle centered at $B$ having radius $BC$ intersects $AC$ and $\omega$ at $D$ and $E$, respectively. The line through $D$ parallel to $CE$ intersects $AB$ at $F$.
a) Prove that $BD$ and $EF$ intersect on $\omega$
b) Let $G$ be the intersection point of $AB$ and $CE$. Prove that $DEFG$ is a rhombus.
2016 Thailand TST 10.3 (also 2011 TST 3.2)
Let $\vartriangle ABC$ be an acute triangle with orthocenter $H$. Points $Y$ and $Z$ are chosen on $AC, AB$ such that $\angle HYC = \angle HZB = 60^o$. Let $U$ be the circumcenter of $\vartriangle HYZ$, and let $N$ be the nine-point center of $\vartriangle ABC$. Prove that $A, U, N$ are collinear.
Let $ABCD$ be a convex quadrilateral such that $AC \perp BD$. Prove that there exist points $P, Q, R, S$, on the sides $AB, BC, CD, DA$ respectively, such that $PR \perp QS$ and the area of the quadrilateral $PQRS$ is exactly half of that of the quadrilateral $ABCD$.
Given a triangle $ABC$ with $AC = BC > AB$, let $E$ and $F$ be the midpoints of $AC$ and $AB$, respectively. The perpendicular bisector $\ell$ of $AC$ meets $AB$ at $K$ and the line parallel to $KC$ and passing through the point $B$ intersects $AC$ at a point $L$. For a point $P$ on the line segment $BF$, let $H$ be the orthocenter of the triangle ACP. The line segments $BH$ and $CP$ meet at a point $J$ and the lines $FJ$ and $\ell$ meet at a point $M$. Let $W$ be the intersection point of $FL$ and ℓ. Show that $AW = BW$ if and only if the points $B,E, F,M M$ are concyclic.
In a convex pentagon, five altitudes are drawn from each vertex to its opposite side. Prove that if four of them meet at a single point, then the fifth altitude must pass through that point.
Two distinct circle $\Gamma_1, \Gamma_2$ meet at two distinct points $M,N$. A line $\ell$ intersects $\Gamma_1$ at $A, C$ and intersects $\Gamma_2$ at $B,D$ such that points $A,B,C,D$ are all distinct and lie on $\ell$ in this order. Let $X$ be a point on the line $MN$ such that $M$ lies between $X$ and $N$. Lines $AX,BM$ intersect at $P$ and lines $DX,CM$ intersect at $Q$. Finally, let $K,L$ be the midpoints of $AD, BC$ respectively. Prove that $XK,ML,PQ$ are concurrent.
Let $ABC$ be a triangle with $\angle A= 60^{\circ}$ and $AB>AC$. Let $O$ be its circumcenter, $F$ be the foot of the altitude from $C$ , and $D$ be a point on the side $AB$ such that $BD=AC$. Suppose that the points $O,F, $ and $D$ are distinct. Prove that the circumcircle of the triangle $OFD$ intersects the circle centered at $O$ with radius $OF$ on the altitude of the triangle $ABC$ from $B$.
Let $P$ be a point inside $\Delta ABC$. Let $A_1, B_1, C_1$ be points in the interiors of the segments $PA,PB,PC$ respectively. Let $\overline{BC_1}\cap\overline{CB_1}=\{A_2\}$, $\overline{CA_1}\cap\overline{AC_1}=\{B_2\}$, and $\overline{AB_1}\cap\overline{BA_1}=\{C_2\}$. Let $U$ be the intersection of the lines $A_1B_1$ and $A_2B_2$, and $V$ be the intersection of the lines $A_1C_1$ and $A_2C_2$. Show that the lines $UC_2$, $VB_2$, and $AP$ are concurrent.
Let $\vartriangle ABC$ be an acute triangle with altitudes $AA_1, BB_1, CC_1$ and orthocenter $H$. Let $K, L$ be the midpoints of $BC_1, CB_1$. Let $\ell_A$ be the external angle bisector of $\angle BAC$. Let $\ell_B, \ell_C$ be the lines through $B, C$ perpendicular to $\ell_A$. Let $\ell_H$ be the line through $H$ parallel to $\ell_A$. Prove that the centers of the circumcircles of $\vartriangle A_1B_1C_1, \vartriangle AKL$ and the rectangle formed by $\ell_A, \ell_B, \ell_C, \ell_H$ lie on the same line.
Let $E$ and $F$ be points on side $BC$ of a triangle $\vartriangle ABC$. Points $K$ and $L$ are chosen on segments $AB$ and $AC$, respectively, so that $EK \parallel AC$ and $FL \parallel AB$. The incircles of $\vartriangle BEK$ and $\vartriangle CFL$ touches segments $AB$ and $AC$ at $X$ and $Y$ , respectively. Lines $AC$ and $EX$ intersect at $M$, and lines $AB$ and $FY$ intersect at $N$. Given that $AX = AY$, prove that $MN \parallel BC$.
Let $\omega$ be the incircle of $\vartriangle ABC$. The circle $\omega$ touches sides $BC$ and $AB$ at the points $D$ and $F$ respectively. Denote by $D'$ the reflection of the point $D$ with respect to the point $C$. The circle passing through the points $B$ and $C$ is tangent to $\omega$ at the point $T$. Let $J$ be the $A$-excenter of $\vartriangle ABC$. Prove that the points $T, F, B, J$ and $D'$ lie on the one circle.
2018 Thailand - March Problem Solving p4 (Russian Regional)
Triangle $ABC$ is equilateral. $\Omega$ is circumcircle and $\omega$ is incircle of $ABC$ with common center $O$. Points $P$ and $Q$ are on sides $AC$ and $AB$ and $O \in PQ$. $\Gamma_b$ and $\Gamma_c$ - are circles, with diameters $BP$ and $CQ$. Prove, that one point of intersection of $\Gamma_b$ and $\Gamma_c$ lies on $\omega$ and second point lies on $\Omega$
2018 Thailand - March Problem Solving p5 (Komal A.716)
Let $ABC$ be a triangle and let $D$ be a point in the interior of the triangle which lies on the angle bisector of $\angle$$BAC$. Suppose that lines $BD$ and $AC$ meet at $E$, and that lines $CD$ and $AB$ meet at $F$. The circumcircle of $ABC$ intersects line $EF$ at points $P$ and $Q$. Show that if $O$ is the circumcenter of $DPQ$, then $OD$ is perpendicular to $BC$.
2018 Thailand - March Problem Solving p6 (Komal A.705)
Triangle $ABC$ has orthocenter $H$. Let $D$ be a point distinct from the vertices on the circumcircle of $ABC$. Suppose that circle $BHD$ meets $AB$ at $P!=B$, and circle $CHD$ meets $AC$ at $Q!=C$. Prove that as $D$ moves on the circumcircle, the reflection of $D$ across line $PQ$ also moves on a fixed circle.
2018 Thailand - March Problem Solving p7 (Komal A.702)
Fix a triangle $ABC$. We say that triangle $XYZ$ is elegant if $X$ lies on segment $BC$, $Y$ lies on segment $CA$, $Z$ lies on segment $AB$, and $XYZ$ is similar to $ABC$ (i.e., $\angle A=\angle X, \angle B=\angle Y, \angle C=\angle Z$). Of all the elegant triangles, which one has the smallest perimeter?
Let $ABC$ be any triangle with $\angle BAC \le \angle ACB \le \angle CBA$. Let $D, E$ and $F$ be the midpoints of $BC, CA$ and $AB$, respectively, and let $\epsilon$ be a positive real number. Suppose there is an ant (represented by a point $T$ ) and two spiders (represented by points $P_1$ and $P_2$, respectively) walking on the sides $BC, CA, AB, EF, FD$ and $DE$. The ant and the spiders may vary their speeds, turn at an intersection point, stand still, or turn back at any point; moreover, they are aware of their and the others’ positions at all time.
Assume that the ant’s speed does not exceed $1$ mm/s, the first spider’s speed does not exceed $\frac{\sin A}{2 \sin A+\sin B}$ mm/s, and the second spider’s speed does not exceed $\epsilon$ mm/s. Show that the spiders always have a strategy to catch the ant regardless of the starting points of the ant and the spiders.
Note: the two spiders can discuss a plan before the hunt starts and after seeing all three starting points, but cannot communicate during the hunt.
Given a convex quadrilateral, is it always possible to determine a point in its interior such that the four line segments joining the point to the midpoints of the sides divide the quadrilateral into four regions of equal area? If such a point exists, is it unique?
Suppose that $ABCDEZ$ is a regular octahedron whose pairs of opposite vertices are $(A,Z),(B,D)$ and $(C,E)$. The points $F, G,H$ are chosen on the segments $AB,AC,AD$ respectively such that $AF = AG = AH$.
(i) Show that $EF$ and $DG$ must intersect at a point $K$, and that $BG$ and $EH$ must intersect at a point $L$.
(ii) Let $EG$ meet the plane of $AKL$ at a point $M$. Show that $AKML$ is a square.
Let $ABC$ be a triangle with $\angle A = 90^o$, and let $D$ be the foot of the altitude from $A$. A variable point $M$ traces the interior of the minor arc $AB$ of the circle $ABC$. The internal bisector of $\angle DAM$ crosses $CM$ at $N$. The line through $N$ perpendicular to $CM$ crosses the line $AD$ at $P$. Determine all possible intersection points of $BN$ and $CP$.
Let $ABC$ be a triangle with circumcircle $\Gamma$. Let $\omega_0$ be a circle tangent to chord $AB$ and arc $ACB$. For each $i = 1, 2$, let $\omega_i$ be a circle tangent to $AB$ at $T_i$ , to $\omega_0$ at $S_i$ , and to arc $ACB$. Suppose $\omega_1 \ne \omega_2$. Prove that there is a circle passing through $S_1, S_2, T_1$, and $T_2$, and tangent to $\Gamma$ if and only if $\angle ACB = 90^o$.
A triangle $ABC$ with $AB < AC < BC$ is given. The point $P$ is the center of an excircle touching the line segment $AB$ at $D$. The point $Q$ is the center of an excircle touching the line segment $AC$ at $E$. The circumcircle of the triangle $ADE$ intersects $\overline{PE}$ and $\overline{QD}$ again at $G$ and $H$ respectively. The line perpendicular to $\overline{AG}$ at $G$ intersects the side $AB$ at $R$. The line perpendicular to $\overline{AH}$ at $H$ intersects the side $AC$ at $S$. Prove that $\overline{DE}$ and $\overline{RS}$ are parallel.
October Camp 2013-19
In a triangle $ABC, AC = BC$ and $D$ is the midpoint of $AB$. Let $E$ be an arbitrary point on line $AB$ which is not $B$ or $D$. Let $O$ be the circumcenter of $\vartriangle ACE$ and $F$ the intersection of the perpendicular from $E$ to $BC$ and the perpendicular to $DO$ at $D$. Prove that the acute angle between $BC$ and $BF$ does not depend on the choice of point $E$.E.
In a triangle $ABC$, the incircle with incenter $I$ is tangent to $BC$ at $A_1, CA$ at $B_1$, and $AB$ at $C_1$. Denote the intersection of $AA_1$ and $BB_1$ by $G$, the intersection of $AC$ and $A_1C_1$ by $X$, and the intersection of $BC$ and $B_1C_1$ by $Y$ . Prove that $IG \perp XY$ .
Let $O$ be the incenter of a tangential quadrilateral $ABCD$. Prove that the orthocenters of $\vartriangle AOB$, $\vartriangle BOC$, $\vartriangle COD$, $\vartriangle DOA$ lie on a line.
Let $O$ be the circumcenter of an acute $\vartriangle ABC$ which has altitude $AD$. Let $AO$ intersect the circumcircle of $\vartriangle BOC$ again at $X$. If $E$ and $F$ are points on lines $AB$ and $AC$ such that $\angle XEA = \angle XFA = 90^o$ , then prove that the line $DX$ bisects the segment $EF$.
Let $ABCDEF$ be a hexagon inscribed in a circle (with vertices in that order) with $\angle B + \angle C > 180^o$ and $\angle E + \angle F > 180^o$. Let the lines $AB$ and $CD$ intersect at $X$ and the lines $AF$ and $DE$ intersect at $S$. Let $XY$ and $ST$ be the diameters of the circumcircles of $\vartriangle BCX$ and $\vartriangle EFS$ respectively. If $U$ is the intersection point of the lines $BX$ and $ES$ and $V$ is the intersection point of the lines $BY$ and $ET,$ prove that the lines $UV, XY$ and $ST$ are all parallel.
The circles $S_{1}$ and $S_{2}$ intersect at $M$ and $N$.Show that if vertices $A$ and $C$ of a rectangle $ABCD$ lie on $S_{1}$ while vertices $B$ and $D$ lie on $S_{2}$,then the intersection of the diagonals of the rectangle lies on the line $MN$.
Let $D$ be a point inside an acute triangle $ABC$ such that $\angle ADC = \angle A +\angle B$, $\angle BDA = \angle B + \angle C$ and $\angle CDB = \angle C + \angle A$. Prove that $\frac{AB \cdot CD}{AD} = \frac{AC \cdot CB} {AB}$.
In any $\vartriangle ABC, \ell$ is any line through $C$ and points $P, Q$. If $BP, AQ$ are perpendicular to the line $\ell$ and $M$ is the midpoint of the line segment $AB$, then prove that $MP = MQ$
Let $\omega$ be a circle touching two parallel lines $\ell_1, \ell_2$, $\omega_1$ a circle touching $\ell_1$ at $A$ and $\omega$ externally at $C$, and $\omega_2$ a circle touching $\ell_2$ at $B$, $\omega$ externally at $D$, and $\omega_1$ externally at $E$. Prove that $AD, BC$ intersect at the circumcenter of $\vartriangle CDE$.
Let $H$ be the orthocenter of acute-angled $\vartriangle ABC$, and $X, Y$ points on the ray $AB, AC$. ($B$ lies between $X, A$, and $C$ lies between $Y, A$.) Lines $HX, HY$ intersect $BC$ at $D, E$ respectively. Let the line through $D$ parallel to $AC$ intersect $XY$ at $Z$. Prove that $\angle XHY = 90^o$ if and only if $ZE \parallel AB$.
In $\vartriangle ABC, D, E, F$ are the midpoints of $AB, BC, CA$ respectively. Denote by $O_A, O_B, O_C$ the incenters of $\vartriangle ADF, \vartriangle BED, \vartriangle CFE$ respectively. Prove that $O_AE, O_BF, O_CD$ are concurrent.
In $\vartriangle ABC$ with $AB > AC$, the tangent to the circumcircle at $A$ intersects line $BC$ at $P$. Let $Q$ be the point on $AB$ such that $AQ = AC$, and $A$ lies between $B$ and $Q$. Let $R$ be the point on ray $AP$ such that $AR = CP$. Let $X, Y$ be the midpoints of $AP, CQ$ respectively. Prove that $CR = 2XY$ .
Let $\omega_1, \omega_2$ be two circles with different radii, and let $H$ be the exsimilicenter of the two circles. A point X outside both circles is given. The tangents from $X$ to $\omega_1$ touch $\omega_1$ at $P, Q$, and the tangents from $X$ to $\omega_2$ touch $\omega_2$ at $R, S$. If $PR$ passes through $H$ and is not a common tangent line of $\omega_1, \omega_2$, prove that $QS$ also passes through $H$.
Circles $O_1, O_2$ intersects at $A, B$. The circumcircle of $O_1BO_2$ intersects $O_1, O_2$ and line $AB$ at $R, S, T$ respectively. Prove that $TR = TS$
2017 Thailand October Camp 1.6 (IGO Elementary 2016 4)
In a right-angled triangle $ABC$ ($\angle A = 90^o$), the perpendicular bisector of $BC$ intersects the line $AC$ in $K$ and the perpendicular bisector of $BK$ intersects the line $AB$ in $L$. If the line $CL$ be the internal bisector of angle $C$, find all possible values for angles $B$ and $C$.
Let $BC$ be a chord not passing through the center of a circle $\omega$. Point $A$ varies on the major arc $BC$. Let $E$ and $F$ be the projection of $B$ onto $AC$, and of $C$ onto $AB$ respectively. The tangents to the circumcircle of $\vartriangle AEF$ at $E, F$ intersect at $P$.
(a) Prove that $P$ is independent of the choice of $A$.
(b) Let $H$ be the orthocenter of $\vartriangle ABC$, and let $T$ be the intersection of $EF$ and $BC$. Prove that $TH \perp AP$.
In triangle \vartriangle $ABC$, $\angle BAC = 135^o$. $M$ is the midpoint of $BC$, and $N \ne M$ is on $BC$ such that $AN = AM$. The line $AM$ meets the circumcircle of $\vartriangle ABC$ at $D$. Point $E$ is chosen on segment $AN$ such that $AE = MD$. Show that $ME = BC$.
Let $\Omega$ be the inscribed circle of a triangle $\vartriangle ABC$. Let $D, E$ and $F$ be the tangency points of $\Omega$ and the sides $BC, CA$ and $AB$, respectively, and let $AD, BE$ and $CF$ intersect $\Omega$ at $K, L$ and $M$, respectively, such that $D, E, F, K, L$ and $M$ are all distinct. The tangent line of $\Omega$ at $K$ intersects $EF$ at $X$, the tangent line of $\Omega$ at $L$ intersects $DE$ at $Y$ , and the tangent line of $\Omega$ at M intersects $DF$ at $Z$. Prove that $X,Y$ and $Z$ are collinear.
Let $ABC$ be an acute triangle with $AX, BY$ and $CZ$ as its altitudes.
$\bullet$ Line $\ell_A$, which is parallel to $YZ$, intersects $CA$ at $A_1$ between $C$ and $A$, and intersects $AB$ at $A_2$ between $A$ and $B$.
$\bullet$ Line $\ell_B$, which is parallel to $ZX$, intersects $AB$ at $B_1$ between $A$ and $B$, and intersects $BC$ at $B_2$ between $B$ and $C$.
$\bullet$ Line $\ell_C$, which is parallel to $XY$ , intersects $BC$ at $C_1$ between $B$ and $C$, and intersects $CA$ at $C_2$ between $C$ and $A$.
Suppose that the perimeters of the triangles $\vartriangle AA_1A_2$, $\vartriangle BB_1B_2$ and $\vartriangle CC_1C_2$ are equal to $CA+AB,AB +BC$ and $BC +CA$, respectively. Prove that $\ell_A, \ell_B$ and $\ell_C$ are concurrent.
Let $ABC$ be an acute triangle and $\Gamma$ be its circumcircle. Line $\ell$ is tangent to $\Gamma$ at $A$ and let $D$ and $E$ be distinct points on $\ell$ such that $AD = AE$. Suppose that $B$ and $D$ lie on the same side of line $AC$. The circumcircle $\Omega_1$ of $\vartriangle ABD$ meets $AC$ again at $F$. The circumcircle $\Omega_2$ of $\vartriangle ACE$ meets $AB$ again at $G$. The common chord of $\Omega_1$ and $\Omega_2$ meets $\Gamma$ again at $H$. Let $K$ be the reflection of $H$ across line $BC$ and let $L$ be the intersection of $BF$ and $CG$. Prove that $A, K$ and $L$ are collinear.
Let $P$ be an interior point of a circle $\Gamma$ centered at $O$ where $P \ne O$. Let $A$ and $B$ be distinct points on $\Gamma$. Lines $AP$ and $BP$ meet $\Gamma$ again at $C$ and $D$, respectively. Let $S$ be any interior point on line segment $PC$. The circumcircle of $\vartriangle ABS$ intersects line segment $PD$ at $T$. The line through $S$ perpendicular to $AC$ intersects $\Gamma$ at $U$ and $V$ . The line through $T$ perpendicular to $BD$ intersects $\Gamma$ at $X$ and $Y$ . Let $M$ and $N$ be the midpoints of $UV$ and $XY$ , respectively. Let $AM$ and $BN$ meet at $Q$. Suppose that $AB$ is not parallel to $CD$. Show that $P, Q$, and $O$ are collinear if and only if $S$ is the midpoint of $PC$.
random before 2010 TST problems
mentioned in aops
at most 2005 Thailand TST (posted at Jan 4 2005)
let ABC be a triangle such that AB<BC , AC not equal to BC and K be its circumcircle. The tangent line to K at the point A intersects the line BC in the point D. Let k be the circle tangent to K and to the segment AD and BD. We denote by M,N the points where k touches BD and AD respectively. Let J be the center of the exscribed circle which is tangent to the side AB. Prove that J,M,N are collinear.
2005 Thailand TST (also as 2006 Swiss TST)
Let $ABC$ an acute triangle so that $AB\not= AC$ and let $H$ the orthocenter of $ABC$. Points $D$ and $E$ are in the sides $AB,AC$ respectively so that $D,H,E$ are collinear and $AE=AD$. Prove that if $M$ is the midpoint of $BC$, then $MH$ is perpendicular to the straight line that joins $A$ with the second intersections (different of $A$) of the circumcircles of the triangles $AED$ and $ABC$.
source: https://www.infinitydots.org/
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