USA TST 2000-19 37p

geometry problems from USA Team Selection Tests  (USA TSTST)
with aops links in the names


2000 USA TST  problem 2
Let $ ABCD$ be a cyclic quadrilateral and let $ E$ and $ F$ be the feet of perpendiculars from the intersection of diagonals $ AC$ and $ BD$ to $ AB$ and $ CD$, respectively. Prove that $ EF$ is perpendicular to the line through the midpoints of $ AD$ and $ BC$

2001 USA TST  problem 5
In triangle $ABC$, $\angle B = 2\angle C$. Let $P$ and $Q$ be points on the perpendicular bisector of segment $BC$ such that rays $AP$ and $AQ$ trisect $\angle A$. Prove that $PQ < AB$ if and only if $\angle B$ is obtuse.

2001 USA TST problem7
Let $ABCD$ be a convex quadrilateral such that $\angle ABC = \angle ADC = 135^{\circ}$ and $AC^2\cdot BD^2 = 2\cdot AB\cdot BC\cdot CD\cdot DA.$ Prove that the diagonals of the quadrilateral $ABCD$ are perpendicular.

2002 USA TST problem 5
Consider the family of  nonisosceles triangles $ABC$ satisfying the property $AC^2 + BC^2 = 2 AB^2$. Points $M$ and $D$ lie on side $AB$ such that $AM = BM$ and $\angle ACD = \angle BCD$. Point $E$ is in the plane such that $D$ is the incenter of triangle $CEM$. Prove that exactly one of the ratios $\frac{CE}{EM}, \quad \frac{EM}{MC}, \quad \frac{MC}{CE}  $ is constant.

2003 USA TST problem 3
Let $ABC$ be a triangle and let $P$ be a point in its interior. Lines $PA$, $PB$, $PC$ intersect sides $BC$, $CA$, $AB$ at $D$, $E$, $F$, respectively. Prove that $ [PAF]+[PBD]+[PCE]=\frac{1}{2}[ABC] $  if and only if $P$ lies on at least one of the medians of triangle $ABC$.

(Here $[XYZ]$ denotes the area of triangle $XYZ$.)

2003 USA TST problem 6
Let $\overline{AH_1}, \overline{BH_2}$, and $\overline{CH_3}$ be the altitudes of an acute scalene triangle $ABC$. The incircle of triangle $ABC$ is tangent to $\overline{BC}, \overline{CA},$ and $\overline{AB}$ at $T_1, T_2,$ and $T_3$, respectively. For $k = 1, 2, 3$, let $P_i$ be the point on line $H_iH_{i+1}$ (where $H_4 = H_1$) such that $H_iT_iP_i$ is an acute isosceles triangle with $H_iT_i = H_iP_i$. Prove that the circumcircles of triangles $T_1P_1T_2$, $T_2P_2T_3$, $T_3P_3T_1$ pass through a common point.

2004 USA TST problem 4
Let $ABC$ be a triangle. Choose a point $D$ in its interior. Let $\omega_1$ be a circle passing through $B$ and $D$ and $\omega_2$ be a circle passing through $C$ and $D$ so that the other point of intersection of the two circles lies on $AD$. Let $\omega_1$ and $\omega_2$ intersect side $BC$ at $E$ and $F$, respectively. Denote by $X$ the intersection of $DF$, $AB$ and $Y$ the intersection of $DE, AC$. Show that $XY \parallel BC$.

2005 USA TST problem 2
Let $A_{1}A_{2}A_{3}$ be an acute triangle, and let $O$ and $H$ be its circumcenter and orthocenter, respectively. For $1\leq i \leq 3$, points $P_{i}$ and $Q_{i}$ lie on lines $OA_{i}$ and $A_{i+1}A_{i+2}$ (where $A_{i+3}=A_{i}$), respectively, such that $OP_{i}HQ_{i}$ is a parallelogram. Prove that  $ \frac{OQ_{1}}{OP_{1}}+\frac{OQ_{2}}{OP_{2}}+\frac{OQ_{3}}{OP_{3}}\geq 3.$

2005 USA TST problem 6
Let $ABC$ be an acute scalene triangle with $O$ as its circumcenter. Point $P$ lies inside triangle $ABC$ with $\angle PAB = \angle PBC$ and $\angle PAC = \angle PCB$. Point $Q$ lies on line $BC$ with $QA = QP$. Prove that $\angle AQP = 2\angle OQB$. 

2006 USA TST problem 2
In acute triangle $ABC$ , segments $AD; BE$ , and $CF$ are its altitudes, and $H$ is its orthocenter. Circle $\omega$, centered at $O$, passes through $A$ and $H$ and intersects sides $AB$ and $AC$ again at $Q$ and $P$ (other than $A$), respectively. The circumcircle of triangle $OPQ$ is tangent to segment $BC$ at $R$. Prove that $\frac{CR}{BR}=\frac{ED}{FD}.$

2006 USA TST problem 6
Let $ABC$ be a triangle. Triangles $PAB$ and $QAC$ are constructed outside of triangle $ABC$ such that $AP = AB$ and $AQ = AC$ and $\angle{BAP}= \angle{CAQ}$. Segments $BQ$ and $CP$ meet at $R$. Let $O$ be the circumcenter of triangle $BCR$. Prove that $AO \perp PQ.$

2007 USA TST problem 1
Circles $ \omega_1$ and $ \omega_2$ meet at $ P$ and $ Q$. Segments $ AC$ and $ BD$ are chords of $ \omega_1$ and $ \omega_2$ respectively, such that segment $ AB$ and ray $ CD$ meet at $ P$. Ray $ BD$ and segment $ AC$ meet at $ X$. Point $ Y$ lies on $ \omega_1$ such that $ PY \parallel BD$. Point $ Z$ lies on $ \omega_2$ such that $ PZ \parallel AC$. Prove that points $ Q,X,Y,Z$ are collinear.

2007 USA TST problem 5
Triangle $ ABC$ is inscribed in circle $ \omega$. The tangent lines to $ \omega$ at $ B$ and $ C$ meet at $ T$. Point $ S$ lies on ray $ BC$ such that $ AS \perp AT$. Points $ B_1$ and $ C_1$ lie on ray $ ST$ (with $ C_1$ in between $ B_1$ and $ S$) such that $ B_1T= BT = C_1T$. Prove that triangles $ ABC$ and $ AB_1C_1$ are similar to each other.

2008 USA TST problem 2
Let $ P$, $ Q$, and $ R$ be the points on sides $ BC$, $ CA$, and $ AB$ of an acute triangle $ ABC$ such that triangle $ PQR$ is equilateral and has minimal area among all such equilateral triangles. Prove that the perpendiculars from $ A$ to line $ QR$, from $ B$ to line $ RP$, and from $ C$ to line $ PQ$ are concurrent.

2008 USA TST problem 6
Determine the smallest positive real number $ k$ with the following property. Let $ ABCD$ be a convex quadrilateral, and let points $ A_1$, $ B_1$, $ C_1$, and $ D_1$ lie on sides $ AB$, $ BC$, $ CD$, and $ DA$, respectively. Consider the areas of triangles $ AA_1D_1$, $ BB_1A_1$, $ CC_1B_1$ and $ DD_1C_1$; let $ S$ be the sum of the two smallest ones, and let $ S_1$ be the area of quadrilateral $ A_1B_1C_1D_1$. Then we always have $ kS_1\ge S$.

by Zuming Feng & Oleg Golberg

2009 USA TST  problem 2
Let $ ABC$ be an acute triangle.  Point $ D$ lies on side $ BC$.  Let $ O_B, O_C$ be the circumcenters of triangles $ ABD$ and $ ACD$, respectively.  Suppose that the points $ B, C, O_B, O_C$ lies on a circle centered at $ X$.  Let $ H$ be the orthocenter of triangle $ ABC$.  Prove that $ \angle{DAX}= \angle{DAH}$.

by Zuming Feng
2009 USA TST problem 4
Let $ ABP, BCQ, CAR$ be three non-overlapping triangles erected outside of acute triangle $ ABC$. Let $ M$ be the midpoint of segment $ AP$. Given that $ \angle PAB  = \angle CQB  = 45^\circ$, $ \angle ABP = \angle QBC  75^\circ$, $ \angle RAC  = 105^\circ$, and $ RQ^2 = 6CM^2$, compute $ AC^2/AR^2$.


by Zuming Feng
2010 USA TST problem 5
Let $ABC$ be a triangle. Point $M$ and $N$ lie on sides $AC$ and $BC$ respectively such that $MN || AB$. Points $P$ and $Q$ lie on sides $AB$ and $CB$ respectively such that $PQ || AC$. The incircle of triangle $CMN$ touches segment $AC$ at $E$. The incircle of triangle $BPQ$ touches segment $AB$ at $F$. Line $EN$ and $AB$ meet at $R$, and lines $FQ$ and $AC$ meet at $S$. Given that $AE = AF$, prove that the incenter of triangle $AEF$ lies on the incircle of triangle $ARS$.

2010 USA TST problem 7
In triangle ABC, let $P$ and $Q$ be two interior points such that $\angle ABP = \angle QBC$ and $\angle ACP = \angle QCB$. Point $D$ lies on segment $BC$. Prove that $\angle APB + \angle DPC = 180^\circ$ if and only if $\angle AQC + \angle DQB = 180^\circ$.

2011 USA TST problem 1
In an acute scalene triangle $ABC$, points $D,E,F$ lie on sides $BC, CA, AB$, respectively, such that $AD \perp BC, BE \perp CA, CF \perp AB$. Altitudes $AD, BE, CF$ meet at orthocenter $H$. Points $P$ and $Q$ lie on segment $EF$ such that $AP \perp EF$ and $HQ \perp EF$. Lines $DP$ and $QH$ intersect at point $R$. Compute $HQ/HR$.

by Zuming Feng
Let $ABC$ be an acute scalene triangle inscribed in circle $\Omega$. Circle $\omega$, centered at $O$, passes through $B$ and $C$ and intersects sides $AB$ and $AC$ at $E$ and $D$, respectively. Point $P$ lies on major arc $BAC$ of $\Omega$. Prove that lines $BD, CE, OP$ are concurrent if and only if triangles $PBD$ and $PCE$ have the same incenter.

In acute triangle $ABC$, $\angle{A}<\angle{B}$ and $\angle{A}<\angle{C}$. Let $P$ be a variable point on side $BC$. Points $D$ and $E$ lie on sides $AB$ and $AC$, respectively, such that $BP=PD$ and $CP=PE$. Prove that as $P$ moves along side $BC$, the circumcircle of triangle $ADE$ passes through a fixed point other than $A$.

In cyclic quadrilateral $ABCD$, diagonals $AC$ and $BD$ intersect at $P$. Let $E$ and $F$ be the respective feet of the perpendiculars from $P$ to lines $AB$ and $CD$. Segments $BF$ and $CE$ meet at $Q$. Prove that lines $PQ$ and $EF$ are perpendicular to each other.

Let $ABC$ be a scalene triangle with $\angle BCA = 90^{\circ}$, and let $D$ be the foot of the altitude from $C$.  Let $X$ be a point in the interior of the segment $CD$.  Let $K$ be the point on the segment $AX$ such that $BK = BC$.  Similarly, let $L$ be the point on the segment $BX$ such that $AL = AC$.  The circumcircle of triangle $DKL$ intersects segment $AB$ at a second point $T$ (other than $D$).  Prove that $\angle ACT = \angle BCT$

Let $ABC$ be an acute triangle.  Circle $\omega_1$, with diameter $AC$, intersects side $BC$ at $F$ (other than $C$).  Circle $\omega_2$, with diameter $BC$, intersects side $AC$ at $E$ (other than $C$).  Ray $AF$ intersects $\omega_2$ at $K$ and $M$ with $AK < AM$.  Ray $BE$ intersects $\omega_1$ at $L$ and $N$ with $BL < BN$.  Prove that lines $AB$, $ML$, $NK$ are concurrent.

Let $ABC$ be an acute triangle, and let $X$ be a variable interior point on the minor arc $BC$ of its circumcircle. Let $P$ and $Q$ be the feet of the perpendiculars from $X$ to lines $CA$ and $CB$, respectively. Let $R$ be the intersection of line $PQ$ and the perpendicular from $B$ to $AC$. Let $\ell$ be the line through $P$ parallel to $XR$. Prove that as $X$ varies along minor arc $BC$, the line $\ell$ always passes through a fixed point. (Specifically: prove that there is a point $F$, determined by triangle $ABC$, such that no matter where $X$ is on arc $BC$, line $\ell$ passes through $F$.)

by Robert Simson et al.
Let $ABCD$ be a cyclic quadrilateral, and let $E$, $F$, $G$, and $H$ be the midpoints of $AB$, $BC$, $CD$, and $DA$ respectively. Let $W$, $X$, $Y$ and $Z$ be the orthocenters of triangles $AHE$, $BEF$, $CFG$ and $DGH$, respectively. Prove that the quadrilaterals $ABCD$ and $WXYZ$ have the same area.

Let $ABC$ be a non-isosceles triangle with incenter $I$ whose incircle is tangent to $\overline{BC}$, $\overline{CA}$, $\overline{AB}$ at $D$, $E$, $F$, respectively. Denote by $M$ the midpoint of $\overline{BC}$. Let $Q$ be a point on the incircle such that $\angle AQD = 90^{\circ}$.  Let $P$ be the point inside the triangle on line $AI$ for which $MD = MP$.  Prove that either $\angle PQE = 90^{\circ}$ or $\angle PQF = 90^{\circ}$.

by Evan Chen
Let $ABC$ be a non-equilateral triangle and let $M_a$, $M_b$, $M_c$ be the midpoints of the sides $BC$, $CA$, $AB$, respectively.  Let $S$ be a point lying on the Euler line. Denote by $X$, $Y$, $Z$ the second intersections of $M_aS$, $M_bS$, $M_cS$ with the nine-point circle. Prove that $AX$, $BY$, $CZ$ are concurrent.

Let $ABC$ be a scalene triangle with circumcircle $\Omega$, and suppose the incircle of $ABC$ touches $BC$ at $D$.  The angle bisector of $\angle A$ meets $BC$ and $\Omega$ at $E$ and $F$.  The circumcircle of $\triangle DEF$ intersects the $A$-excircle at $S_1$, $S_2$, and $\Omega$ at $T \neq F$.  Prove that line $AT$ passes through either $S_1$ or $S_2$.

by Evan Chen
Let $ABC$ be an acute scalene triangle and let $P$ be a point in its interior. Let $A_1$, $B_1$, $C_1$ be projections of $P$ onto triangle sides $BC$, $CA$, $AB$, respectively. Find the locus of points $P$ such that $AA_1$, $BB_1$, $CC_1$ are concurrent and $\angle PAB + \angle PBC + \angle PCA = 90^{\circ}$.

Let $ABC$ be an acute scalene triangle with circumcenter $O$, and let $T$ be on line $BC$ such that $\angle TAO = 90^{\circ}$. The circle with diameter $\overline{AT}$ intersects the circumcircle of $\triangle BOC$ at two points $A_1$ and $A_2$, where $OA_1 < OA_2$. Points $B_1$, $B_2$, $C_1$, $C_2$ are defined analogously.
a) Prove that $\overline{AA_1}$, $\overline{BB_1}$, $\overline{CC_1}$ are concurrent.
b) Prove that $\overline{AA_2}$, $\overline{BB_2}$, $\overline{CC_2}$ are concurrent on the Euler line of triangle $ABC$. 
by Evan Chen
Let $ABC$ be a triangle with altitude $\overline{AE}$. The $A$-excircle touches $\overline{BC}$ at $D$, and intersects the circumcircle at two points $F$ and $G$. Prove that one can select points $V$ and $N$ on lines $DG$ and $DF$ such that quadrilateral $EVAN$ is a rhombus.

by Danielle Wang & Evan Chen

Let $ABCD$ be a convex cyclic quadrilateral which is not a kite, but whose diagonals are perpendicular and meet at $H$. Denote by $M$ and $N$ the midpoints of $\overline{BC}$ and $\overline{CD}$. Rays $MH$ and $NH$ meet $\overline{AD}$ and $\overline{AB}$ at $S$ and $T$, respectively. Prove that there exists a point $E$, lying outside quadrilateral $ABCD$, such that

  • ray $EH$ bisects both angles $\angle BES$, $\angle TED$, and
  • $\angle BEN = \angle MED$.

 by Evan Chen

Let $ABC$ be a triangle and let $M$ and $N$ denote the midpoints of $\overline{AB}$ and $\overline{AC}$, respectively. Let $X$ be a point such that $\overline{AX}$ is tangent to the circumcircle of triangle $ABC$. Denote by $\omega_B$ the circle through $M$ and $B$ tangent to $\overline{MX}$, and by $\omega_C$ the circle through $N$ and $C$ tangent to $\overline{NX}$. Show that $\omega_B$ and $\omega_C$ intersect on line $BC$.

by Merlijn Staps

Let $ABC$ be a triangle with incenter $I$, and let $D$ be a point on line $BC$ satisfying $\angle AID=90^{\circ}$. Let the excircle of triangle $ABC$ opposite the vertex $A$ be tangent to $\overline{BC}$ at $A_1$. Define points $B_1$ on $\overline{CA}$ and $C_1$ on $\overline{AB}$ analogously, using the excircles opposite $B$ and $C$, respectively.

Prove that if quadrilateral $AB_1A_1C_1$ is cyclic, then $\overline{AD}$ is tangent to the circumcircle of $\triangle DB_1C_1$.

by Ankan Bhattacharya

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