USA TSTST 2011-18 17p

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

more USA Competitions in appendix: UK USA Canada


2011 - 

Two circles $\omega_1$ and $\omega_2$ intersect at points $A$ and $B$. Line $\ell$ is tangent to $\omega_1$ at $P$ and to $\omega_2$ at $Q$ so that $A$ is closer to $\ell$ than $B$. Let $X$ and $Y$ be points on major arcs ${PA}$ (on $\omega_1$) and $AQ$ (on $\omega_2$), respectively, such that $AX/PX = AY/QY = c$. Extend segments $PA$ and $QA$ through $A$ to $R$ and $S$, respectively, such that $AR = AS = c\cdot PQ$. Given that the circumcenter of triangle $ARS$ lies on line $XY$, prove that $\angle XPA = \angle AQY$.

Acute triangle $ABC$ is inscribed in circle $\omega$. Let $H$ and $O$ denote its orthocenter and circumcenter, respectively. Let $M$ and $N$ be the midpoints of sides $AB$ and $AC$, respectively. Rays $MH$ and $NH$ meet $\omega$ at $P$ and $Q$, respectively. Lines $MN$ and $PQ$ meet at $R$. Prove that $OA\perp RA$.

Let $ABC$ be a triangle. Its excircles touch sides $BC, CA, AB$ at $D, E, F$, respectively. Prove that the perimeter of triangle $ABC$ is at most twice that of triangle $DEF$.

Let $ABCD$ be a quadrilateral with $AC = BD$.  Diagonals $AC$ and $BD$ meet at $P$.  Let $\omega_1$ and $O_1$ denote the circumcircle and the circumcenter of triangle $ABP$.  Let $\omega_2$ and $O_2$ denote the circumcircle and circumcenter of triangle $CDP$.  Segment $BC$ meets $\omega_1$ and $\omega_2$ again at $S$ and $T$ (other than $B$ and $C$), respectively.  Let $M$ and $N$ be the midpoints of minor arcs $\widehat {SP}$ (not including $B$) and $\widehat {TP}$ (not including $C$).  Prove that $MN \parallel O_1O_2$.

In scalene triangle $ABC$, let the feet of the perpendiculars from $A$ to $BC$, $B$ to $CA$, $C$ to $AB$ be $A_1, B_1, C_1$, respectively.  Denote by $A_2$ the intersection of lines $BC$ and $B_1C_1$.  Define $B_2$ and $C_2$ analogously.  Let $D, E, F$ be the respective midpoints of sides $BC, CA, AB$.  Show that the perpendiculars from $D$ to $AA_2$, $E$ to $BB_2$ and $F$ to $CC_2$ are concurrent.

Triangle $ABC$ is inscribed in circle $\Omega$.  The interior angle bisector of angle $A$ intersects side $BC$ and $\Omega$ at $D$ and $L$ (other than $A$), respectively.  Let $M$ be the midpoint of side $BC$.  The circumcircle of triangle $ADM$ intersects sides $AB$ and $AC$ again at $Q$ and $P$ (other than $A$), respectively.  Let $N$ be the midpoint of segment $PQ$, and let $H$ be the foot of the perpendicular from $L$ to line $ND$.  Prove that line $ML$ is tangent to the circumcircle of triangle $HMN$.

Let $ABC$ be a triangle and $D$, $E$, $F$ be the midpoints of arcs $BC$, $CA$, $AB$ on the circumcircle.  Line $\ell_a$ passes through the feet of the perpendiculars from $A$ to $DB$ and $DC$.  Line $m_a$ passes through the feet of the perpendiculars from $D$ to $AB$ and $AC$.  Let $A_1$ denote the intersection of lines $\ell_a$ and $m_a$.  Define points $B_1$ and $C_1$ similarly.  Prove that triangle $DEF$ and $A_1B_1C_1$ are similar to each other.

Circle $\omega$, centered at $X$, is internally tangent to circle $\Omega$, centered at $Y$, at $T$.  Let $P$ and $S$ be variable points on $\Omega$ and $\omega$, respectively, such that line $PS$ is tangent to $\omega$ (at $S$).  Determine the locus of $O$ -- the circumcenter of triangle $PST$.

Consider a convex pentagon circumscribed about a circle. We name the lines that connect vertices of the pentagon with the opposite points of tangency with the circle  gergonnians .
(a) Prove that if four gergonnians are conncurrent, the all five of them are concurrent.
(b) Prove that if there is a triple of gergonnians that are concurrent, then there is another triple of gergonnians that are concurrent.

Let ABC be a scalene triangle. Let $K_a$, $L_a$ and $M_a$ be the respective intersections with BC of the internal angle bisector, external angle bisector, and the median from A. The circumcircle of $AK_aL_a$ intersects $AM_a$ a second time at point $X_a$ different from A. Define $X_b$ and $X_c$ analogously. Prove that the circumcenter of $X_aX_bX_c$ lies on the Euler line of ABC.

(The Euler line of ABC is the line passing through the circumcenter, centroid, and orthocenter of ABC.)

by Ivan Borsenco
Let $ABC$ be a scalene triangle with orthocenter $H$ and circumcenter $O$.  Denote by $M$, $N$ the midpoints of $\overline{AH}$, $\overline{BC}$.  Suppose the circle $\gamma$ with diameter $\overline{AH}$ meets the circumcircle of $ABC$ at $G \neq A$, and meets line $AN$ at a point $Q \neq A$.  The tangent to $\gamma$ at $G$ meets line $OM$ at $P$.  Show that the circumcircles of $\triangle GNQ$ and $\triangle MBC$ intersect at a point $T$ on $\overline{PN}$.

by Evan Chen
Let $ABC$ be a triangle with incenter $I$, and whose incircle is tangent to $\overline{BC}$, $\overline{CA}$, $\overline{AB}$ at $D$, $E$, $F$, respectively.  Let $K$ be the foot of the altitude from $D$ to $\overline{EF}$.  Suppose that the circumcircle of $\triangle AIB$ meets the incircle at two distinct points $C_1$ and $C_2$, while the circumcircle of $\triangle AIC$ meets the incircle at two distinct points $B_1$ and $B_2$.  Prove that the radical axis of the circumcircles of $\triangle BB_1B_2$ and $\triangle CC_1C_2$ passes through the midpoint $M$ of $\overline{DK}$.

by Danielle Wang
Let $ABC$ be a triangle with circumcircle $\Gamma$, circumcenter $O$, and orthocenter $H$. Assume that $AB\neq AC$ and that $\angle A \neq 90^{\circ}$. Let $M$ and $N$ be the midpoints of sides $AB$ and $AC$, respectively, and let $E$ and $F$ be the feet of the altitudes from $B$ and $C$ in $\triangle ABC$, respectively. Let $P$ be the intersection of line $MN$ with the tangent line to $\Gamma$ at $A$. Let $Q$ be the intersection point, other than $A$, of $\Gamma$ with the circumcircle of $\triangle AEF$. Let $R$ be the intersection of lines $AQ$ and $EF$. Prove that $PR\perp OH$.

 by Ray Li


Let $ABC$ be a triangle with incenter $I$. Let $D$ be a point on side $BC$ and let $\omega_B$ and $\omega_C$ be the incircles of $\triangle ABD$ and $\triangle ACD$, respectively. Suppose that $\omega_B$ and $\omega_C$ are tangent to segment $BC$ at points $E$ and $F$, respectively. Let $P$ be the intersection of segment $AD$ with the line joining the centers of $\omega_B$ and $\omega_C$. Let $X$ be the intersection point of lines $BI$ and $CP$ and let $Y$ be the intersection point of lines $CI$ and $BP$. Prove that lines $EX$ and $FY$ meet on the incircle of $\triangle ABC$.

by Ray Li
Let $ABC$ be an acute triangle with incenter $I$, circumcenter $O$, and circumcircle $\Gamma$. Let $M$ be the midpoint of $\overline{AB}$. Ray $AI$ meets $\overline{BC}$ at $D$. Denote by $\omega$ and $\gamma$ the circumcircles of $\triangle BIC$ and $\triangle BAD$, respectively. Line $MO$ meets $\omega$ at $X$ and $Y$, while line $CO$ meets $\omega$ at $C$ and $Q$. Assume that $Q$ lies inside $\triangle ABC$ and $\angle AQM = \angle ACB$. Consider the tangents to $\omega$ at $X$ and $Y$ and the tangents to $\gamma$ at $A$ and $D$. Given that $\angle BAC \neq 60^{\circ}$, prove that these four lines are concurrent on $\Gamma$.

by Evan Chen &Yannick Yao
Let $ABC$ be an acute triangle with circumcircle $\omega$, and let $H$ be the foot of the altitude from $A$ to $\overline{BC}$. Let $P$ and $Q$ be the points on $\omega$ with $PA = PH$ and $QA = QH$. The tangent to $\omega$ at $P$ intersects lines $AC$ and $AB$ at $E_1$ and $F_1$ respectively; the tangent to $\omega$ at $Q$ intersects lines $AC$ and $AB$ at $E_2$ and $F_2$ respectively. Show that the circumcircles of $\triangle AE_1F_1$ and $\triangle AE_2F_2$ are congruent, and the line through their centers is parallel to the tangent to $\omega$ at $A$.

by Ankan Bhattacharya & Evan Chen

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