geometry shortlists from Central American and Caribbean Mathematical Olympiads (OMCC) with aops links in the names
Olimpiada Matemática de Centroamérica y el Caribe
geometry shortlists collected inside aops: here
from years: 2006-07 , 2012-18, 2020
2006-07, 2012-18, 2020
2006 shortlist
Let $\Gamma$ and $\Gamma'$ be two congruent circles centered at $O$ and $O'$, respectively, and let $A$ be one of their two points of intersection. $B$ is a point on $\Gamma$, $C$ is the second point of intersection of $AB$ and $\Gamma'$, and $D$ is a point on $\Gamma'$ such that $OBDO'$ is a parallelogram. Show that the length of $CD$ does not depend on the position of $B$.
Let $ABC$ be any triangle. Construct the equilateral triangles $BCD$, $CAE$ and $ABF$ where $D$, $E$ and $F$ are in half planes opposite $A$, $B$ and $C$ wrt lines $BC$, $CA$ and $AB$ respectively. Let $P$ be a point on the extension of $DA$ beyond $A$ such that $DA = AP$. Prove that the triangle $EFP$ is equilateral.
Let $ABCD$ be a convex quadrilateral. $I=AC\cap BD$, and $E$, $H$, $F$ and $G$ are points on $AB$, $BC$, $CD$ and $DA$ respectively, such that $EF \cap GH= I$. If $M=EG \cap AC$, $N=HF \cap AC$, show that\[\frac{AM}{IM}\cdot \frac{IN}{CN}=\frac{IA}{IC}.\]
Given a circle with center $O$ and diameter $AB$. Let $ P$ be a point on the circle, construct the bisector of the angle $\angle PAB$ that intersects the circle at $Q$. Let $R$ be the intersection point of the perpendicular from $Q$ on $PA$, $M$ the midpoint of $PQ$ and $S$ the intersection of $OQ$ with $RM$. Determine the locus of point $S$ as $ P$ moves along the circle.
Let $S_1$,$S_2$ be two circles that intersect at two points $M$ and $N$. Let $P$ be a point on the segment $MN$. Consider points $A, B, C$ and $D$ such that $A$ and $C$ lie at $S_1$, while $B$ and $D$ lie on $S_2$, and segments $AC$ and $BD$ intersect at $P$. Let $\ell$ and $m$ be the perpendicular bisectors of segments $AP$ and $DP$. Prove that if $\ell$, $m$ and the line $AD$ concur at point $X$, then the line $XP$ is perpendicular on the line $BC$ .
Let $ABC$ be a triangle and $P$ be an interior point of the triangle. We consider the feet $D$, $E$ and $F$ of the perpendiculars on the sides $BC$, $CA$ and $AB$ drawn from $P$ . Draw the perpendicular on the line $EF$ from $A$, the perpendicular on the line $FD$ from $B$ and the perpendicular on the line $DE$ from $C$ and paint them red.
a) Prove that the three red lines concur.
b) Given a point $P$ inside the triangle, we denote by $P'$ the point of concurrency of the red lines drawn from $ P$. If $P'$ also lies inside the triangle, determine the intersection of the three red lines constructed in a similar way starting from $P'$.
2007 shortlist
In a triangle $ABC$, the angle bisector of $A$ and the cevians $BD$ and $CE$ concur at a point $P$ inside the triangle. Show that the quadrilateral $ADPE$ has an incircle if and only if $AB=AC$.
Let $ABC$ be an acute triangle and let $O$ be its circumcenter. Let $D, E, F$ be the feet of the altitudes from $A, B, C$, and $M,N,P$ the midpoints of the sides $BC$, $CA$, $AB$, respectively. Consider the circle $C_a$ that passes through $O$ and is tangent to $BC$ at $M$. Similarly, $C_b$ and $C_c$ are defined. Let $I_a$ be the intersection of $C_b$ and $C_c$, other than $O$. Similarly, $I_b$ and $I_c$ are defined. Prove that $I_aD$, $I_bE$ and $I_CF$ concur in the centroid of triangle $ABC$.
In a triangle $ABC$, we consider the circle $\Gamma$ of radius $AC$ and center $C$. Let $D$ be the second intersection point of the bisector of $\angle BAC$ with $\Gamma$ , $E$ the second intersection point of $CD$ with $\Gamma$ , and let $F$ be the midpoint of $AB$. Prove that $AD$, $BE$ and $CF$ are concurrent.
Let $ABC$ be a non-isosceles triangle with$ \angle BAC = 60^o$, circumcenter $O$ and incenter $I$. Prove that $BC$, $OI$, and the perpendicular bisector of $AI$ are concurrent.
Let $ABC$ be a triangle. The perpendicular bisector of side $AB$ intersects the other two sides at $D$ and $E$ . The perpendicular bisector of side $AC$ intersects the other two sides at$ F$ and $G$. Prove that points $D$, $E$, $F$, and $G$ are concyclic.
Let $ABC$ be a scalene triangle, $BC$ being its smallest side. Let $ P$ and $Q$ be points on $AB$ and $AC$ respectively such that $BQ = CP =BC$. The circumcircles of $AQB$ and $APC$ intersect again at a point $X$. $BX$ intersects $AC$ at $R$ and $CX$ intersects $AB$ at $S$. Show that $A$, $R$, $X$, and $S$ are concyclic.
Consider a circle $S$, and a point $P$ outside it. The tangent lines from $P$ meet $S$ at $A$ and $B$, respectively. Let $M$ be the midpoint of $AB$. The perpendicular bisector of $AM$ meets $S$ in a point $C$ lying inside the triangle $ABP$. $AC$ intersects $PM$ at $G$, and $PM$ meets $S$ in a point $D$ lying outside the triangle $ABP$. If $BD$ is parallel to $AC$, show that $G$ is the centroid of the triangle $ABP$.
2012 shortlist
In triangle $\vartriangle ABC$ , $\angle A$ is the greatest internal angle. Let $H$ be its orthocenter. The midpoints of the sides $BC$, $CA$ and $AB$ are denoted by $D$, $E$ and $F$, respectively. The triangle $\vartriangle ABC$ has altitudes $AP$, $BQ$ and $CR$. Points $ J$ and $ L$ are taken on $AB$ and $AC$ such that $AD = EJ = FL$. Points $K$ and $M$ belong to $AB$ and $AC$ such that $CK$ is parallel to $DJ$ and $BM$ is parallel to $DL$.
a) Prove that $HK = HM$.
b) The lines $PR$ and $CK$ meet at $T$. Prove that $AB$ is parallel to $QT$.
Let $\vartriangle ABC$ be a triangle, $I$ its incenter and $O$ its circumcenter. Lines $AI$, $BI$ and $CI$ meet the circumcircle of $\vartriangle ABC$ at $D$, $E$ and $F$ . The midpoints of $EF$, $FD$ and $DE$ are denoted by $ L$, $M$ and $N$, respectively. Let $P$ be the the circumcenter of $\vartriangle LMN$.
Prove that $P$, $O$ and $I$ are collinear.
Let $\gamma$ be the circumcircle of the acute triangle $ABC$. Let $P$ be the midpoint of the minor arc $BC$. The parallel to $AB$ through $P$ cuts $BC, AC$ and $\gamma$ at points $R,S$ and $T$, respectively. Let $K \equiv AP \cap BT$ and $L \equiv BS \cap AR$. Show that $KL$ passes through the midpoint of $AB$ if and only if $CS = PR$.
Let $\Gamma_1$, $\Gamma_2$, $\Gamma_3$, and $\Gamma_4$, be four different circumferences such that they have a common point $T$ and there are not two circumferences tangent to each other. A line $m$ trough $T$ meets again $\Gamma_1$ and $\Gamma_3$, at $M_1$ and $M_3$, respectively. Another line $n$ through $T$ meets again $\Gamma_2$, and $\Gamma_4$ at $N_2$ and $N_4$, respectively.
Prove that it is always possible (and show how) to construct conveniently lines $m$ and $n$ such that the circumcircles of the triangles $\vartriangle TM_1N_2$ and $\vartriangle TM_3N_4$ are tangent..
Let $ABCD$ be a convex cyclic quadrilateral with non-parallel opposite sides such that $AC$ is a diameter of the circumcircle of $ABCD$. Lines $AB$ and $CD$ meet at $S$. Lines $BC$ and $DA$ meet at $T$. The internal bisector of $\angle BSC$ meets $BC$ and $DA$ at $F$ and $H$, respectively. The internal bisector of $\angle CTD$ meets $CD$ and $AB$ at $G$ and $E$, respectively.
Prove that the circumcircles of the triangles $\vartriangle BEF$ and $\vartriangle DGH$ are tangent.
In triangle $\vartriangle ABC$, let $H$ be the intersection point of the lines perpendicular to $BA$ and $CA$ through $C$ and $ B$, respectively. The midpoint of $BC$ is $M$. The foot of the perpendicular to $MH$ through $ A$ is $ P$. Let $Q$ be the reflection point of $ P$ through the midpoint of $AH$. The line perpendicular to $CP$ through $Q$ meets $PA$ and $PC$ at $S$ and $T$, respectively. Lines $BA$ and $PM$ meet at $X$. Lines $CP$ and $AH$ meet at $Y$. The circumcircle of $\vartriangle CQT$ meets $CS$ again at $Z$.
Prove that $\angle AXY = \angle AZS$.
Let $P$ be a point on the internal bisector of angle $\angle BAC$ in the triangle $\vartriangle ABC$. Let $D$ and $F$ be the intersection points of the circumcircle of the triangles $\vartriangle PBA$ and $\vartriangle PCA$ with the external bisector of angle $\angle BAC$. The midpoint of $DE$ is $M$.
Prove that $M$ belongs to the circumcircle of triangle $\vartriangle ABC$.
Let $ABC$ be a triangle with $AB < BC$, and let $E$ and $F$ be points in $AC$ and $AB$ such that $BF = BC = CE$, both on the same halfplane as $A$ with respect to $BC$. Let $G$ be the intersection of $BE$ and $CF$. Let $H$ be a point in the parallel through $G$ to $AC$ such that $HG = AF$ (with $H$ and $C$ in opposite halfplanes with respect to $BG$). Show that $\angle EHG = \frac{\angle BAC}{2}$.
Two circumferences $\omega_1$ and $\omega_2$ with centers $O_1$ and $O_2$ meet at $J$ and $M$. The line $BC$ is a common tangent such that $B$ belongs to $\omega_1$ and C belongs to $\omega_2$ . The line $BC$ meets $O_1O_2$ at $F$. The line $FM$ meets again $\omega_1$ and $\omega_2$ at $ A$ and $D$, respectively. Point $G$ is the intersection of $AB$ and $CD$. Let $O$ be the circumcenter of $\vartriangle AGD$.
Prove that $ \angle OJM = 90^o$.
2013 shortlist
Let $ABCD$ be a convex quadrilateral and $E$ a point on the extension of side $AD$. Suppose $AB = BC = CD$, $\angle BAD = 5x$, $\angle BCD= 6x$, and $\angle CDE = 7x$. Find the measure of $x$ .
Let $T$ be the vertex of an angle, $R$ a point in its interior and $Q$ the intersection of one side of the angle, with a ray passing through $R$, the ray is reflected at $Q$ and intersects on the other side of the angle at $V$, the ray is reflected again at $V$ and is such that it passes through $R$. Let $\ell_1$ and $\ell_2$ be the respective bisectors of the angles $VQR$ and $RVQ$. Let $M$ be the intersection point of $\ell_1$ with the line $TV$. Let $N$ be the intersection point of $\ell_2$ with the line $TQ$. Prove that points $M, R$ and $N$ are collinear.
Let $ABCD$ be a rhombus such that $BD = AB$. A point $E$ is taken on $BD$, different from $B, D$ and the midpoint of $BD$. Let $ BN$ be a altitude in $BDC$. If $AE$ cuts $BC$ at $F$ and $EC$ cuts $ BN$ at $Q$, show that $FQ$ passes through a fixed point as $E$ varies in $BD$.
Let $ABC$ be an acute triangle and let $\Gamma$ be its circumcircle. The bisector of $\angle{A}$ intersects $BC$ at $D$, $\Gamma$ at $K$ (different from $A$), and the line through $B$ tangent to $\Gamma$ at $X$. Show that $K$ is the midpoint of $AX$ if and only if $\frac{AD}{DC}=\sqrt{2}$.
Given a convex quadrilateral, find necessary and sufficient conditions so that twice its area, the semidifference of the sum of the squares of the two pairs of consecutive sides, and the product of their diagonals form a Pythagorean triple in that order.
Note: A Pythagorean triple in the following order: $a, b$ and $c$, means that $a^2 + b^2 = c^2$
Let $ABCD$ be a right trapezoid, with right angles at $B$ and $C$, also with $CD <BC$. Using a ruler and compass, locate a point $P$ in the interior of the segment $BC$ such that $\angle APB = 2\angle DPC$.
Let $ABCD$ be a convex quadrilateral and let $M$ be the midpoint of side $AB$. The circle passing through $D$ and tangent to $AB$ at $A$ intersects the segment $DM$ at $E$. The circle passing through $C$ and tangent to $AB$ at $B$ intersects the segment $CM$ at $F$. Suppose that the lines $AF$ and $BE$ intersect at a point which belongs to the perpendicular bisector of side $AB$. Prove that $A$, $E$, and $C$ are collinear if and only if $B$, $F$, and $D$ are collinear.
Let $I$ be the incenter of a triangle $ABC$. Points $D, E, F, G$ are considered on $BC$, $CA$, $AB$, $EF$ such that $DI$, $El$, $Fl$, $Gl$ are perpendicular on $BC$, $Cl$, $Bl$, $EF$ respectively. If $ED$, $BG$ intersect at $H$, prove that $AH$ is parallel to $BC$.
2014 shortlist
Segments $AC$ and $BD$ intersect at point $P$ such that $PA = PD$, $PB = PC$. Let $O$ be the circumcenter of the triangle $\vartriangle PAB$. Prove that lines $OP$ and $CD$ are perpendicular
Let $ABCD$ be a trapezoid with bases $AB$ and $CD$, inscribed in a circle of center $O$. Let $P$ be the intersection of the lines $BC$ and $AD$. A circle through $O$ and $P$ intersects the segments $BC$ and $AD$ at interior points $F$ and $G$, respectively. Show that $BF=DG$.
Let $\vartriangle ABC$ be an isosceles triangle with $AC = BC$. Let $O$ be the circumcenter of $\vartriangle ABC$, and let $D$ be the point in the plane such that $B$ is the midpoint of segment $AD$. Let $E$ be a point such that $OD = OE$ and $CE$ is parallel to $AB$. Show that $\frac{CE}{AB} = \sqrt2$.
Points $A$, $B$, $C$ and $D$ are chosen on a line in that order, with $AB$ and $CD$ greater than $BC$. Equilateral triangles $APB$, $BCQ$ and $CDR$ are constructed so that $P$, $Q$ and $R$ are on the same side with respect to $AD$. If $\angle PQR=120^\circ$, show that \[\frac{1}{AB}+\frac{1}{CD}=\frac{1}{BC}.\]
Let $\vartriangle ABC$ be a triangle and points $D, E, F$ in the interior of the sides $BC$, $CA$, $AB$. Let $P$ be the intersection of $BE$ and $CF$, $Q$ the intersection of $CF$ and $AD$, and $R$ the intersection of $AD$ and $BE$. Show that if the triangles $\vartriangle ARE$, $\vartriangle BPF$ and $\vartriangle CQD$ are similar (not necessarily in that order of vertices) then they are in fact three triangles congruent with each other.
On the three sides of a triangle $\vartriangle ABC$ squares are built on the outside of the triangle with lengths $AB$, $BC$ and $CA$ respectively. Let $A’$, $B’$ and $C’.$ be the centers of the squares built on $BC$, $CA$ and $BA$ respectively. Prove that $AA’= B’C ‘$ and $AA’ \perp B’C’$.
Segment $AB$ is diameter of a circle $\Gamma$ . On $\Gamma$ three other points $X$, $Y$ and $Z$ are chosen. The lines $XB$ and $YZ$ intersect at $C$. The bisectors of the angles $\angle XAB$ and $\angle YAZ$ intersect again $\Gamma$ at points $P$ and $Q$ respectively. Line $PQ$ intersects lines $AX$ and $BX$ at points $R$ and $S$, respectively. Show that $AC$ is the bisector of the $\angle XCY$ if and only if the lines $AS$ and $RC$ are perpendicular.
Consider the triangle $ABC$, obtuse at $A$. The altitudes on the opposite sides are $AD$, $BE$, and $CF$ (with $D, E, F$ in the respective segments $BC$, $AC$, and $AB$ or their extensions). Let $E’$ and $F’$ be the feet of the respective perpendiculars from $E$ and $F$ on $BC$. Suppose $2E’F’= 2AD + BC$. Determine $\angle A$.
2015 shortlist
Let $ABCD$ be a cyclic quadrilateral, with $O$ the point of intersection of the diagonals $AC$ and $BD$. Prove that
$\angle ABC = \angle AOB$ if and only if $DA = AB$.
Let $ABC$ be a triangle such that $AC=2AB$. Let $D$ be the point of intersection of the angle bisector of the angle $CAB$ with $BC$. Let $F$ be the point of intersection of the line parallel to $AB$ passing through $C$ with the perpendicular line to $AD$ passing through $A$. Prove that $FD$ passes through the midpoint of $AC$.
Let $\Gamma_1$ and $\Gamma_2$ be circles that intersect at two different points $ A$ and $ B$. Let $C$ and $D$ be points on $\Gamma_1$ and $\Gamma_2$ such that $CB$ and $DB$ are tangent to $\Gamma_2$ and $\Gamma_1$, respectively. Let $F$ be a point on $\Gamma_1$ such that $AB = CF$ and $G$ a point on $\Gamma_2$ such that $BG = DA$. Let $ P$ be the intersection point of $CF$ with $BA$, and $Q$ the intersection point of $BG$ with $DA$. Prove that the circumcircle of the triangle $QFP$ is tangent to $\Gamma_2$.
Show that if it is possible to inscribe a quadrilateral $A'B'C'D'$ in a quadrilateral $ABCD$ (putting a single vertex of $A'B'C'D'$ on each side of $ABCD$) of minimum perimeter, then the quadrilateral $ABCD$ is inscribed in a circle.
Let $ABC$ be an acute triangle of circumcenter $O$ and let $D, E, F$ be the feet of the altitudes from $A, B, C$, respectively. Let $ P$ be the foot of the perpendicular from $ B$ on $DE$ and $Q$ the foot of the perpendicular from $C$ on $DF$. Prove that $OD$ is perpendicular to $PQ$.
Let $ABCD$ be a cyclic quadrilateral with $AB<CD$, and let $P$ be the point of intersection of the lines $AD$ and $BC$.The circumcircle of the triangle $PCD$ intersects the line $AB$ at the points $Q$ and $R$. Let $S$ and $T$ be the points where the tangents from $P$ to the circumcircle of $ABCD$ touch that circle.
(a) Prove that $PQ=PR$.
(b) Prove that $QRST$ is a cyclic quadrilateral.
Let $ABC$ be a triangle and let $\omega_A$ and $\omega_B$ be the circles that pass through $C$ and are tangent to $AB$ at $A$ and $B$, respectively. Circles $\omega_A$ and $\omega_B$ intersect at $C$ and $N$. Let $M$ be the intersection of $CN$ with $AB$. Let $\Omega$ be the circle that passes through $C$ and is tangent to $AB$ at $M$, and let $\omega$ be the circle tangent to $\Omega$ passing through $C$ and $N$. The tangents to $\omega$ from $M$ intersect $\omega$ at $X$ and $Y$. Let $Z$ be the intersection point of $AX$ and $BY$. Prove that $Z$ lies on $\omega$.
2016 shortlist
Let $ABC$ be a triangle and $D$ be the midpoint of $AC$. Let $H$ be the intersection of the parallel to $BD$ by $A$ with the parallel to $BA$ by $D$. Let $E$ be the closest to $C$ of the points that trisect $BC$. Prove that $ED$ bisects $AH$.
Let $ABC$ be an acute triangle. Let $D$, $E$, and $F$ be points on the sides $BC$, $AC$, and $AB$, respectively, such that $AD$, $BE$, and $CF$ are concurrent. Let $X, Y$, and $Z$ be points on sides $BC$, $AC$, and $AB$, respectively, such that the triangles $AFY$, $ZBD$, and $EXC$ are similar. Show that $AX$, $BY$, and $CZ$ are concurrent.
Let $ABC$ be an acute-angled triangle, $\Gamma$ its circumcircle and $M$ the midpoint of $BC$. Let $N$ be a point in the arc $BC$ of $\Gamma$ not containing $A$ such that $\angle NAC= \angle BAM$. Let $R$ be the midpoint of $AM$, $S$ the midpoint of $AN$ and $T$ the foot of the altitude through $A$. Prove that $R$, $S$ and $T$ are collinear.
Let $\triangle ABC$ be triangle with incenter $I$ and circumcircle $\Gamma$. Let $M=BI\cap \Gamma$ and $N=CI\cap \Gamma$, the line parallel to $MN$ through $I$ cuts $AB$, $AC$ in $P$ and $Q$. Prove that the circumradius of $\odot (BNP)$ and $\odot (CMQ)$ are equal.
Let $A_1A_2A_3$ be an acute triangle. The bisector of side $A_1A_2$ and the one parallel to $A_1A_3$ that passes through $A_2$ intersect at $P_1$, the bisector of side $A_1A_3$ and the one parallel to $A_1A_2$ that passes through $A_3$ intersect at $Q_1$ and let $M_1$ be the midpoint of the segment $P_1Q_1$. Similarly points $M_2$ and $M_3$ are defined. Prove that the incircle of triangle $M_1M_2M_3$ passes through the midpoints of the sides of $A_1A_2A_3$.
2017 shortlist
Let $\Gamma$ be a circle with center $O$ and diameter $AB$, and $C$ the midpoint of an arc $AB$. $AB$ extends beyond $ B$ to a point $D$. $ P$ is a point such that $OP = CD$ and the line $OP$ passes through the midpoint $M$ of $CD$. Let $R$ be the point of tangency from $ P$ to $\Gamma$ in the arc $AB$ that does not contain $C$. The intersections of $MR$ with $AD$ and $MD$ with $PR$ are $E$ and $F$ respectively. Show that $EF$ and $DR$ are parallel.
Two given lines $\ell_1$ and $\ell_2$ intersect at $X$. Two circles $C_1$ and $C_2$ intersect at two points other than $\ell_2$ called $ A$ and $ B$. Line $\ell_1$ intersects $C_1$ at $C$ and $D$, and $C_2$ at $E$ and $F$, respectively. Take an arbitrary point $Y$ on $\ell_2$ and draw the circumcircles $C_3$ of $YCD$ and $C_4$ of $YEF$. Prove that $C_3$ and $C_4$ intersect over $\ell_2$.
In triangle $ABC$, points $E$ and $F$ are the feet in altitudes from $ B$ and $C$ respectively. Let $D$ be a point such that $ABCD$ is a parallelogram with $A$ and $D$ in different half planes with respect to the line $BC$, and $T$ a point such that $AEFT$ is a parallelogram with $T$ and $A$ in different half-planes with respect to the line $EF$. Prove that $T, D$, and the orthocenter of the triangle $ABC$ are collinear.
Let $ABC$ be a scalene acute triangle and $\Gamma$ be its circumcircle. Point $D$ is the midpoint of the small arc $BC$. Points $M$ and $N$ are the feet of the perpendiculars on the lines $AB$ and $AC$ from $D$, respectively. Let $X$ and $Y$ be the points where the lines $DM$ and $DN$ cut for second time to $\Gamma$, respectively. Prove that $MN$ bisects $XY$.
Let $ABC$ be a triangle and $D$ be the foot of the altitude from $A$. Let $l$ be the line that passes through the midpoints of $BC$ and $AC$. $E$ is the reflection of $D$ over $l$. Prove that the circumcentre of $\triangle ABC$ lies on the line $AE$.
Let $ABC$ be a triangle with a circumcircle $\Gamma$. The line perpendicular on $BC$ through $A$ intersects $BC$ and $\Gamma$ at $D$ and $E$, respectively. Let $X$ and $Y$ be the feet of the altitudes from $D$ on $AB$ and $AC$, respectively. The perpendicular on $XY$ from $A$ cuts $\Gamma$ at $F$. Let $R, M, N$ and $Q$ be midpoints of the segments $AF$, $BE$, $AC$ and $DR$ . Show that $M$, $N$, and $L$ are collinear.
Alternative version:
Let $ABC$ be a triangle with circumcircle $\Gamma$ and circumcenter $R$. The line perpendicular on $BC$ from $A$ intersects $BC$ and $\Gamma$ at $D$ and $E$, respectively. Let $X$ and $Y$ be feet of the altitudes from $D$ to $AB$ and $AC$, respectively. Let $M$ and $Q$ be midpoints of the segments $BE$ and $DR$ . Show that $Q$ is circumcenter of $MXY$.
Let $PQ_1R_1$ and $PQ_2R_2$ be two triangles with the same orientation, such that $PQ_1 = PR_1$, $PQ_2 = P R_2$ and $\angle Q_1P R_1 = \angle Q_2PR_2$. Let $X$ be the point of intersection of the lines $Q_1R_1$ and $Q_2R_2$. Show that the segment joining the circumcenters of the triangles $Q_1PR_2$ and $R_1XQ_2$ passes through $ P$.
2018 shortlist
In triangle $ABC$, points $D$ and $E$ are the second intersections of the circle of center $A$ and radius $AC$ with the circumcircle of $ABC$ and the line perpendicular to $AB$ passing through $C$, respectively. Prove that $ B$, $E$, and $D$ lie on the same line.
Let $ABC$ be a triangle such that $\angle BAC = 90^o$ and $AB = AC$. Let $M$ be the midpoint of the segment $BC$. Consider a point $D$ on the semicircle of diameter $BC$ that contains $A$. The circumcircle of triangle $DAM$ intersects segments $DB$ and $DC$ at points $E$ and $F$ respectively. Prove that $BE = CF$.
Let $ABCD$ be a quadrilateral, and $P$, $Q$, $R$ and $S$ be the center of gravity of the triangles $BCD$, $ACD$, $ABD$ and $ABC$ respectively. Show that lines $AP$, $BQ$, $CR$, and $DS$ are concurrent.
On the circle of center $O$ and diameter $AB$, a point $C$ is considered in the ray $AB$, with $A$ between $C$ and $B$, such that: $\frac{CO}{BO} =\frac{5}{4}.$ Let $M$ be the midpoint of segment $AO$. Suppose that $P$ is a point on the circle of center $C$ and radius $CM$, which does not belong to the line AB. Consider a point $Q$ on the extension of the segment $PM$ such that $M$ is the midpoint of $PQ$. Prove that the triangle $PBQ$ is right .
Let $\Delta ABC$ be a triangle inscribed in the circumference $\omega$ of center $O$. Let $T$ be the symmetric of $C$ respect to $O$ and $T'$ be the reflection of $T$ respect to line $AB$. Line $BT'$ intersects $\omega$ again at $R$. The perpendicular to $CT$ through $O$ intersects line $AC$ at $L$. Let $N$ be the intersection of lines $TR$ and $AC$. Prove that $\overline{CN}=2\overline{AL}$.
Let $ABC$ be an acute triangle with orthocenter $H$ and $AB \ne AC$. Let $D$ and $E$ be the intersection points of $BH$ and $CH$ with lines $AC$ and $AB$ respectively, and let $P$ be the foot of the perpendicular drawn from $A$ on $DE$. The circumcircle of triangle $BPC$ intersects $DE$ at a point $Q \ne P$. Show that lines $AP$ and $QH$ intersect on the circumcircle of triangle $ABC$.
Two circles $S_1$ and $S_2$ have centers $O_1$ and $O_2$ respectively, they do not intersect and neither is inside the other. Line $\ell_1$ is tangent to $S_1$ at $A$ and $S_2$ at $ B$, and does not intersect at segment $O_1O_2$. Line $\ell_2$ is tangent to $S_1$ at $C$ and $S_2$ at $D$, and intersects segment $O_1O_2$. Prove that lines $AC$, $BD$, and $O_1O_2$ concur.
Let $ABC$ be an acute triangle with circumcenter $O$, and let $D$ be the midpoint of the segment $OA$. The circumcircle of triangle $BOC$ again cuts the lines $AB$ and $AC$ at the points $E$ and $F$ respectively, with $E \ne B$ and $F \ne C$. Suppose the lines $OF$ and $AB$ intersect at $G$, and lines $OE$ and $AC$ intersect at $H$. The circumcircle of triangle $GDH$ intersects back to lines $AB$ and $AC$ at points $M$ and $N$ respectively. Prove that $A$ and intersection points of $MN$ with $GH$ and of $AB$ with $EF$ are collinear.
Alternative wording:
Let $ABC$ be an acute triangle with circumcenter $O$, and let $D$ the midpoint of segment$ OA$. The circumcircle of the $BOC$ triangle cuts back to lines $AB$ and $AC$ at points $E$ and $F$ respectively, with $E \ne B$ and $F \ne C$. Suppose that lines $OF$ and $AB$ intersect at $G$, and lines $OE$ and $AC$ intersect at $H$. circumcircle of triangle $GDH$ again intersects lines $AB$ and $AC$ at points $M$ and $N$ respectively. Let $J$ be the other point of intersection of the circumcircles of the triangles $AEF$ and $ABC$. Prove that $A$, $J$ and the intersection point of $MN$ with $GH$ are collinear.
Let $ABC$ be a triangle with circumcircle $\Gamma$ (the vertices have been taken counterclockwise). Let $M$ be the midpoint of segment $BC$. Let $D$ and $E$ be points on sides $BC$ and $CA$ respectively, such that $AD$ and $DE$ are the bisectors of the angles $\angle CAB$ and $\angle ADC$. Consider the point $F$ as the intersection between the line $DE$ and $\Gamma$, located in the same half plane that $ A$ with respect to the line $BC$. The circumcircle of triangle $CDF$ intersects segment $CA$ at $G$ and is tangent to $AD$. Suppose that lines $AD$ and $GM$ intersect on $\Gamma$ . Prove that $\Gamma$ and the circumcircles of the triangles $ADM$ and $EFG$ intersect on the line $GM$.
Let $ABC$ be a triangle with circumcircle $\Gamma$ and orthocenter $H$, such that $AB <AC$. Let $D, E$ and $F$ be the feet of the altitudes plotted from vertices $A, B$, and $C$ to opposite sides respectively. Consider $M$ to be the midpoint of side $BC$. Circle of diameter $AH$ intersects $\Gamma$ at point $P$ and segment $AM$ at point $Q$. Show that the triangles $PDQ$ and $DEF$ share the same incircle if and only if $AB = AM$.
Let $ABC$ be a right triangle at $A$, with circumcircle $\Omega$ and circumcenter $O$. A circle $\Gamma_1$ is tangent to $OB$ at $P$, to $\Omega$ at $R$, and to line $OA$; another circle $\Gamma_2$ is tangent to $OC$ at $Q$, to $\Omega$ at $S$ and also to the line $OA$. The line $RS$ intersects $\Gamma_1$ and $\Gamma_2$ at points $X$ and $Y$ respectively. Let $T$ be the intersection points of the tangents at $R$ and $S$ to the circumcircles of the triangles $RBQ$ and $PCS$ respectively. Lines $BR$ and $CS$ intersect at $W$, and lines $PX$ and $QY$ intersect at $Z$. Let us assume that the centers of the circles $\Gamma_1$ and $\Gamma_2$ are they are located in the same half plane as $A$ with respect to line $BC$. Show that points $T, W$, and $Z$ are collinear.
2020 shortlist
Consider a triangle $ABC$ with $BC>AC$. The circle with center $C$ and radius $AC$ intersects the segment $BC$ in $D$. Let $I$ be the incenter of triangle $ABC$ and $\Gamma$ be the circle that passes through $I$ and is tangent to the line $CA$ at $A$. The line $AB$ and $\Gamma$ intersect at a point $F$ with $F \neq A$. Prove that $BF=BD$.
Let $ABC$ be an acute triangle, $O$ its circumcenter and $\Gamma$ its circumcircle. $BO$ cuts $\Gamma$ again at $D$. The tangents of $\Gamma$ at $A$ , $D$ intersect at $T$. The perpendicular on $OC$ from $T$ cuts $AC$ at$ E$ and $\Gamma$ at $R$ and $S$. $DE$ cuts $\Gamma$ again at $F$. Prove that $F$ is the midpoint of the arc $RS$ of$\Gamma$ that does not contain $C$.
Let $ABC$ be a scalene acute triangle and $H$ its orthocenter. Let $X_A$ be the point on the circumcircle of triangle $BHC$, different from $H$, such that $AH = AX_A$. Similarly are defined points $X_B$ and $X_C$. Show that quadrilateral $HX_AX_BX_C$ is cyclic.
Given a triangle $ABC$ whose circumscribed circle is $\Gamma$. A point $P$ is chosen at arc $BAC$ other than the vertices of the triangle. Let $Q$ be the intersection of lines $AP$ and $BC$. The internal bisector of the angle $\angle AQB$ intersects $\Gamma$ at two points, $K$ and $L$. Let R and S be points on $\Gamma$ such that $KA = KR$ and $LA = LS$. Lines $PR$ and $PS$ intersect segment $BC$ at $X$ and $Y$ respectively. Show that $KX$ and $LY$ intersect on circle $\Gamma$.
Let $ABC$ be a triangle such that $AB> AC$, and let $\Gamma$ be its circumcircle. Tangents of $\Gamma$ at $B ,C$ intersect at $ P$. The perpendicular on $AP$ from $A$ intersects $BC$ at $R$. Let $S$ be a point on the segment $PR$ such that $PS = PC$. Show that lines $CS$ and $AR$ intersect on $\Gamma$ .
Let $ABC$ be a triangle with $\angle CAB = 90^o$, with $\Gamma$ its circumcircle and $O$ its circumcenter. A circle $\Gamma_1$ tangent to line $OA$, is tangent to $OB$ at $P$, and is internally tangent to circle $\Gamma$ at $R$. Another circle $\Gamma_2$ is tangent to $OA$, is tangent to $OC$ at $Q$, and is internally tangent to circle $\Gamma$ at $S$. The line $RS$ intersects circles $\Gamma_1$ and $\Gamma_2$ at $X$ and $Y$, respectively. Let $T$ be the intersection point of the tangents to the circumcircles of the triangles $RBQ$ and $PCS$ at $R$ and $S$, respectively. Line $BR$ cuts $CS$ at $W$ and line $PX$ cuts $QY$ at $Z$. Prove that points $T$, $W$, and $Z$ are collinear.
Let $\vartriangle A_1A_2A_3$ be an acute triangle. The perpendicular bisector of side $A_1A_2$ and the line parallel to $A_1A_3$ that passes through $A_2$ intersect at $P_1$. The perpendicular bisector on side $A_1A_3$ and the line parallel to $A_1A_2$ that passes through $A_3$ intersect at $Q_1$. Let $M_1$ be the midpoint of segment $P_1Q_1$. In an analogous way they define points $P_2$, $P_3$, $Q_2$, $Q_3$, $M_2$ and $M_3$. Prove that the incircle of triangle $\vartriangle M_1M_2M_3$ is the circumcircle of the medial triangle $\vartriangle A_1A_2A_3$.
Note: The medial triangle is the one whose vertices are the midpoints of the sides of a triangle.
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