Mathematical Ashes 2008-19 4p / 12p

geometry problems from Mathematical Ashes,
with aops links in the names

a binational contest, IMO Team Training Exam
between UK and Australia

2008 - 2019
(only 4 out of the 11 are not from Shortlist)

Mathematical Ashes 2008.3  (2007 IMO Shortlist G8)
Point $P$ lies on side $AB$ of a convex quadrilateral $ABCD$. Let $I$ be the incircle of triangle $CPD$, and let $I$ be its incentre. Suppose that $I$ is tangent to the incircles of triangles $APD$ and $BPC$ at points $K$ and $L$, respectively. Let lines $AC$ and $BD$ meet at $E$, and let lines $AK$ and $BL$ meet at $F$. Prove that points $E, I$ and $F$ are collinear.

Mathematical Ashes 2009.2 (2008 IMO Shortlist G3)
Let $ABCD$ be a convex quadrilateral and let $P$ and $Q$ be points in $ABCD$ such that $PQDA$ and $QPBC$ are cyclic quadrilaterals. Suppose that there exists a point $E$ on the line segment $PQ$ such that $\angle PAE = \angle QDE$ and $\angle PBE = \angle QCE$. Show that the quadrilateral $ABCD$ is cyclic.

Mathematical Ashes 2010.2 (2009 IMO Shortlist G4)
Given a cyclic quadrilateral $ABCD$, let the diagonals $AC$ and $BD$ meet at $E$ and the lines $AD$ and $BC$ meet at $F$. The midpoints of $AB$ and $CD$ are $G$ and $H$, respectively. Show that $EF$ is tangent at $E$ to the circle through the points $E, G$ and $H$.

Mathematical Ashes 2011.1
Let  $\gamma$ be a fixed circle in the plane and let $P$ be a fixed point not on $\gamma$. Two variable lines  l and  l'  through $P$ intersect $\gamma$  at points $X$ and $Y$ , and $X'$ and $Y'$, respectively. Let $M$ and $N$ be the points directly opposite $P$ in circles $PXX'$ and $PY Y'$, respectively. Prove that as the lines  l  and  l'  vary, there is a fixed point through which line $MN$ always passes.

Mathematical Ashes 2012.2 (2011 IMO Shortlist G3)
Let $ABCD$ be a convex quadrilateral whose sides $AD$ and $BC$ are not parallel. Suppose that the circles with diameters $AB$ and $CD$ meet at points $E$ and $F$ inside the quadrilateral. Let $\omega_E$ be the circle through the feet of the perpendiculars from $E$ to the lines $AB, BC$, and $CD$. Let $\omega_F$ be the circle through the feet of the perpendiculars from $F$ to the lines $CD, DA$, and $AB$. Prove that the midpoint of the segment $EF$ lies on the line through the two intersection points of $\omega_E$ and $\omega_F$ .

Mathematical Ashes 2013.3 (2012 IMO Shortlist G8)
Let $ABC$ be a triangle with circumcircle $\omega$ and let l be a line which does not intersect $\omega$ . Let $P$ be the foot of the perpendicular from the centre of $\omega$ to  l . The side-lines $BC, CA$ and $AB$ intersect l , respectively, at the points $X, Y$ and $Z$ different from $P$. Prove that the circumcircles of triangles $AXP, BY P$ and $CZP$ have a common point different from $P$ or are mutually tangent at $P$.

Mathematical Ashes 2014.1
Let $D$ be the point on side $BC$ such that $AD$ bisects angle $\angle BAC$. Let $E$ and $F$ be the incentres of triangles $ADC$ and $ADB$, respectively. Let $\omega$ be the circumcircle of triangle $DEF$. Let $Q$ be the point of intersection of the lines $BE$ and $CF$. Let $H, J, K$ and $M$ be the second points of intersection of $\omega$ with the lines $CE, CF, BE$ and $BF$, respectively. Circles $HQJ$ and $KQM$ intersect at the two points $Q$ and $T$. Prove that $T$ lies on line $AD$.

Mathematical Ashes 2015.2 (2014 IMO Shortlist G3)
Let  $\Omega$ and $O$ be the circumcircle and the circumcentre of an acute-angled triangle $ABC$ with $AB > BC$. The angle bisector of $\angle ABC$ intersects   at $M \ne B$. Let $\Gamma$  be the circle with diameter $BM$. The angle bisectors of $\angle AOB$ and $\angle BOC$ intersect $\Gamma$ a points $P$ and $Q$, respectively. The point $R$ is chosen on the line $PQ$ so that $BR = MR$. Prove that $BR // AC$. (In this problem, we assume that an angle bisector is a ray.)

Mathematical Ashes 2016.1 (2015 IMO Shortlist G1)
Let $ABC$ be an acute triangle with orthocenter $H$. Let $G$ be the point such that the quadrilateral $ABGH$ is a parallelogram ($AB//GH$ and $BG//AH$). Let $I$ be the point on the line $GH$ such that $AC$ bisects $HI$. Suppose that the line $AC$ intersects the circumcircle of the triangle $GCI$ at $C$ and $J$. Prove that $IJ=AH$.

Mathematical Ashes 2017.1
Point $A_1$ lies inside acute scalene triangle $ABC$ and satisfies $\angle A_1AB = \angle A_1BC$ and $\angle A_1AC = \angle A_1CB$.  Points $B_1$ and $C_1$ are similarly defined. Let $G$ and $H$ be the centroid and orthocentre, repsectively, of triangle $ABC$. Prove that $A_1, B_1, C_1, G$, and $H$ all lie on a common circle.

Mathematical Ashes 2018.1
Let $ABCD$ be a cyclic quadrilateral. Rays $AD$ and $BC$ meet at $P$. In the interior of the triangle $DCP$ a point $M$ is given, such that the line $PM$ bisects $\angle CMD$. Line $CM$ meets the circumcircle of triangle $DMP$ again at $Q$. Line $DM$ meets the circumcircle of triangle $CMP$ again at $R$. The circumcircles of triangles $APR$ and $BPQ$ meet for a second time at $S$. Prove that $PS$ bisects $\angle BSA$.

Mathematical Ashes 2019.3  (2018 IMO Shortlist G5 DEN)
Let $ABC$ be a triangle with circumcircle $\omega$ and incentre $I$. A line $\ell$ intersects the lines $AI$, $BI$, and $CI$ at points $D$, $E$, and $F$, respectively, distinct from the points $A$, $B$, $C$, and $I$. The perpendicular bisectors $x$, $y$, and $z$ of the segments $AD$, $BE$, and $CF$, respectively determine a triangle $\Theta$. Show that the circumcircle of the triangle $\Theta$ is tangent to $\Omega$.


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