Olympiad problems

2022 SG Originals, Q1

For $\triangle ABC$ and its circumcircle $\omega$, draw the tangents at $B,C$ to $\omega$ meeting at $D$. Let the line $AD$ meet the circle with center $D$ and radius $DB$ at $E$ inside $\triangle ABC$. Let $F$ be the point on the extension of $EB$ and $G$ be the point on the segment $EC$ such that $\angle AFB=\angle AGE=\angle A$. Prove that the tangent at $A$ to the circumcircle of $\triangle AFG$ is parallel to $BC$. [i]Proposed by 61plus[/i]

2004 Turkey MO (2nd round), 1

Tags: parallelogram , geometry , rectangle , trapezoid , symmedian

In a triangle $\triangle ABC$ with$\angle B>\angle C$, the altitude, the angle bisector, and the median from $A$ intersect $BC$ at $H, L$ and $D$, respectively. Show that $\angle HAL=\angle DAL$ if and only if $\angle BAC=90^{\circ}$.

2024 Myanmar IMO Training, 8

Tags: config geo , symmedian , excircle , geometry

Let $ABC$ be a triangle and let $X$ and $Y$ be points on the $A$-symmedian such that $AX = XB$ and $AY = YC$. Let $BX$ and $CY$ meet at $Z$. Let the $Z$-excircle of triangle $XYZ$ touch $ZX$ and $ZY$ at $E$ and $F$. Show that $A$, $E$, $F$ are collinear.

2021 Baltic Way, 14

Tags: geometry , circumcenter , harmonic quadrilateral , symmedian , circumcircle

Let $ABC$ be a triangle with circumcircle $\Gamma$ and circumcentre $O$. Denote by $M$ the midpoint of $BC$. The point $D$ is the reflection of $A$ over $BC$, and the point $E$ is the intersection of $\Gamma$ and the ray $MD$. Let $S$ be the circumcentre of the triangle $ADE$. Prove that the points $A$, $E$, $M$, $O$, and $S$ lie on the same circle.

2023 Turkey MO (2nd round), 6

Tags: circumcircle , symmedian , geometry

On a triangle $ABC$, points $D$, $E$, $F$ are given on the segments $BC$, $AC$, $AB$ respectively such that $DE \parallel AB$, $DF \parallel AC$ and $\frac{BD}{DC}=\frac{AB^2}{AC^2}$ holds. Let the circumcircle of $AEF$ meet $AD$ at $R$ and the line that is tangent to the circumcircle of $ABC$ at $A$ at $S$ again. Let the line $EF$ intersect $BC$ at $L$ and $SR$ at $T$. Prove that $SR$ bisects $AB$ if and only if $BS$ bisects $TL$.

2009 Oral Moscow Geometry Olympiad, 6

Tags: circles , geometry , symmedian

To two circles $r_1$ and $r_2$, intersecting at points $A$ and $B$, their common tangent $CD$ is drawn ($C$ and $D$ are tangency points, respectively, point $B$ is closer to line $CB$ than $A$). Line passing through $A$ , intersects $r_1$ and $r_2$ for second time at points $K$ and $L$, respectively ($A$ lies between $K$ and $L$). Lines $KC$ and $LD$ intersect at point $P$. Prove that $PB$ is the symmedian of triangle $KPL$. (Yu. Blinkov)

2023 Brazil Team Selection Test, 2

Tags: geometry , symmedian

Let $ABCD$ be a parallelogram. The tangent to the circumcircle of triangle $BCD$ at $C$ intersects $AB$ at $P$ and intersects $AD$ at $Q$. The tangents to the circumcircle of triangle $APQ$ at $P$ and $Q$ meet at $R$. Show that points $A$, $C$, and $R$ are collinear.

2023 Sharygin Geometry Olympiad, 20

Tags: geometry , isogonal conjugate , tangent , symmedian , parallel lines

Let a point $D$ lie on the median $AM$ of a triangle $ABC$. The tangents to the circumcircle of triangle $BDC$ at points $B$ and $C$ meet at point $K$. Prove that $DD'$ is parallel to $AK$, where $D'$ is isogonally conjugated to $D$ with respect to $ABC$.

2011 Balkan MO Shortlist, G2

Tags: geometry , circumcircle , angle bisector , circles , symmedian , equal angles

Let $ABC$ be a triangle and let $O$ be its circumcentre. The internal and external bisectrices of the angle $BAC$ meet the line $BC$ at points $D$ and $E$, respectively. Let further $M$ and $L$ respectively denote the midpoints of the segments $BC$ and $DE$. The circles $ABC$ and $ALO$ meet again at point $N$. Show that the angles $BAN$ and $CAM$ are equal.

2023 Israel TST, P2

Tags: geometry , incenter , symmedian

Let $ABC$ be an isosceles triangle, $AB=AC$ inscribed in a circle $\omega$. The $B$-symmedian intersects $\omega$ again at $D$. The circle through $C,D$ and tangent to $BC$ and the circle through $A,D$ and tangent to $CD$ intersect at points $D,X$. The incenter of $ABC$ is denoted $I$. Prove that $B,C,I,X$ are concyclic.

2018 IMO Shortlist, G6

Tags: geometry , isogonal conjugate , symmedian

A convex quadrilateral $ABCD$ satisfies $AB\cdot CD = BC\cdot DA$. Point $X$ lies inside $ABCD$ so that \[\angle{XAB} = \angle{XCD}\quad\,\,\text{and}\quad\,\,\angle{XBC} = \angle{XDA}.\] Prove that $\angle{BXA} + \angle{DXC} = 180^\circ$. [i]Proposed by Tomasz Ciesla, Poland[/i]

Geometry Mathley 2011-12, 15.3

Tags: perpendicular bisector , circumcircle , geometry , symmedian

Triangle $ABC$ has circumcircle $(O,R)$, and orthocenter $H$. The symmedians through $A,B,C$ meet the perpendicular bisectors of $BC,CA,AB$ at $D,E, F$ respectively. Let $M,N, P$ be the perpendicular projections of H on the line $OD,OE,OF.$ Prove that $$\frac{OH^2}{R^2} =\frac{\overline{OM}}{\overline{OD}}+\frac{\overline{ON}}{\overline{OE}} +\frac{\overline{OP}}{\overline{OF}}$$ Đỗ Thanh Sơn

2019 Olympic Revenge, 1

Tags: symmedian , concurrency , geometry

Let $ABC$ be a scalene acute-angled triangle and $D$ be the point on its circumcircle such that $AD$ is a symmedian of triangle $ABC$. Let $E$ be the reflection of $D$ about $BC$, $C_0$ the reflection of $E$ about $AB$ and $B_0$ the reflection of $E$ about $AC$. Prove that the lines $AD$, $BB_0$ and $CC_0$ are concurrent if and only if $\angle BAC = 60^{\circ}.$

2023 Bulgaria EGMO TST, 1

Tags: circumcircle , similar triangles , geometry , symmedian , angle chasing

Let $ABC$ be a triangle with circumcircle $k$. The tangents at $A$ and $C$ intersect at $T$. The circumcircle of triangle $ABT$ intersects the line $CT$ at $X$ and $Y$ is the midpoint of $CX$. Prove that the lines $AX$ and $BY$ intersect on $k$.