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IMO ISL Geometry 2023+

ISL IMO Shortlist Geometry

2023g1

Let $ABCDE$ be a convex pentagon such that $\angle ABC = \angle AED = 90^\circ$ . Suppose that the midpoint of $CD$ is the circumcenter of triangle $ABE$ . Let $O$ be the circumcenter of triangle $ACD$ .

Prove that line $AO$ passes through the midpoint of segment $BE$ .

popop614

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2023g2

Let $ABC$ be a triangle with $AC > BC,$ let $\omega$ be the circumcircle of $\triangle ABC,$ and let $r$ be its radius. Point $P$ is chosen on $\overline{AC}$ such taht $BC=CP,$ and point $S$ is the foot of the perpendicular from $P$ to $\overline{AB}$ . Ray $BP$ mets $\omega$ again at $D$ . Point $Q$ is chosen on line $SP$ such that $PQ = r$ and $S,P,Q$ lie on a line in that order. Finally, let $E$ be a point satisfying $\overline{AE} \perp \overline{CQ}$ and $\overline{BE} \perp \overline{DQ}$ . Prove that $E$ lies on $\omega$ .

avisioner

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2023g3

Let $ABCD$ be a cyclic quadrilateral with $\angle BAD < \angle ADC$ . Let $M$ be the midpoint of the arc $CD$ not containing $A$ . Suppose there is a point $P$ inside $ABCD$ such that $\angle ADB = \angle CPD$ and $\angle ADP = \angle PCB$ .

Prove that lines $AD, PM$ , and $BC$ are concurrent.

GrantStar

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2023g4

Let $ABC$ be an acute-angled triangle with $AB < AC$ . Let $\Omega$ be the circumcircle of $ABC$ . Let $S$ be the midpoint of the arc $CB$ of $\Omega$ containing $A$ . The perpendicular from $A$ to $BC$ meets $BS$ at $D$ and meets $\Omega$ again at $E \neq A$ . The line through $D$ parallel to $BC$ meets line $BE$ at $L$ . Denote the circumcircle of triangle $BDL$ by $\omega$ . Let $\omega$ meet $\Omega$ again at $P \neq B$ . Prove that the line tangent to $\omega$ at $P$ meets line $BS$ on the internal angle bisector of $\angle BAC$ .

799786

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2023g5

Let $ABC$ be an acute-angled triangle with circumcircle $\omega$ and circumcentre $O$ . Points $D\neq B$ and $E\neq C$ lie on $\omega$ such that $BD\perp AC$ and $CE\perp AB$ . Let $CO$ meet $AB$ at $X$ , and $BO$ meet $AC$ at $Y$ .

Prove that the circumcircles of triangles $BXD$ and $CYE$ have an intersection lie on line $AO$ .

Ivan Chan Kai Chin, Malaysia

MarkBcc168

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2023g6

Let $ABC$ be an acute-angled triangle with circumcircle $\omega$ . A circle $\Gamma$ is internally tangent to $\omega$ at $A$ and also tangent to $BC$ at $D$ . Let $AB$ and $AC$ intersect $\Gamma$ at $P$ and $Q$ respectively. Let $M$ and $N$ be points on line $BC$ such that $B$ is the midpoint of $DM$ and $C$ is the midpoint of $DN$ . Lines $MP$ and $NQ$ meet at $K$ and intersect $\Gamma$ again at $I$ and $J$ respectively. The ray $KA$ meets the circumcircle of triangle $IJK$ again at $X\neq K$ .

Prove that $\angle BXP = \angle CXQ$ .

Kian Moshiri, United Kingdom

MarkBcc168

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2023g7

Let $ABC$ be an acute, scalene triangle with orthocentre $H$ . Let $\ell_a$ be the line through the reflection of $B$ with respect to $CH$ and the reflection of $C$ with respect to $BH$ . Lines $\ell_b$ and $\ell_c$ are defined similarly. Suppose lines $\ell_a$ , $\ell_b$ , and $\ell_c$ determine a triangle $\mathcal T$ .

Prove that the orthocentre of $\mathcal T$ , the circumcentre of $\mathcal T$ , and $H$ are collinear.

Fedir Yudin, Ukraine

MarkBcc168

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2023g8

Let $ABC$ be an equilateral triangle. Let $A_1,B_1,C_1$ be interior points of $ABC$ such that $BA_1=A_1C$ , $CB_1=B_1A$ , $AC_1=C_1B$ , and
$\angle BA_1C+\angle CB_1A+\angle AC_1B=480^\circ$ Let $BC_1$ and $CB_1$ meet at $A_2,$ let $CA_1$ and $AC_1$ meet at $B_2,$ and let $AB_1$ and $BA_1$ meet at $C_2.$
Prove that if triangle $A_1B_1C_1$ is scalene, then the three circumcircles of triangles $AA_1A_2, BB_1B_2$
and $CC_1C_2$ all pass through two common points.

(Note: a scalene triangle is one where no two sides have equal length.)

Proposed by Ankan Bhattacharya, USA

mathleticguyyy

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2024.p4

Let $ABC$ be a triangle with $AB < AC < BC$ . Let the incenter and incircle of triangle $ABC$ be $I$ and $\omega$ , respectively. Let $X$ be the point on line $BC$ different from $C$ such that the line through $X$ parallel to $AC$ is tangent to $\omega$ . Similarly, let $Y$ be the point on line $BC$ different from $B$ such that the line through $Y$ parallel to $AB$ is tangent to $\omega$ . Let $AI$ intersect the circumcircle of triangle $ABC$ at $P \ne A$ . Let $K$ and $L$ be the midpoints of $AC$ and $AB$ , respectively.
Prove that $\angle KIL + \angle YPX = 180^{\circ}$ .

Proposed by Dominik Burek, Poland

sarjinius

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