Olympiad problems

2024 Thailand TSTST, 9

Let triangle $ ABC $ be an acute-angled triangle. Square $ AEFB $ and $ ADGC $ lie outside triangle $ ABC $. $ BD $ intersects $ CE $ at point $ H $, and $ BG $ intersects $ CF $ at point $ I $. The circumcircle of triangle $ BFI $ intersects the circumcircle of triangle $ CGI $ again at point $ K $. Prove that line segment $ HK $ bisects $ BC $.

2024 Thailand October Camp, 3

Tags: cartesian coordinates , spiral similarity , complex bash , inversion , geometry

Let triangle $ ABC $ be an acute-angled triangle. Square $ AEFB $ and $ ADGC $ lie outside triangle $ ABC $. $ BD $ intersects $ CE $ at point $ H $, and $ BG $ intersects $ CF $ at point $ I $. The circumcircle of triangle $ BFI $ intersects the circumcircle of triangle $ CGI $ again at point $ K $. Prove that line segment $ HK $ bisects $ BC $.

1998 IMO Shortlist, 5

Tags: complex bash , reflection , homothety , geometry

Let $ABC$ be a triangle, $H$ its orthocenter, $O$ its circumcenter, and $R$ its circumradius. Let $D$ be the reflection of the point $A$ across the line $BC$, let $E$ be the reflection of the point $B$ across the line $CA$, and let $F$ be the reflection of the point $C$ across the line $AB$. Prove that the points $D$, $E$ and $F$ are collinear if and only if $OH=2R$.

2025 Thailand Mathematical Olympiad, 4

Tags: geometry , circumcircle , complex bash , ceva , incenter , trigonometry

Let $D,E$ and $F$ be touch points of the incenter of $\triangle ABC$ at $BC, CA$ and $AB$, respectively. Let $P,Q$ and $R$ be the circumcenter of triangles $AFE, BDF$ and $CED$, respectively. Show that $DP, EQ$ and $FR$ concurrent.

2014 USAMO, 5

Tags: vector , trigonometry , circumcircle , geometric transformation , parallelogram , complex bash , geometry

Let $ABC$ be a triangle with orthocenter $H$ and let $P$ be the second intersection of the circumcircle of triangle $AHC$ with the internal bisector of the angle $\angle BAC$. Let $X$ be the circumcenter of triangle $APB$ and $Y$ the orthocenter of triangle $APC$. Prove that the length of segment $XY$ is equal to the circumradius of triangle $ABC$.

1999 Romania Team Selection Test, 6

Tags: homothety , complex bash , reflection , geometry

Let $ABC$ be a triangle, $H$ its orthocenter, $O$ its circumcenter, and $R$ its circumradius. Let $D$ be the reflection of the point $A$ across the line $BC$, let $E$ be the reflection of the point $B$ across the line $CA$, and let $F$ be the reflection of the point $C$ across the line $AB$. Prove that the points $D$, $E$ and $F$ are collinear if and only if $OH=2R$.