IMO Training, Problem 11
In $\triangle ABC$ with altitudes $AD, BE, CF$, the Euler lines of the triangles $\triangle AEF, \triangle BFD, \triangle CDE$ concur at a point $J$. Let $Fe...
IMO Training
IMO Training, Problem 10
Let $ABCD$ be a convex quadrilateral inscribed in a circle $(O)$. Denote $E, F, G$ as the intersections of the pairs of lines $AB, CD$; $AD, BC$; and $AC, BD...
IMO Training, Problem 9
Given a triangle $ABC$ with $I$ as the incenter. A line passing through $A$ intersects the circle $(IBC)$ at points $J$ and $L$. The line $\ell$, passing thr...
IMO Training, Problem 8
Given a triangle $ABC$ with the Kosnita point $K$. Let $AK$ intersect the circle $(KBC)$ again at $L$. On the tangent line through $A$ to the circumcircle $(...
IMO Training, Problem 7
Given a triangle $ABC$ inscribed in a circle $(O)$. Let $D$ be a point on $(O)$ such that the line $AD$ intersects $(BOC)$ at points $E$ and $F$. Let $P$ be ...
IMO Training, Problem 6
Prove that the convex hexagon $ABCDEF$ will have parallel opposite sides if \((AB+DE)^2+(BC+EF)^2+(CD+FA)^2=AD^2+BE^2+CF^2.\)
IMO Training, Problem 5
Let $ABC$ be a triangle with $L$ as the reflection of the orthocenter over the circumcenter. Let $D, E, F$ be the projections of $L$ onto $BC, CA, AB$ respec...
IMO Training, Problem 4
Given a triangle $ABC$. A circle $\Gamma$ is tangent to $(ABC)$ at $A$ and tangent to side $BC$ at $D$. The circle $\Gamma$ intersects $CA$ and $AB$ at $E$ a...
IMO Training, Problem 1
Let $ABC$ be a triangle inscribed in the circle $(O)$, with the altitudes $AD$ and $CF$ intersecting at the orthocenter $H$. Let $M$ be the midpoint of $AH$....