Question 1
Let $ABC$ be a triangle with incircle $(I)$ and circumcircle $(O)$. Let $D,E,F$ be the points where $(I)$ touches $BC,CA,AB$, respectively. Let $\omega_A,\omega_B,\omega_C$ be circles tangent to both $(I)$ and $(O)$, respectively at $D,K$ for $\omega_A$, at $E,M$ for $\omega_B$, and at $F,N$ for $\omega_C$. Prove that:
1. The lines $DK,EM,FN$ concur at $P$.
2. The orthocenter of triangle $DEF$ lies on the segment $OP$.
The original statement is retained, including the word “segment” and the unspecified tangency branches. It is false on the branch used by the supplied proof; see the explicit counterexample.
The supplied PDF does not state a time limit. This edition uses the user-approved 240 minutes per session, 480 minutes total for both sessions; this is an editorial convention. The supplied source has no question point values. Each problem uses 1 editorial point for import compatibility; this is not an official scoring scale. Essay grading is manual; the rubric is editorial. Import multipliers: correct 1, incorrect 0, blank 0.
1. The lines $DK,EM,FN$ concur at $P$.
2. The orthocenter of triangle $DEF$ lies on the segment $OP$.
The original statement is retained, including the word “segment” and the unspecified tangency branches. It is false on the branch used by the supplied proof; see the explicit counterexample.
The supplied PDF does not state a time limit. This edition uses the user-approved 240 minutes per session, 480 minutes total for both sessions; this is an editorial convention. The supplied source has no question point values. Each problem uses 1 editorial point for import compatibility; this is not an official scoring scale. Essay grading is manual; the rubric is editorial. Import multipliers: correct 1, incorrect 0, blank 0.