Question 1
Find all triangles $ABC$ with $\angle BCA$ acute such that the perpendicular bisector of $BC$ meets rays $Ax,Ay$ trisecting $\angle BAC$, with $\angle BAx=\angle xAy=\angle yAC$, at $M,N$ satisfying
$$AB=NP=2HM,$$
where $H$ is the perpendicular projection of $A$ on $BC$ and $M$ is the midpoint of $BC$.
The notation above reproduces the supplied statement literally; see the explicit defect and conditional interpretation in the solution.
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; this rubric is editorial. Import multipliers: correct 1, incorrect 0, blank 0.
$$AB=NP=2HM,$$
where $H$ is the perpendicular projection of $A$ on $BC$ and $M$ is the midpoint of $BC$.
The notation above reproduces the supplied statement literally; see the explicit defect and conditional interpretation in the solution.
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; this rubric is editorial. Import multipliers: correct 1, incorrect 0, blank 0.