Philosophy of "Garbage" or "Decoration" Removal

Sometimes, problem proposers at competitions are not too creative, and instead of coming up with new ideas, they give a well-known problem a makeover to scare people off. Or when researching a complicated theory, a simple step turns out to be an easy consequence of one of the building blocks of this theory.

At a competition, one can tell that a problem is a redecorated problem if it appears very complicated, hideous, or scary. You might feel that something should be easy, possibly a déjà vu feeling, an idea you still can't quite put your finger around.

In conclusion, a complicated problem environment should tell you to try removing irritating elements so you could see the big picture.

The circles $C_1$ and $C_2$ with diameters $AB$ and $AC$, and centers $O_1$ and $O_2$ respectively, intersect each other at $A$ and $P$.

The perpendicular to $O_2P$ through $B$ meets the perpendicular to $O_1P$ through $C$ at $Q$. The parallel to $AB$ through $O_2$ meets the parallel to $AC$ through $O_1$ at $M$.

Let the parallel to $BC$ through $Q$ meet the circumcircle of $\triangle ABC$ at $Q$ and $R\neq Q$. Finally, let $U$ be the intersection of lines $RM$ and $AP$.

Prove that if $S$ is on $AB$ and $T$ on $AC$ such that $QS,CU$ are parallel and $QT,BU$ are parallel, then $S$, $P$, and $T$ lie on a line.

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