I would like to share how I got the correct answer to #3 on the 2022 qualifying exam. I set $MaxPiecewiseCases to 10000 and then

\!\(
\*SubsuperscriptBox[\(\[Integral]\), \(0\), \(2022\)]\(\((
\*SuperscriptBox[\(x\), \(2\)] - \[LeftFloor]x\[RightFloor] \
\[LeftCeiling]x\[RightCeiling])\) \[DifferentialD]x\)\)

worked to output 674. Edit: I didn't realize I already solved this in my first post.

Stephen Wolfram writes about the advances Mathematica has made in his blog post "Don't Forget Integrals!" for 13.0. I think its exciting to be able to simplify an integral that was previously expressible only in terms of elliptic integrals into ArcTanh, for example.

I would like to mention that there is a lot of theory behind finding the simplest form of an integral.