What is so special about Arnold's Trivium? I have read from Experimental Mathematics by V. I. Arnold that if someone wants to study with Vladimir Arnold, they must solve Arnold's Trivium first in order to "understand mathematics". Arnold himself had said that solving the Trivium was a way to truly understand mathematics.
Why? What does Arnold mean with "truly understand mathematics"? 
 A: (Too long for a comment.)  Quoting author's motivation from the source (page 2 of the linked pdf):

In  Feynman's  words,  these students  understand  nothing,  but  never  ask  questions, so  that  they
  appear  to understand  everything. [...] The  students  reach  a  state of  "self-propagating  pseudo-education"  and  can teach  future
  generations in  the same way.  But  all  this  activity  is completely  senseless,  and  in  fact  our  output  of
  specialists is to  a  significant  extent a fraud,  an  illusion  and  a sham:  these so-called specialists are not
  in  a position  to  solve  the  simplest  problems,  and  do  not  possess the  rudiments  of  their  trade.
Thus, to put  an end to  this spurious enhancement of  the results, we  must  specify  not  a  list  of
  theorems, but  a collection of problems  which students should be able to solve.  [...] The  compilation  of  model  problems  is  a laborious job,  but  I think  it  must  be  done.  As  an  attempt
  I  give  below  a list  of  one hundred  problems  forming  a mathematical  minimum  for  a physics  student.

A: There's nothing inherently special about the problems (and they're not even, say, math-contest-style problems); they're just intended to be reasonably comprehensive (with respect to an undergrad math education) and demonstrate the virtues of a written rather than oral examination. There's nothing there that anyone with a undergrad math degree should have any problem with, possibly modulo looking up a couple of specific formulas. He published a note after the problem set comparing it to other such examinations, if that's helpful to you.
That having been said, the subjects chosen aren't what I would expect. There's a lot of involved calculation, including some questions on numeric approximations. There are pages and pages and pages of questions about differential equations; in fact, the questions are heavily skewed to applied math in general. There are a couple of desultory questions about group theory and probability at the end, but that's about it; the rest is real and complex analysis (including a bunch of computations of specific integrals) and dozens of questions about differential equations. There's nothing about set theory, ring theory, commutative algebra, Galois theory, topology (beyond one question about Betti numbers and a couple about Riemann surfaces), Lie algebras (Problems #89 and #90 don't count), representation theory, any group theory that one wouldn't get in a physics class, etc. That would make more sense if Arnol'd was using it as a way to weed out prospective grad students working specifically under him, but it focuses on an extraordinarily narrow and applied curriculum. It doesn't strike me as a great way of demonstrating mathematical understanding.
