As someone who tends to prefer problem-solving to theory-building, I find the following particularly interesting (my translation):
A student who takes more than five minutes to calculate the average of sin 100 x with a precision of 10% has no mastery of mathematics, even if he has studied nonstandard analysis, universal algebra, supervarieties, or plongement [I don't know this word] theorems.
Go ahead, try that problem! (Curiously, it's not one on the list.) More generally, it's an interesting bunch of questions in various branches of mathematics, biased towards but by no means exclusively in calculus and differential geometry; this is no surprise as it's meant for physicists.
(Why the French version, you ask? The article was originally in Russian; there's an English translation but it's not free. Links to free versions in Russian or English, if they exist, would be appreciated.)
Edit, 12:07 am Sunday: An anonymous commentator provides the English version, and Dmitri Pavlov the Russian.