You could consider calculating a "difficulty" score for problems and use that for ranking. The number of solved problems could be used as a tiebreaker if two people share a score. A simple difficulty ranking would be to take:

  difficulty = #registered/#solved

Which approaches 1 (lowest difficulty) as more people solve a particular problem. You can also account for the number of people who have tried to solve it but not yet succeeded:

  difficulty = (#registered + #unique failed attempts)/#solved

Unique failed attempts means that if I try it 10 times it still counts as 1 failed attempt. You can either keep the failed attempts in the score, or reduce that count as people move from failure to success. In the former case, the score for hard problems (where people consistently submit failed solutions on their first try) will have a difficulty approaching 2 in the limit, and in the latter it will again approach 1.

With numbers, the three schemes:

  #registered = 100
  #unique failed = 50
  #solved = 25
  #solved with an earlier failure = 20

  1) difficulty = 100/25 = 4
  2) difficulty = (100+50)/25 = 6
  3) difficulty = (100+50-20)/25 = 5.2

A person's score is the sum of the difficulty ranking of every problem they've solved. Since the challenges are released at a particular time, you could give a small bump for something like the first 5, 10, whatever participants but be careful of making it too big. Otherwise late joiners will always be in the bottom ranks.

I like the idea of weighting the problems by a difficulty score that is calculated from the number of users solving it.

I think the idea as presented ("A person's score is the sum of the difficulty ranking of every problem they've solved") would not work great on its own because there will be a large number of people who have solved every single problem.

But certainly we could do the thing about adding up people's per-problem ranking with a weighting based on the problem difficulty! I will experiment with this.

The thing I don't like is that it can result in the leaderboard changing, even for people who have solved all problems, simply because more new people attempt a particular problem, which can alter the relative weightings:

Imagine Alice has 1st place on problem 1 and 2nd place on problem 2. Bob has 2nd place on problem 1 and 1st place on problem 2. Problem 1 is considered twice as difficult as problem 2, so Alice is ahead of Bob overall.

Bob doesn't like this. He wants to be ahead of Alice. He signs up for thousands of accounts to attempt (but not solve) problem 2. This makes problem 2 look very difficult, so now problem 2 is worth more than problem 1 and Bob is ranked ahead of Alice.