The prototypical hard IVT problem is utf06-007, proving that any continuous function f with values between 0 and 1 has a fixed point on [0,1]. The trick is to apply the IVT to f(x) - x.
Some will graph a function and handwave the theorem that the function must cross the line y=x somewhere in the unit square. You can decide for yourself if this is acceptable. I consider it unacceptable because they are invoking a theorem that is equivalent to the IVT (even if it looks more intuitive), but I will praise them to the skies for cleverness and have them present their solution to the class. But I'll send them back and ask them to think about what Theorems they know that express properties of continuity without having to rely on physical intuition. Even good students often need more of a hint. They can often think of the IVT itself but not see the trick of applying the IVT to the difference. The hint I usually give is, what functions could you possibly apply it to, out of all the ones out there?
If it's a reasonably smart group with a free flow of ideas, someone's going to get it around then. If it's a slower group, one that doesn't work well together or one with weak backgrounds, this problem may be too hard. (Eric Hsu, 8-9-98)