I have tried creating many such problems with ideas that there is some condition on the range [L, R]. Most of them are hard to solve, unless you have some special properties e.g. if two ranges [L1, R1] intersect with [L2, R2], then the common range also follows that property. e.g. see the problem CLOST on codechef, in this problem, the property is whether the string is balanced or not, for this particular property the above stated intersection condition holds.
You can create some interesting question if you ask whether there exists a situation with given property. e.g. see this problem in which inverse of RMQ problem over a permutation is asked.
Also, I would like to point out one thing, if we have following condition that the if ranges intersect, then one range will lie completely inside the other or they will be disjoint, then we can create a tree like structure in which the desired problem might not be hard to solve. e.g. see one of the problems set by me.
So, overall think about many variations of the above problem, some of them might be feasible to solve others are much harder to solve in general.