Actually, I think it is rewarding to look some things up, even if you use Google and Co.. For a student it is also a psychological effect to find solutions on the Internet. Namely, if you have some problem in symplectic geometry courses, these problems are normally rather specific, so you will have trouble finding solutions on the web easily. However, when using Google books, you can easily find parts of books which may contain relevant information for your proof or even give a good starting point for a proof. If someone simply uses the Internet to copy a proof then he/she will have problems to really understand mathematics. But, admittedly, there are proofs that I haven't understood or, this more often the case, wouldn't be able to reproduce offhand. In this case, the internet proves to be very useful, especially if you also have students that try to autodidactically learn some further topics in mathematics. Personally, I would design problem sheets as follows: -Make 3 to 4 easy problems which simply consist of getting familiar with the definitions and the rules in the respective field. -Then make one problem that is rather technical and requires the student to make some longer steps in proving the statement. These calculations shouldn't be too complicated as otherwise the student will probably lose patience and simply look the solution up. -Then design 2-3 problems that are more far-leading and require using calculation rules, and a bit creativity. Still, they shouldn't be too complicated.
Why am i always telling you about the complexity of the problems? Well, at least in Germany, we have only 30% of the students obtaining a degree in mathematics. I don't know how things are handles in the U.S. or elsewhere, but most of our students are frustrated becuase they simply don't find a starting point for the exercises. And this shouldn't happen.
I think, most of the students have the will to solve problems on their own. Especially mathematics and physics are subjects you study because you're passionate about them. Biological research has shown indeed that motivated apes are more thankful and curious then demotivated apes. I think this applies to students as well (and to us as well). The typical student involved in a biologically complicates process. The post-adolesence. The years 15-30 are the years where you are requires to pass a lot of tests. And for doing so you have to be motivated and self-confident. At least in Germany, for a lot of students, this is a problem (But I don't think it's much different elsewhere, at least I hope so, because otherwise, we're doing something wrong here).
Most cheating can be avoided if questions are motivating and asked in sch a manner that the first ones are easy to answer, and the following ones increase slightly in diffculty.
This is my opinion. I talk to a lot of students in physics which suffer from depressions because they believe they won't get anything. This belief is apparenbtly that deep that they simply copy homework or use Google. Mostly, when these students continue their studies they are very succesful later on. But a lot of them simply stops studying math and physics. A lot of potential is wasted here.
All I've written may sound a bit offtopic, but I think that there is the real problem. If you motivate students to think on their own, they will rejoice in proofs and all that. This was also my (personal) experience.