Could a collection of data be engineered to encrypt predictably using standard encryption algorithms? Thinking about the rather new problem of Cryptolocker and other crypto-randomware, huge amounts of damage are being caused by malicious actors simply using modern encryption algorithms to encrypt a drive. But this presents an interesting question, and the hint of a possible solution:

Could a collection of data be engineered such that when it is encrypted using a given encryption algorithm, say RSA, patterns in the encryption reveal details that could be used to discover the keys, or at least greatly narrow the search space?
What you're describing is a chosen-plaintext attack.  In this scenario the attacker (you, in this case) creates a specific plaintext for the cryptosystem to encrypt, in an attempt to manipulate it in specific way...In your scenario, to get it to give up information about the key.   Chosen plaintext attacks should not be possible in a semantically secure cryptosystem.  If they are possible, the system is broken, and should not be used. 

So, to directly answer your question, no, this should not be possible, and certainly is not generically possible.   There may be specific implementations of crypto-ransomware that are flawed in a way that would make this possible, but you would have to specifically target an attack for that specific system, and it would be useless against any other, including another system flawed in even a slightly different way.