One thing I've been noticing this year as I find "challenge" problems online or give sets to my own students is that the first thing I acknowledge is the trust.
Often, we ask students to "Prove that..." or "Solve" in problems. All well and good in a classroom setting, but in a more "real-world" situation, the first thing that needs to be assessed is whether that problem CAN be solved. I think my own problem-solving strategies are sometimes directed by knowing that there is an answer before I even begin the work.
I trust that my teacher or problem-giver is giving me the opportunity to solve the problem when it is posed. That is, if I am not allowed a calculator, I trust that it CAN be solved without a calculator and that my answer will most likely be a whole number or at least a simple fraction. If I am in a 6th grade classroom then my problem will be "6th grade appropriate" (ie should be solved without calculus or other advanced methods).
Of course, this is not the case in higher mathematics where the main question can often be, "Is this solvable?" Certainly less so in every day life when we are faced with innumerable unsolvable problems. I'm not suggesting we give students unsolvable problems, but it might be interesting to mention this to students that we are in an artificial, safe environment in this classroom where these problems have already been solved and I (the teacher) am giving them to you knowing you should be able to complete them.