<p>T.A.D., my S is an admitted student who plans on going to Caltech next year. I sent him your problems (which are all Greek to me) and got the following responses: “3 is not that hard (I’m not sure if you have to prove that the square roots are irrational themselves but that’s a very famous result), 2 is easy and 1 is tedious but all it really takes is some jiggling.” Not sure if he is blowing smoke, but now that I have gotten his interest, how about a couple more?</p>