Problem 1
Prove that for every irrational real number a, there are irrational real numbers b and b' such that a+b and ab' are rational, while ab and a+b' are irrational.
Step 4 of 5: At least one candidate has irrational sum
Detailed analysis
The two sums are a+1/a and a+2/a. Their difference is -1/a, which is irrational. If both sums were rational, their difference would be rational, a contradiction. Choose b' to be the candidate whose sum with a is irrational.