(In reply to solution (spoiler)
Nice work, Daniel, except for 2 quibbles:
1/s<=s DOES NOT IMPLY that s^2 >= 1, because you have to reverse the inequality when multiplying by a negative number.
What it does imply is that s >= 1 or 0 > s >= -1
s>=1 q>=1 q<=s DOES NOT IMPLY s=q=1
It does not alter the solution, however, because (as you pointed out) when rs = 1, p + q = 2, so q must be >= 1. Thus r and s must be >= 1, which can only occur if r = s = 1. I think this simplifies your solution (as well as making it correct).
But it does not affect the final solution.