f(k) = (1-k)*n + k*p = 0.
k = 1/(1-n/p)
A few mathematicians actually reject *all* non-constructive arguments as invalid; this means, for instance, that the law of the excluded middle (either P or not-P must hold, whatever P is) has to go; this makes proof by contradiction invalid. See intuitionistic logic.
Constructive proofs are popular in theoretical computer science, both because computer scientists are less given to abstraction than mathematicians and because intuitionistic logic turns out to be an appropriate theoretical treatment of the foundations of computer science.