IMO 2013 Shortlist A3
Let Qą0 be the set of positive rational numbers. Let f : Qą0 Ñ R be a function satisfying the conditions fpxqfpyq ě fpxy...
Category: Algebra
Problem
Let Qą0 be the set of positive rational numbers. Let f : Qą0 Ñ R be a function satisfying
the conditions
fpxqfpyq ě fpxyq and fpx yq ě fpxq fpyq
for all x,y P Qą0. Given that fpaq “ a for some rational a ą 1, prove that fpxq “ x for all
x P Qą0.
(Bulgaria)