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Playing around with a two variable function [ILL 1977]

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January 11, 2011
functionalgebra proposedalgebra

Problem Statement

Let ff be a function defined on the set of pairs of nonzero rational numbers whose values are positive real numbers. Suppose that ff satisfies the following conditions:
(1) f(ab,c)=f(a,c)f(b,c), f(c,ab)=f(c,a)f(c,b);f(ab,c)=f(a,c)f(b,c),\ f(c,ab)=f(c,a)f(c,b);
(2) f(a,1−a)=1f(a,1-a)=1
Prove that f(a,a)=f(a,−a)=1,f(a,b)f(b,a)=1f(a,a)=f(a,-a)=1,f(a,b)f(b,a)=1.
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