IMO 2008 Q4 #
Find all functions f : (0,∞) → (0,∞)
(so, f
is a function from the positive real
numbers to the positive real numbers) such that
lean (f(w)^2 + f(x)^2)/(f(y^2) + f(z^2)) = (w^2 + x^2)/(y^2 + z^2)
for all positive real numbers w
, x
, y
, z
, satisfying wx = yz
.
Solution #
The desired theorem is that either f = λ x, x
or f = λ x, 1/x