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f(x)f(y)+f\(\lambda /x})f(\lambda/y)=2f(xy) w f(\lambda)=1

Source: 2006 Spanish Mathematical Olympiad P4

July 20, 2018
functional equationreal numberalgebra

Problem Statement

Find all the functions f:(0,+∞)→Rf:(0,+\infty) \to R that satisfy the equation
f(x)f(y)+f(λx)f(λy)=2f(xy)f(x)f(y)+f\big(\frac{\lambda}{x})f(\frac{\lambda}{y})=2f(xy) for all pairs of x,yx,y real and positive numbers, where λ\lambda is a positive real number such that f(λ)=1f(\lambda )=1
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