Two Cool FEs
Posted: Sat Sep 06, 2014 4:15 pm
1. Find all polynomials \(P:\mathbb{R}-\{0\}\to\mathbb{R}\) satisfying \[P(x)~P\left(\dfrac{1}{x}\right)=P(x)+P\left(\dfrac{1}{x}\right).\]
2. Consider all functions \(f:\mathbb{R}-\{0\}\to\mathbb{R}\) satisfying \[f\left(x+\dfrac 1 x\right)~f\left(x^3+\dfrac{1}{x^3}\right)-f\left(x^2+\dfrac{1}{x^2}\right)^2=\left(x-\dfrac 1 x\right)^2.\] Find the sum of all possible values of \(f(2014)\). Hint Below.
2. Consider all functions \(f:\mathbb{R}-\{0\}\to\mathbb{R}\) satisfying \[f\left(x+\dfrac 1 x\right)~f\left(x^3+\dfrac{1}{x^3}\right)-f\left(x^2+\dfrac{1}{x^2}\right)^2=\left(x-\dfrac 1 x\right)^2.\] Find the sum of all possible values of \(f(2014)\). Hint Below.