MathDB

3

Part of 2003 Italy TST

Problems(2)

Functional Equation

Source: Italy TST 2003

11/9/2010
Determine all functions f:Rā†’Rf:\mathbb{R}\rightarrow\mathbb{R} that satisfy f(f(x)+y)=2x+f(f(y)-x) \text{for all real}\ x,y.
functionalgebra
p(p(n))=n

Source: Italy TST 2003

11/9/2010
Let p(x)p(x) be a polynomial with integer coefficients and let nn be an integer. Suppose that there is a positive integer kk for which f(k)(n)=nf^{(k)}(n) = n, where f(k)(x)f^{(k)}(x) is the polynomial obtained as the composition of kk polynomials ff. Prove that p(p(n))=np(p(n)) = n.
algebrapolynomialalgebra unsolved