MathDB

1

Part of 1997 Pre-Preparation Course Examination

Problems(2)

Show that f is a periodic function

Source: Iran Pre-Preparation Course Examination 1997, E1, P1

10/17/2005
Let f:R→Rf: \mathbb R \to\mathbb R be a function such that ∣f(x)∣≤1|f(x)| \leq 1 for all x∈Rx \in \mathbb R and f \left( x + \frac{13}{42} \right) + f(x) = f \left( x + \frac 17 \right) + f \left( x + \frac 16 \right),   \forall x \in \mathbb R. Show that ff is a periodic function.
functionalgebra proposedalgebra
Show that there exist polynomials f, g for each n

Source: Iran Pre-Preparation Course Examination 1997, E3, P1

3/28/2011
Let nn be a positive integer. Prove that there exist polynomialsf(x)f(x)and g(xg(x) with integer coefficients such that f(x)(x+1)2n+g(x)(x2n+1)=2.f(x)\left(x + 1 \right)^{2^n}+ g(x) \left(x^{2^n}+ 1 \right) = 2.
algebrapolynomialgreatest common divisoralgebra proposed