MathDB

3

Part of 1998 Vietnam National Olympiad

Problems(2)

a_{n+h}-a_n

Source: 36-th Vietnamese Mathematical Olympiad 1998

2/17/2007
The sequence {an}n≥0\{a_{n}\}_{n\geq 0} is defined by a0=20,a1=100,an+2=4an+1+5an+20(n=0,1,2,...)a_{0}=20,a_{1}=100,a_{n+2}=4a_{n+1}+5a_{n}+20(n=0,1,2,...). Find the smallest positive integer hh satisfying 1998∣an+h−an∀n=0,1,2,...1998|a_{n+h}-a_{n}\forall n=0,1,2,...
modular arithmeticalgebra unsolvedalgebra
polynomial

Source: 36-th Vietnamese Mathematical Olympiad 1998

2/17/2007
Find all positive integer nn such that there exists a P∈R[x]P\in\mathbb{R}[x] satisfying P(x1998−x−1998)=xn−x−n∀x∈R−{0}P(x^{1998}-x^{-1998})=x^{n}-x^{-n}\forall x\in\mathbb{R}-\{0\}.
algebrapolynomialalgebra unsolved