MathDB

2

Part of 1984 Vietnam National Olympiad

Problems(2)

A sequence satisfying linear recurrence and limit.

Source: Vietnam MO 1984 P2

3/20/2011
The sequence (un)(u_n) is defined by u1=1,u2=2u_1 = 1, u_2 = 2 and un+1=3un−un−1u_{n+1} = 3u_n - u_{n-1} for n≥2n \ge 2. Set vn=∑k=1narccot ukv_n =\sum_{k=1}^n \text{arccot }u_k. Compute lim⁡n→∞vn\lim_{n\to\infty} v_n.
limittrigonometryalgebra unsolvedalgebra
Polynomial satisfying xP(x - a) = (x - b)P(x) for given a,b.

Source: Vietnam MO 1984 P5

3/20/2011
Given two real numbers a,ba, b with a≠0a \neq 0, find all polynomials P(x)P(x) which satisfy xP(x−a)=(x−b)P(x).xP(x - a) = (x - b)P(x).
algebrapolynomialalgebra unsolved