MathDB

Problem 1

Part of 1999 Yugoslav Team Selection Test

Problems(1)

polynomial of 2n-th degree

Source: Yugoslav TST 1999

8/8/2005
For a natural number nn, let P(x)P(x) be the polynomial of 2n2n−th degree such that: P(0)=1P(0) = 1 and P(k)=2k−1P(k) = 2^{k-1} for k=1,2,...,2nk = 1, 2, . . . , 2n. Prove that 2P(2n+1)−P(2n+2)=12P(2n + 1) - P(2n + 2) = 1. P.S. I tried to prove it by firstly expressing this polynomial using Lagrange interpolation but get bored of computations - it seems like it can be done this way, but I'd like to see more 'clever' solution. :)
algebrapolynomialalgebra unsolved