MathDB

12

Part of 1989 IMO Longlists

Problems(1)

For all n polynomial value P(n) shall be positive

Source: IMO Longlist 1989, Problem 12

9/18/2008
Let P(x) P(x) be a polynomial such that the following inequalities are satisfied: P(0)>0; P(0) > 0;P(1)>P(0); P(1) > P(0); P(2) > 2P(1) \minus{} P(0); P(3) > 3P(2) \minus{} 3P(1) \plus{} P(0); and also for every natural number n, n, P(n\plus{}4) > 4P(n\plus{}3) \minus{} 6P(n\plus{}2)\plus{}4P(n \plus{} 1) \minus{} P(n). Prove that for every positive natural number n, n, P(n) P(n) is positive.
algebrapolynomialinequalitiesalgebra unsolved